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 A048901 Indices of hexagonal numbers which are also heptagonal. 3

%I

%S 1,247,79453,25583539,8237820025,2652552464431,854113655726677,

%T 275021944591525483,88556212044815478769,28514825256485992638055,

%U 9181685176376444813974861,2956474111967958744107267107

%N Indices of hexagonal numbers which are also heptagonal.

%C 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 Vincenzo Librandi, <a href="/A048901/b048901.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeptagonalHexagonalNumber.html">Heptagonal hexagonal number.</a>

%H <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (323,-323,1).

%F G.f. x*(-1+76*x+5*x^2) / ( (x-1)*(x^2-322*x+1) ). - R. J. Mathar, Dec 21 2011

%F Contribution from Ant King, Dec 24 2011: (Start)

%F a(n) = 322*a(n-1)-a(n-2)-80.

%F a(n) = 1/40*sqrt(5)*((1+sqrt(5))*(sqrt(5)+2)^(4n-3)+(1-sqrt(5))*(sqrt(5)-2)^(4n-3)+2*sqrt(5)).

%F a(n) = ceiling(1/40*sqrt(5)*(1+sqrt(5))*(sqrt(5)+2)^(4n-3)).

%F (End)

%t LinearRecurrence[{323, -323, 1}, {1, 247, 79453}, 12]; (* Ant King, Dec 24 2011 *)

%o (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 Cf. A048902, A048903.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_

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