A305539 a(n) is a generalized pentagonal number such that 2*a(n) is also a generalized pentagonal number.

0, 1, 35, 1190, 40426, 1373295, 46651605, 1584781276, 53835911780, 1828836219245, 62126595542551, 2110475412227490, 71694037420192110, 2435486796874304251, 82734857056306152425, 2810549653117534878200, 95475953348939879706376, 3243371864210838375138585

Offset

1,3

Comments

Enge, Hart and Johansson prove that every generalized pentagonal number c >= 5 is the sum of a smaller one and twice a smaller one, that is, there are generalized pentagonal numbers a, b < c such that c = 2a + b (see link, theorem 5). We look here at those c >= 0 which have b = 0. A305538 lists the smallest b > 0 for a given c.

Links

G. C. Greubel, Table of n, a(n) for n = 1..650

Andreas Enge, William Hart and Fredrik Johansson, Short addition sequences for theta functions, Journal of Integer Sequences, Vol. 21 (2018), Article 18.2.4. Also available as arXiv:1608.06810 [math.NT], 2016-2018.

Simon Plouffe Conjectures of the OEIS, as of June 20, 2018.

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Formula

If for given n there is an integer k such that k*(3*k + 2) - 6*n^2 - 4*n = (n mod 2)*(4*n + 2) then A001318(n) is in this sequence.

G.f.: x^2/(1 - 35*x + 35*x^2 -x^3). - Simon Plouffe, Jun 20 2018

a(n) = (Pell(2*n-1)^2 - 1)/24, n > 0. - G. C. Greubel, Jun 05 2023

Example

For n = 56 and k = -80 there is k*(3*k + 2) - 6*n^2 - 4*n = 0, hence A001318(56) = 1190 is in this sequence. And indeed also 2380 is a generalized pentagonal number, A001318(79).

Maple

a := proc(searchlimit) local L, g, n, s; L := NULL;
g := n -> ((6*n^2+6*n+1)-(2*n+1)*(-1)^n)/16;
for n from 0 to searchlimit do
    s := isolve(k*(3*k+2)-6*n^2-4*n = irem(n, 2)*(4*n+2));
    if s <> NULL then L:=L, g(n); fi
od: L end:
a(12000);

Mathematica

LinearRecurrence[{35, -35, 1}, {0, 1, 35}, 18] (* Jean-François Alcover, Jul 14 2019, after A029546 *)
(Fibonacci[2*Range[40]-1, 2]^2 -1)/24 (* G. C. Greubel, Jun 05 2023 *)

Prog

(Magma) I:=[0, 1, 35]; [n le 3 select I[n] else 35*Self(n-1) -35*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Jun 05 2023
(SageMath) [(lucas_number1(2*n-1, 2, -1)^2 -1)/24 for n in range(1, 41)] # G. C. Greubel, Jun 05 2023

Crossrefs

Essentially A029546.

Cf. A001318, A305538.

Sequence in context: A352183 A162847 A029546 * A163218 A163600 A164068

Adjacent sequences: A305536 A305537 A305538 * A305540 A305541 A305542

Keyword

nonn,easy

Author

Peter Luschny, Jun 04 2018

Status

approved