How to resolve the algorithm Faulhaber's triangle step by step in the Prolog programming language

Published on 12 May 2024 09:40 PM
#Prolog

How to resolve the algorithm Faulhaber's triangle step by step in the Prolog programming language

Table of Contents

Problem Statement

Named after Johann Faulhaber, the rows of Faulhaber's triangle are the coefficients of polynomials that represent sums of integer powers, which are extracted from Faulhaber's formula:

where

B

n

{\displaystyle B_{n}}

is the nth-Bernoulli number.

The first 5 rows of Faulhaber's triangle, are:

Using the third row of the triangle, we have:

∑

k

1

n

k

2

=

1 6

n +

1 2

n

2

1 3

n

3

{\displaystyle \sum _{k=1}^{n}k^{2}={1 \over 6}n+{1 \over 2}n^{2}+{1 \over 3}n^{3}}

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Faulhaber's triangle step by step in the Prolog programming language

Source code in the prolog programming language

ft_rows(Lz) :-
    lazy_list(ft_row, [], Lz).

ft_row([], R1, R1) :- R1 = [1].
ft_row(R0, R2, R2) :-
    length(R0, P),
    Jmax is 1 + P, numlist(2, Jmax, Qs),
    maplist(term(P), Qs, R0, R1),
    sum_list(R1, S), Bk is 1 - S, % Bk is Bernoulli number
    R2 = [Bk | R1].

term(P, Q, R, S) :- S is R * (P rdiv Q).

show(N) :-
    ft_rows(Rs),
    length(Rows, N), prefix(Rows, Rs),
    forall(
        member(R, Rows),
            (format(string(S), "~w", [R]),
             re_replace(" rdiv "/g, "/", S, T),
             re_replace(","/g, ", ", T, U),
             write(U), nl)).

sum(N, K, S) :- % sum I=1,N (I ** K)
    ft_rows(Rows), drop(K, Rows, [Coefs|_]),
    reverse([0|Coefs], Poly),
    foldl(horner(N), Poly, 0, S).

horner(N, A, S0, S1) :-
    S1 is N*S0 + A.

drop(N, Lz1, Lz2) :-
    append(Pfx, Lz2, Lz1), length(Pfx, N), !.

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