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A048901 Indices of hexagonal numbers which are also heptagonal.

%I A048901 #25 Feb 16 2025 08:32:40
%S A048901 1,247,79453,25583539,8237820025,2652552464431,854113655726677,
%T A048901 275021944591525483,88556212044815478769,28514825256485992638055,
%U A048901 9181685176376444813974861,2956474111967958744107267107
%N A048901 Indices of hexagonal numbers which are also heptagonal.
%C A048901 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 24 2011
%H A048901 Vincenzo Librandi, <a href="/A048901/b048901.txt">Table of n, a(n) for n = 1..200</a>
%H A048901 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeptagonalHexagonalNumber.html">Heptagonal hexagonal number.</a>
%H A048901 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (323,-323,1).
%F A048901 G.f.: x*(-1 + 76*x + 5*x^2) / ( (x-1)*(x^2 - 322*x + 1) ). - _R. J. Mathar_, Dec 21 2011
%F A048901 From _Ant King_, Dec 24 2011: (Start)
%F A048901 a(n) = 322*a(n-1) - a(n-2) - 80.
%F A048901 a(n) = (1/40)*sqrt(5)*((1+sqrt(5))*(sqrt(5)+2)^(4*n-3) + (1-sqrt(5))*(sqrt(5)-2)^(4*n-3) + 2*sqrt(5)).
%F A048901 a(n) = ceiling((1/40)*sqrt(5)*(1+sqrt(5))*(sqrt(5)+2)^(4*n-3)).
%F A048901 (End)
%t A048901 LinearRecurrence[{323, -323, 1}, {1, 247, 79453}, 12]; (* _Ant King_, Dec 24 2011 *)
%o A048901 (Magma) I:=[1, 247, 79453]; [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 A048901 Cf. A048902, A048903.
%K A048901 nonn,easy
%O A048901 1,2
%A A048901 _Eric W. Weisstein_