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

%I A305539 #45 Jun 06 2023 08:17:52
%S A305539 0,1,35,1190,40426,1373295,46651605,1584781276,53835911780,
%T A305539 1828836219245,62126595542551,2110475412227490,71694037420192110,
%U A305539 2435486796874304251,82734857056306152425,2810549653117534878200,95475953348939879706376,3243371864210838375138585
%N A305539 a(n) is a generalized pentagonal number such that 2*a(n) is also a generalized pentagonal number.
%C A305539 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.
%H A305539 G. C. Greubel, <a href="/A305539/b305539.txt">Table of n, a(n) for n = 1..650</a>
%H A305539 Andreas Enge, William Hart and Fredrik Johansson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Enge/enge1.html">Short addition sequences for theta functions</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.2.4. Also available as <a href="http://arxiv.org/abs/1608.06810">arXiv:1608.06810</a> [math.NT], 2016-2018.
%H A305539 Simon Plouffe <a href="http://vixra.org/abs/1409.0048">Conjectures of the OEIS, as of June 20, 2018.</a>
%H A305539 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).
%F A305539 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.
%F A305539 G.f.: x^2/(1 - 35*x + 35*x^2 -x^3). - _Simon Plouffe_, Jun 20 2018
%F A305539 a(n) = (Pell(2*n-1)^2 - 1)/24, n > 0. - _G. C. Greubel_, Jun 05 2023
%e A305539 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).
%p A305539 a := proc(searchlimit) local L, g, n, s; L := NULL;
%p A305539 g := n -> ((6*n^2+6*n+1)-(2*n+1)*(-1)^n)/16;
%p A305539 for n from 0 to searchlimit do
%p A305539     s := isolve(k*(3*k+2)-6*n^2-4*n = irem(n,2)*(4*n+2));
%p A305539     if s <> NULL then L:=L,g(n); fi
%p A305539 od: L end:
%p A305539 a(12000);
%t A305539 LinearRecurrence[{35,-35,1}, {0,1,35}, 18] (* _Jean-François Alcover_, Jul 14 2019, after A029546 *)
%t A305539 (Fibonacci[2*Range[40]-1,2]^2 -1)/24 (* _G. C. Greubel_, Jun 05 2023 *)
%o A305539 (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
%o A305539 (SageMath) [(lucas_number1(2*n-1,2,-1)^2 -1)/24 for n in range(1,41)] # _G. C. Greubel_, Jun 05 2023
%Y A305539 Essentially A029546.
%Y A305539 Cf. A001318, A305538.
%K A305539 nonn,easy
%O A305539 1,3
%A A305539 _Peter Luschny_, Jun 04 2018