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A305539 a(n) is a generalized pentagonal number such that 2*a(n) is also a generalized pentagonal number. 1
0, 1, 35, 1190, 40426, 1373295, 46651605, 1584781276, 53835911780, 1828836219245, 62126595542551, 2110475412227490, 71694037420192110, 2435486796874304251, 82734857056306152425, 2810549653117534878200, 95475953348939879706376, 3243371864210838375138585 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..18.

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.

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

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-Fran├žois Alcover, Jul 14 2019, after A029546 *)

CROSSREFS

Essentially A029546.

Cf. A001318, A305538.

Sequence in context: A046176 A162847 A029546 * A163218 A163600 A164068

Adjacent sequences:  A305536 A305537 A305538 * A305540 A305541 A305542

KEYWORD

nonn,easy

AUTHOR

Peter Luschny, Jun 04 2018

STATUS

approved

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Last modified September 26 01:25 EDT 2020. Contains 337346 sequences. (Running on oeis4.)