

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
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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], 20162018.
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^235*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^24*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] (* JeanFrançois 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



