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A048902 Indices of heptagonal numbers (A000566) which are also hexagonal.

%I A048902 #29 Feb 16 2025 08:32:40
%S A048902 1,221,71065,22882613,7368130225,2372515049741,763942477886281,
%T A048902 245987105364332645,79207083984837225313,25504435056012222218045,
%U A048902 8212348880951950716985081,2644350835231472118646977941
%N A048902 Indices of heptagonal numbers (A000566) which are also hexagonal.
%C A048902 As n increases, this sequence is approximately geometric with common ratio r = lim_{n->infinity} a(n)/a(n-1) = (2 + sqrt(5))^4 = 161 + 72*sqrt(5). - _Ant King_, Dec 26 2011
%H A048902 Vincenzo Librandi, <a href="/A048902/b048902.txt">Table of n, a(n) for n = 1..200</a>
%H A048902 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeptagonalHexagonalNumber.html">Heptagonal hexagonal number.</a>
%H A048902 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (323,-323,1).
%F A048902 G.f.: -x*(1 - 102*x + 5*x^2) / ( (x-1)*(x^2 - 322*x + 1) ). - _R. J. Mathar_, Dec 21 2011
%F A048902 From _Ant King_, Dec 26 2011: (Start)
%F A048902 a(n) = 322*a(n-1) - a(n-2) - 96.
%F A048902 a(n) = (1/20)*((sqrt(5)+1)*(sqrt(5)+2)^(4*n-3) + (sqrt(5)-1)*(sqrt(5)-2)^(4*n-3) + 6).
%F A048902 a(n) = ceiling((1/20)*(sqrt(5)+1)*(sqrt(5)+2)^(4*n-3)).
%F A048902 (End)
%t A048902 LinearRecurrence[{323, -323, 1}, {1, 221, 71065}, 12]; (* _Ant King_, Dec 26 2011 *)
%o A048902 (Magma) I:=[1, 221, 71065]; [n le 3 select I[n] else 323*Self(n-1)-323*Self(n-2)+Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Dec 28 2011
%Y A048902 Cf. A048901, A048903.
%K A048902 nonn,easy
%O A048902 1,2
%A A048902 _Eric W. Weisstein_