A048905 -id:A048905 - cp's OEIS frontend

A048906 Octagonal heptagonal numbers.

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

Offset: 1

Eric W. Weisstein

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

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).

Cf. A048904, A048905.

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

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 *)

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)

A048904 Indices of heptagonal numbers (A000566) which are also octagonal.

Original entry on oeis.org

1, 345, 166145, 80081401, 38599068993, 18604671173081, 8967412906355905, 4322274416192372985, 2083327301191817422721, 1004159436900039805378393, 484002765258517994374962561, 233288328695168773248926575865

Offset: 1

Eric W. Weisstein

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

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Octagonal Heptagonal Number.
Index entries for linear recurrences with constant coefficients, signature (483,-483,1).

Cf. A048905, A048906.

I:=[1, 345, 166145]; [n le 3 select I[n] else 483*Self(n-1)-483*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Dec 28 2011

LinearRecurrence[{483,-483,1},{1,345,166145},30]

G.f.: x*(-1 + 138*x + 7*x^2) / ( (x-1)*(x^2 - 482*x + 1) ). - R. J. Mathar, Dec 21 2011
From Ant King, Dec 30 2011: (Start)
a(n) = 482*a(n-1) - a(n-2) - 144.
a(n) = (1/60)*((3*sqrt(5) + sqrt(6))*(sqrt(5) + sqrt(6))^(4*n-3) + (3*sqrt(5) - sqrt(6))*(sqrt(5) - sqrt(6))^(4*n-3) + 18).
a(n) = ceiling((1/60)*(3*sqrt(5) + sqrt(6))*(sqrt(5) + sqrt(6))^(4*n-3)). (End)

cp's OEIS Frontend

A048906 Octagonal heptagonal numbers.

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