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A046189 Octagonal pentagonal numbers. 4

%I #21 Aug 16 2015 12:03:55

%S 1,176,1575425,234631320,2098015778145,312461813932000,

%T 2793956983975264801,416109772078405066376,3720751630955537773670465,

%U 554139209013308662750166160,4954977037463529073741814611905,737954942591533222733596372781560

%N Octagonal pentagonal numbers.

%C From _Ant King_, Dec 16 2011: (Start)

%C lim(n->Infinity, a(2n+1)/a(2n))=1/49*(219073+154908*sqrt(2)).

%C lim(n->Infinity, a(2n)/a(2n-1))=1/49*(3649+2580*sqrt(2)).

%C (End)

%H Colin Barker, <a href="/A046189/b046189.txt">Table of n, a(n) for n = 1..327</a>

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

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1331714,-1331714,-1,1).

%F From _Ant King_, Dec 16 2011: (Start)

%F a(n) = 1331714*a(n-2) - a(n-4) + 249696.

%F a(n) = a(n-1) + 1331714*a(n-2) - 1331714*a(n-3) - a(n-4) + a(n-5).

%F a(n) = 1/96*((11-6*sqrt(2)*(-1)^n)*(1+sqrt(2))^(8*n-6)+(11+6*sqrt(2)*(-1)^n)*(1-sqrt(2))^(8*n-6)-18).

%F a(n) = floor(1/96*(11-6*sqrt(2)*(-1)^n)*(1+sqrt(2))^(8*n-6)).

%F G.f.: x*(1+175*x+243535*x^2+5945*x^3+40*x^4)/((1-x)*(1-1154*x+x^2)*(1+1154*x+x^2)).

%F (End)

%t LinearRecurrence[{1, 1331714, -1331714, -1, 1}, {1, 176, 1575425, 234631320, 2098015778145}, 11] (* _Ant King_, Dec 16 2011 *)

%o (PARI) Vec(x*(1+175*x+243535*x^2+5945*x^3+40*x^4)/((1-x)*(1-1154*x+x^2)*(1+1154*x+x^2)) + O(x^20)) \\ _Colin Barker_, Jun 23 2015

%Y Cf. A046187, A046188.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_

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Last modified March 29 09:14 EDT 2024. Contains 371268 sequences. (Running on oeis4.)