A355774 An extension of the generalized pentagonal numbers such that every positive integer can be represented as the sum of at most two terms of the sequence.

0, 1, 2, 5, 7, 11, 12, 15, 21, 22, 25, 26, 35, 39, 40, 49, 51, 57, 67, 70, 77, 87, 92, 100, 117, 120, 123, 126, 145, 153, 155, 173, 176, 182, 186, 187, 205, 210, 214, 222, 228, 241, 247, 251, 260, 283, 287, 301, 319, 330, 345, 376, 382, 392, 425, 435, 442, 448

Offset

0,3

Comments

The sequence is defined inductively. Starting from the empty sequence, the terms are added one after the other. A term is added if it is a generalized pentagonal number or if it cannot be represented as the sum of two preceding terms. Note that these exceptions form a proper subsequence of A093519.

Thus any positive number can be expressed as the sum of at most two positive terms by Euler's Pentagonal Number Theorem. Every pentagonal number and every generalized pentagonal number is in this sequence.

Links

Table of n, a(n) for n=0..57.

Andreas Enge, William Hart and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016.

Burkard Polster (Mathloger), The hardest 'What comes next?' (Euler's pentagonal formula), YouTube video, 2020.

Example

32 = 7 + 25; 195 = 22 + 173.

Maple

A355774_list := proc(upto) local P, k, issum, isgpn; P := [];
isgpn := k -> ormap(n -> 0 = 8*k-(n+irem(n, 2))*(3*n+2-irem(n, 2)), [$0..k]);
issum := k -> ormap(p -> member(k - p, P), P);
for k from 0 to upto do
    if isgpn(k) or not issum(k) then P := [op(P), k] fi od;
P end: print(A355774_list(448));

Mathematica

isgpn[k_] := AnyTrue[Range[0, k], 0 == 8*k-(#+Mod[#, 2])*(3*#+2-Mod[#, 2])&];
issum[k_] := AnyTrue[P, MemberQ[P, k-#]&];
P = {};
For[k = 0, k <= 448, k++, If[isgpn[k] || !issum[k], AppendTo[P, k]]];
P (* Jean-François Alcover, Mar 07 2024, after Peter Luschny *)

Prog

(Python)
def A355774_list(upto: int) -> list[int]:
    P: list[int] = []
    for k in range(upto + 1):
        if any(
            k == ((n + n % 2) * (3 * n + 2 - n % 2)) >> 3
            for n in range(k + 1)
        ) or not any([(k - p) in P for p in P]):
            P.append(k)
    return P
print(A355774_list(448))

Crossrefs

Cf. A000326, A001318, A093519, A100878, A355717, A176747 (same construction with triangular numbers).

Sequence in context: A360019 A175034 A030498 * A079933 A075610 A057922

Adjacent sequences: A355771 A355772 A355773 * A355775 A355776 A355777

Keyword

nonn

Author

Peter Luschny, Jul 17 2022

Status

approved