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Indices of octagonal numbers which are also heptagonal.
3

%I #27 Feb 16 2025 08:32:40

%S 1,315,151669,73103983,35235967977,16983663460771,8186090552123485,

%T 3945678662460058839,1901808929215196236753,916667958203062126055947,

%U 441832054044946729562729541,212962133381706120587109582655

%N Indices of octagonal 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)) = (sqrt(5)+sqrt(6))^4 = 241+44*sqrt(30). - Ant King, Dec 30 2011

%H Vincenzo Librandi, <a href="/A048905/b048905.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctagonalHeptagonalNumber.html">Octagonal Heptagonal Number</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (483,-483,1).

%F G.f.: -x*(1-168*x+7*x^2) / ( (x-1)*(x^2-482*x+1) ). - _R. J. Mathar_, Dec 21 2011

%F From _Ant King_, Dec 30 2011: (Start)

%F a(n) = 482*a(n-1)-a(n-2)-160.

%F a(n) = 1/120*((2*sqrt(5)+5*sqrt(6))*(sqrt(5)+sqrt(6))^(4n-3)+ (2*sqrt(5)-5*sqrt(6))*(sqrt(5)-sqrt(6))^(4n-3)+40).

%F a(n) = ceiling(1/120*(2*sqrt(5)+5*sqrt(6))*(sqrt(5)+sqrt(6))^(4n-3)). (End)

%t LinearRecurrence[{483,-483,1},{1,315,151669},20] (* _Vincenzo Librandi_, Dec 28 2011 *)

%Y Cf. A048904, A048906.

%K nonn,easy,changed

%O 1,2

%A _Eric W. Weisstein_