# Figurate Numbers v1.6.0 - Sonic Pi Web
# Source: https://github.com/edelveart/figurate_numbers
# Author: Edgar Delgado Vega
# Bundle: multidimensional_figurate_numbers

module FigurateNumbers;module Utils;module_function;def factorial_iter(num)t=1;(1..num).each do |i|t*=i;end;t;end;def binomial_coefficient(n,k)factorial_iter(n)/(factorial_iter(k)*factorial_iter(n-k));end;def rising_factorial(n,k)t=1;(n..(n+k-1)).each do |i|t*=i;end;t;end;def pseudo_rising_factorial(n,k)t=1;(n..(n+k-2)).each do |i|t*=i;end;t;end;def pseudo_pochhammer_function(n,k)(n..(n+k-2)).reduce(:*);end;def _validation_fnomial(n,k)raise ArgumentError,'n must be a non-negative Integer' unless n.is_a?(Integer)&&n>=0;raise ArgumentError,'k must be an Integer between 0 and n' unless k.is_a?(Integer)&&k.between?(0,n);end;def _calculate_fnomial(first,last,k)numerator=1;denominator=1;k.times do |i|numerator*=last[i];denominator*=first[i];gcd=numerator.gcd(denominator);numerator/=gcd;denominator/=gcd;end;[numerator,denominator];end;def figurate_binomial(n,k,seq)_validation_fnomial(n,k);k=[k,n-k].min;first=[];last=[];(1..n).each do |i|value=seq.next;first<<value if i<=k;last<<value if i>n-k;end;numerator,denominator=_calculate_fnomial(first,last,k);denominator==1 ? numerator: Rational(numerator,denominator);end;end;end;module FigurateNumbers;module PAdicUtils;extend self;def prime?(n)return false if n<2;return true if n==2;return false if n.even?;divisor=3;while divisor*divisor<=n;return false if(n%divisor).zero?;divisor+=2;end;true;end;def first_n_primes(n)return [] if n<=0;return [2] if n==1;primes=[2];candidate=3;while primes.length<n;primes<<candidate if prime?(candidate);candidate+=2;end;primes;end;def padic_valuation(base_ten_number,p)raise"#{p} is not a prime number" unless prime?(p);index=0;return index if base_ten_number<1;while base_ten_number%(p**index)==0
index+=1;end;index-1;end;def padic_norm(base_ten_number,p)return 0 if base_ten_number==0
power=padic_valuation(base_ten_number,p);1.0/p**power;end;def padic_expansion(base_ten_number,p,precision=11,reverse_trim=false)
inverse_limit_arr=[];(1...precision).each do |index|power=p**index;inverse_limits=((base_ten_number%power)+power)%power;inverse_limit_arr<<inverse_limits;end;p_adic_expansion_array=[inverse_limit_arr[0]];(0...(precision-2)).each do |index|a1=inverse_limit_arr[index+1];a0=inverse_limit_arr[index];power=p**(index+1);digits=(a1-a0)/power;p_adic_expansion_array<<digits;end;if reverse_trim;reversed=p_adic_expansion_array.reverse;first_nonzero_index=reversed.index{|digit|digit!=0}||0;reversed[first_nonzero_index..-1];else p_adic_expansion_array;end;end;end;end;module FigurateNumbers;module MultiDimensionalFigurateNumbers
extend self;def pentatope;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*(delta+3))/24;end;end;end;alias hypertetrahedral pentatope;alias triangulotriangular pentatope;def reference_of_book_k_dimensional_hypertetrahedron(k)Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<Utils.rising_factorial(delta,k)/Utils.factorial_iter(k);end;end;end;def k_dimensional_hypertetrahedron(k)Enumerator.new do |y|delta=1;bin_coeff=1;loop do y<<bin_coeff;bin_coeff=bin_coeff*(delta+k)/delta;delta+=1;end;end;end;alias k_hypertetrahedron k_dimensional_hypertetrahedron;alias regular_k_polytopic k_dimensional_hypertetrahedron;alias figurate_numbers_of_order_k k_dimensional_hypertetrahedron;def five_dimensional_hypertetrahedron;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<Utils.rising_factorial(delta,5)/Utils.factorial_iter(5);end;end;end;def