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Indices of hexagonal numbers which are also heptagonal.
3

%I #25 Feb 16 2025 08:32:40

%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="https://mathworld.wolfram.com/HeptagonalHexagonalNumber.html">Heptagonal hexagonal number.</a>

%H <a href="/index/Rec">Index entries for 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 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)^(4*n-3) + (1-sqrt(5))*(sqrt(5)-2)^(4*n-3) + 2*sqrt(5)).

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

%O 1,2

%A _Eric W. Weisstein_