cp's OEIS Frontend

A048906 Octagonal heptagonal numbers.

%I A048906 #34 Feb 16 2025 08:32:40
%S A048906 1,297045,69010153345,16032576845184901,3724720317758036481633,
%T A048906 865334473646149974640821781,201036235582696134090746961388705,
%U A048906 46705140322177796790584365589105966085
%N A048906 Octagonal heptagonal numbers.
%C A048906 As n increases, this sequence is approximately geometric with common ratio r = lim(n->Infinity,a(n)/a(n-1)) = (sqrt(5)+sqrt(6))^8 = 116161+21208*sqrt(30). - _Ant King_, Dec 30 2011
%H A048906 Vincenzo Librandi, <a href="/A048906/b048906.txt">Table of n, a(n) for n = 1..100</a>
%H A048906 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctagonalHeptagonalNumber.html">Octagonal Heptagonal Number</a>.
%H A048906 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (232323,-232323,1).
%F A048906 From _Harvey P. Dale_, Dec 09 2011: (Start)
%F A048906 G.f.: x*(-133*x^2-64722*x-1)/(x^3-232323*x^2+232323*x-1).
%F A048906 a(1)=1, a(2)=297045, a(3)=69010153345, a(n) = 232323*a(n-1)-232323*a(n-2)+a(n-3). (End)
%F A048906 From _Ant King_, Dec 30 2011: (Start)
%F A048906 a(n) = 232322*a(n-1)-a(n-2)+64856.
%F A048906 a(n) = 1/480*((17+2*sqrt(30))*(sqrt(5)+sqrt(6))^(8n-6)+(17-2*sqrt(30))*(sqrt(5)-sqrt(6))^(8n-6)-134).
%F A048906 a(n) = floor(1/480*(17+2*sqrt(30))*(sqrt(5)+sqrt(6))^(8n-6)). (End)
%t A048906 CoefficientList[Series[(-133*x^2-64722*x-1)/(x^3-232323*x^2+ 232323*x- 1),{x,0,20}],x] (* or *) LinearRecurrence[{232323,-232323,1},{1,297045,69010153345},21] (* _Harvey P. Dale_, Dec 09 2011 *)
%o A048906 (Magma) I:=[1, 297045, 69010153345]; [n le 3 select I[n] else 232323*Self(n-1)-232323*Self(n-2)+Self(n-3): n in [1..15]]; // _Vincenzo Librandi_, Dec 28 2011
%Y A048906 Cf. A048904, A048905.
%K A048906 nonn,easy
%O A048906 1,2
%A A048906 _Eric W. Weisstein_