six_dimensional_hypertetrahedron;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<Utils.rising_factorial(delta,6)/Utils.factorial_iter(6);end;end;end;def binomial_coefficient_k_dimensional_hypertetrahedron(k)Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<Utils.binomial_coefficient(delta+(k-1),k);end;end;end;def biquadratic;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<delta**4;end;end;end;def k_dimensional_hypercube(k)Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<delta**k;end;end;end;alias k_hypercube k_dimensional_hypercube;def five_dimensional_hypercube;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<delta**5;end;end;end;def six_dimensional_hypercube;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<delta**6;end;end;end;def hyperoctahedral;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta**2*(delta**2+2))/3;end;end;end;alias hexadecachoron hyperoctahedral;alias four_cross_polytope hyperoctahedral;alias four_orthoplex hyperoctahedral;def hypericosahedral;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(145*delta**3-280*delta**2+179*delta-38))/6;end;end;end;alias tetraplex hypericosahedral;alias polytetrahedron hypericosahedral;alias hexacosichoron hypericosahedral;def hyperdodecahedral;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(261*delta**3-504*delta**2+283*delta-38))/2;end;end;end;alias hecatonicosachoron hyperdodecahedral;alias dodecaplex hyperdodecahedral;alias polydodecahedron hyperdodecahedral;def polyoctahedral;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<delta**2*(3*delta**2-4*delta+2);end;end;end;alias icositetrachoron polyoctahedral;alias octaplex polyoctahedral;alias hyperdiamond polyoctahedral;def four_dimensional_hyperoctahedron;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<delta**2*(delta**2+2)/3;end;end;end;def five_dimensional_hyperoctahedron;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<delta*(2*delta**4+10*delta**2+3)/15;end;end;end;def six_dimensional_hyperoctahedron;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<delta**2*(2*delta**4+20*delta**2+23)/45;end;end;end;def seven_dimensional_hyperoctahedron;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(4*delta**6+70*delta**4+196*delta**2+45)*delta/315;end;end;end;def eight_dimensional_hyperoctahedron;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta**6+28*delta**4+154*delta**2+132)*delta**2/315;end;end;end;def nine_dimensional_hyperoctahedron
Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(2*delta**8+84*delta**6+798*delta**4+1636*delta**2+315)*delta/2835;end;end;end;def ten_dimensional_hyperoctahedron
Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(2*delta**8+120*delta**6+1806*delta**4+7180*delta**2+5067)*delta**2/14_175;end;end;end;def k_dimensional_hyperoctahedron(k)Enumerator.new do |y|(1..Float::INFINITY).each do |delta|a=0;(0..(k-1)).each do |i|a+=Utils.binomial_coefficient(k-1,i)*(Utils.rising_factorial(delta-i,k)/Utils.factorial_iter(k));end;y<<a;end;end;end;alias k_cross_polytope k_dimensional_hyperoctahedron;def four_dimensional_mgonal_pyramidal(m)Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*((m-2)*delta-m+6))/24;end;end;end;alias mgonal_pyramidal_numbers_of_the_second_order four_dimensional_mgonal_pyramidal;def four_dimensional_square_pyramidal;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*((4-2)*delta-4+6))/24;end;end;end;def four_dimensional_pentagonal_pyramidal;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*((5-2)*delta-5+6))/24;end;end;end;def