A048906 Octagonal heptagonal numbers.

1, 297045, 69010153345, 16032576845184901, 3724720317758036481633, 865334473646149974640821781, 201036235582696134090746961388705, 46705140322177796790584365589105966085

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• Eric Weisstein's World of Mathematics, Octagonal Heptagonal Number.
• Index entries for linear recurrences with constant coefficients, signature (232323,-232323,1).

Crossrefs

Cf. A048904, A048905.

Programs

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

Mathematica

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 *)

Formula

From Harvey P. Dale, Dec 09 2011: (Start)

G.f.: x*(-133*x^2-64722*x-1)/(x^3-232323*x^2+232323*x-1).

a(1)=1, a(2)=297045, a(3)=69010153345, a(n) = 232323*a(n-1)-232323*a(n-2)+a(n-3). (End)

From Ant King, Dec 30 2011: (Start)

a(n) = 232322*a(n-1)-a(n-2)+64856.

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).

a(n) = floor(1/480*(17+2*sqrt(30))*(sqrt(5)+sqrt(6))^(8n-6)). (End)