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

Author

Peter Luschny, Jul 17 2022

Keywords

• nonn

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.

Examples

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

Links

• 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.

Crossrefs

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

Programs

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 *)

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))