four_dimensional_hexagonal_pyramidal;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*((6-2)*delta-6+6))/24;end;end;end;def four_dimensional_heptagonal_pyramidal;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*((7-2)*delta-7+6))/24;end;end;end;def four_dimensional_octagonal_pyramidal;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*((8-2)*delta-8+6))/24;end;end;end;def four_dimensional_nonagonal_pyramidal;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*((9-2)*delta-9+6))/24;end;end;end;def four_dimensional_decagonal_pyramidal;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*((10-2)*delta-10+6))/24;end;end;end;def four_dimensional_hendecagonal_pyramidal;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*((11-2)*delta-11+6))/24;end;end;end;def four_dimensional_dodecagonal_pyramidal;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*((12-2)*delta-12+6))/24;end;end;end;def k_dimensional_mgonal_pyramidal(k,m)Enumerator.new do |y|(1..Float::INFINITY).each do |n|y<<(Utils.pseudo_pochhammer_function(n,k)*((m-2)*n-m+k+2))/Utils.factorial_iter(k);end;end;end;alias mgonal_pyramidal_numbers_of_the_k_2_th_order k_dimensional_mgonal_pyramidal;def five_dimensional_mgonal_pyramidal(m)Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*(delta+3)*((m-2)*delta-m+7))/120;end;end;end;def five_dimensional_square_pyramidal;Enumerator.new do |y|k=5;m=4;(1..Float::INFINITY).each do |delta|y<<(Utils.pseudo_pochhammer_function(delta,k)*((m-2)*delta-m+k+2))/Utils.factorial_iter(k);end;end;end;def five_dimensional_pentagonal_pyramidal;Enumerator.new do |y|k=5;m=5;(1..Float::INFINITY).each do |delta|y<<(Utils.pseudo_pochhammer_function(delta,k)*((m-2)*delta-m+k+2))/Utils.factorial_iter(k);end;end;end;def five_dimensional_hexagonal_pyramidal;Enumerator.new do |y|k=5;m=6;(1..Float::INFINITY).each do |delta|y<<(Utils.pseudo_pochhammer_function(delta,k)*((m-2)*delta-m+k+2))/Utils.factorial_iter(k);end;end;end;def five_dimensional_heptagonal_pyramidal;Enumerator.new do |y|k=5;m=7;(1..Float::INFINITY).each do |delta|y<<(Utils.pseudo_pochhammer_function(delta,k)*((m-2)*delta-m+k+2))/Utils.factorial_iter(k);end;end;end;def five_dimensional_octagonal_pyramidal;Enumerator.new do |y|k=5;m=8;(1..Float::INFINITY).each do |delta|y<<(Utils.pseudo_pochhammer_function(delta,k)*((m-2)*delta-m+k+2))/Utils.factorial_iter(k);end;end;end;def six_dimensional_mgonal_pyramidal(m)
Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta*(delta+1)*(delta+2)*(delta+3)*(delta+4)*((m-2)*delta-m+8))/720;end;end;end;def six_dimensional_square_pyramidal;Enumerator.new do |y|k=6;m=4;(1..Float::INFINITY).each do |delta|y<<(Utils.pseudo_pochhammer_function(delta,k)*((m-2)*delta-m+k+2))/Utils.factorial_iter(k);end;end;end;def six_dimensional_pentagonal_pyramidal;Enumerator.new do |y|k=6;m=5;(1..Float::INFINITY).each do |delta|y<<(Utils.pseudo_pochhammer_function(delta,k)*((m-2)*delta-m+k+2))/Utils.factorial_iter(k);end;end;end;def six_dimensional_hexagonal_pyramidal;Enumerator.new do |y|k=6;m=6;(1..Float::INFINITY).each do |delta|y<<(Utils.pseudo_pochhammer_function(delta,k)*((m-2)*delta-m+k+2))/Utils.factorial_iter(k);end;end;end;def six_dimensional_heptagonal_pyramidal;Enumerator.new do |y|k=6;m=7;(1..Float::INFINITY).each do |delta|y<<(Utils.pseudo_pochhammer_function(delta,k)*((m-2)*delta-m+k+2))/Utils.factorial_iter(k);end;end;end;def six_dimensional_octagonal_pyramidal;Enumerator.new do |y|k=6;m=8;(1..Float::INFINITY).each do |delta|y<<(Utils.pseudo_pochhammer_function(delta,k)*((m-2)*delta-m+k+2))/Utils.factorial_iter(k);end;end;end;def centered_biquadratic;Enumerator.new do |y|a=0;(1..Float::INFINITY).each do |delta|a+=delta**4-(delta-2)**4;y<<a+1;end;end;end;def k_dimensional_centered_hypercube(k)Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<delta**k+(delta-1)**k;end;end;end;def five_dimensional_centered_hypercube;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<delta**5+(delta-1)**5;end;end;end;def six_dimensional_centered_hypercube;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<delta**6+(delta-1)**6;end;end;end;def centered_polytope;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(5*delta**4-10*delta**3+55*delta**2-50*delta+24)/24;end;end;end;def helper_centered_hypertetrahedron(k,n)return 1 if n==1;t=Utils.binomial_coefficient(k+1,k);return t if n==2;tau=0;(0..k-1).each do |i|tau+=Utils.binomial_coefficient(k+1,k-i)*Utils.binomial_coefficient(n-2,i);end;tau;end;def acc_helper_centered_hypertetrahedron(k,n)a=0;(1..n).each do |j|a+=helper_centered_hypertetrahedron(k,j);end;a;end;def k_dimensional_centered_hypertetrahedron(k)Enumerator.new do |y|(1..Float::INFINITY).each do |n|y<<acc_helper_centered_hypertetrahedron(k,n);end;end;end;def five_dimensional_centered_hypertetrahedron;Enumerator.new do |y|(1..Float::INFINITY).each do |n|y<<acc_helper_centered_hypertetrahedron(5,n);end;end;end;def six_dimensional_centered_hypertetrahedron;Enumerator.new do |y|(1..Float::INFINITY).each do |n|y<<acc_helper_centered_hypertetrahedron(6,n);end;end;end;def centered_hyperoctahedral;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(2*delta**4-4*delta**3+10*delta**2-8*delta+3)/3;end;end;end;alias orthoplex centered_hyperoctahedral;def nexus(k)Enumerator.new do |y|(0..Float::INFINITY).each do |delta|y<<(delta+1)**(k+1)-delta**(k+1);end;end;end;def ext_int_double_summation(k,n)
t=((2**1)*Utils.binomial_coefficient(k,1)*Utils.binomial_coefficient(1,0));a=0;(1..(n-1)).each do |j|(0..(k-1)).each do |i|a+=(2**(1+i))*Utils.binomial_coefficient(k,1+i)*Utils.binomial_coefficient(j,i);end;end;(1+t+a);end;def k_dimensional_centered_hyperoctahedron(k)Enumerator.new do |y|y<<1;(1..Float::INFINITY).each do |n|y<<ext_int_double_summation(k,n);end;end;end;def five_dimensional_centered_hyperoctahedron;Enumerator.new do |y|y<<1;(1..Float::INFINITY).each do |n|y<<ext_int_double_summation(5,n);end;end;end;def six_dimensional_centered_hyperoctahedron;Enumerator.new do |y|y<<1;(1..Float::INFINITY).each do |n|y<<ext_int_double_summation(6,n);end;end;end;def generalized_pentatope(left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |delta|y<<delta*(delta+1)*(delta+2)*(delta+3)/24;end;end;end;def generalized_k_dimensional_hypertetrahedron(k,left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |delta|y<<Utils.rising_factorial(delta,k)/Utils.factorial_iter(k);end;end;end;def generalized_biquadratic(left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |delta|y<<delta**4;end;end;end;def generalized_k_dimensional_hypercube(k=5,left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |delta|y<<delta**k;end;end;end;def generalized_hyperoctahedral(left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |delta|y<<(delta**2*(delta**2+2))/3;end;end;end;def generalized_k_dimensional_hyperoctahedron(k=5,left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |delta|a=0;(0..(k-1)).each do |i|a+=Utils.binomial_coefficient(k-1,i)*(Utils.rising_factorial(delta-i,k)/Utils.factorial_iter(k));end;y<<a;end;end;end;def generalized_hyperdodecahedral(left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |delta|y<<(delta*(261*delta**3-504*delta**2+283*delta-38))/2;end;end;end;def generalized_hypericosahedral(left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |delta|y<<(delta*(145*delta**3-280*delta**2+179*delta-38))/6;end;end;end;def generalized_polyoctahedral(left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |delta|y<<delta**2*(3*delta**2-4*delta+2);end;end;end;def generalized_k_dimensional_mgonal_pyramidal(k,m,left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |n|y<<(Utils.pseudo_pochhammer_function(n,k)*((m-2)*n-m+k+2))/Utils.factorial_iter(k);end;end;end;def generalized_k_dimensional_centered_hypercube(k,left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |delta|y<<delta**k+(delta-1)**k;end;end;end;def gen_acc_helper_centered_hypertetrahedron(k,n)a=0;(1..n.abs).each do |j|a+=helper_centered_hypertetrahedron(k,j);end;n>0 ? a: -a;end;def generalized_k_dimensional_centered_hypertetrahedron(k,left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |n|y<<gen_acc_helper_centered_hypertetrahedron(k,n);end;end;end;def gen_ext_int_double_summation(k,n)
is_positive_n=n;n=n.abs;t=(2*Utils.binomial_coefficient(k,1)*Utils.binomial_coefficient(1,0));a=0;(1..(n-1)).each do |j|(0..(k-1)).each do |i|a+=(2**(1+i))*Utils.binomial_coefficient(k,1+i)*Utils.binomial_coefficient(j,i);end;end;is_positive_n>0 ? (1+t+a): (1+t+a)*-1;end;def generalized_k_dimensional_centered_hyperoctahedron(k,left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |n|y<<1 if n==1;y<<gen_ext_int_double_summation(k,n) if n!=0;end;end;end;def generalized_nexus(k,left_index=0)Enumerator.new do |y|((-1*left_index.abs)..Float::INFINITY).each do |delta|y<<(delta+1)**(k+1)-delta**(k+1);end;end;end;def cuban;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<(delta+1)**3-delta**3;end;end;end;alias cuban_prime cuban;def helper_quartan(delta)seq=[2];(1..delta).each do |x|(x+1..delta).each do |y|q=x**4+y**4;seq<<q if PAdicUtils.prime?(q);end;end;seq.sort;end;def quartan;Enumerator.new do |y|(1..Float::INFINITY).each do |delta|y<<helper_quartan(delta)[delta-1];end;end;end;def pell;pell_numbers=[0,1];Enumerator.new do |y|y<<0;y<<1;(2..Float::INFINITY).each do |delta|y<<pell_numbers[delta]=2*pell_numbers[delta-1]+pell_numbers[delta-2];end;end;end;def helper_carmichael(n)return if PAdicUtils.prime?(n);is_carmichael=(2..Math.sqrt(n)).none? do |a|n.gcd(a)==1&&a.pow(n-1,n)!=1;end;is_carmichael ? n: nil;end;def carmichael;Enumerator.new do |y|(561..Float::INFINITY).each do |delta|num_seq=helper_carmichael(delta);y<<num_seq unless num_seq.nil?;end;end;end;def helper_stern_prime(delta)prime_list=PAdicUtils.first_n_primes(delta);q=prime_list[-1];b=1;while(2*b**2)<q;prime_list[0..-2].each do |p|return nil if q==p+(2*b**2);end;b+=1;end;q;end;def stern_prime(infty=false)
Enumerator.new do |y|max_term=infty==false ? 238: float::INFINITY;(1..max_term).each do |delta|num_seq=helper_stern_prime(delta);y<<num_seq unless num_seq.nil?;end;end;end;def helper_apocalyptic(delta)search_666=(2**delta).to_s;len_666=search_666.length-1
(0..len_666).each do |index|return delta if search_666[index,3]=='666'&&search_666[index+1]=='6'&&search_666[index+2]=='6';end;nil;end;def apocalyptic;Enumerator.new do |y|(157..Float::INFINITY).each do |delta|y<<helper_apocalyptic(delta) unless helper_apocalyptic(delta).nil?;end;end;end;end;end;module FigurateNumbers;extend MultiDimensionalFigurateNumbers;end;MultiDimensionalFigurateNumbers=FigurateNumbers::MultiDimensionalFigurateNumbers