A046182 Indices of triangular numbers which are also octagonal.

1, 6, 153, 638, 15041, 62566, 1473913, 6130878, 144428481, 600763526, 14152517273, 58868694718, 1386802264321, 5768531318886, 135892469386233, 565257200556158, 13316075197586561, 55389437123184646

Offset

1,2

Comments

From Ant King, Nov 01 2011: (Start)

lim(n->Infinity,a(2n+1)/a(2n))=1/5*(59+24*sqrt(6)).

lim(n->Infinity,a(2n)/a(2n-1))=1/5*(11+4*sqrt(6)).

(End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Octagonal Triangular Number

Index entries for linear recurrences with constant coefficients, signature (1,98,-98,-1,1).

Formula

For n odd, a(n+2) = 98*a(n+1)-a(n)+48; for n even, a(n+1) = 49*a(n)+24+10*(24*a(n)^2+24*a(n)+16)^0.5 - Richard Choulet, Oct 03 2007, Oct 09 2007

From Ant King, Nov 01 2011: (Start)

a(n) = a(n-1) + 98*a(n-2) - 98*a(n-3) - a(n-4) + a(n-5).

a(n) = 98*a(n-2) - a(n-4) + 48.

a(n) = 1/12*sqrt(3)(( sqrt(6)-(-1)^n)*(sqrt(3)+sqrt(2))^(2*n-1)-(sqrt(6)+(-1)^n)*(sqrt(3)-sqrt(2))^(2*n-1)-2*sqrt(3)).

a(n) = floor(1/12* sqrt(3)*(sqrt(6)-(-1)^n)*(sqrt(3)+sqrt(2))^(2*n-1)).

G.f.: x(1+5*x+49*x^2-5*x^3-2*x^4)/((1-x)(1-10*x+x^2)(1+10*x+x^2)).

(End)

Mathematica

LinearRecurrence[{1, 98, -98, -1, 1}, {1, 6, 153, 638, 15041}, 18](* Ant King, Nov 01 2011 *)

Prog

(PARI) Vec((1+5*x+49*x^2-5*x^3-2*x^4)/((1-x)*(1-10*x+x^2)*(1+10*x+x^2))+O(x^99)) \\ Charles R Greathouse IV, Nov 01 2011
(Magma) I:=[1, 6, 153, 638, 15041]; [n le 5 select I[n] else Self(n-1)+98*Self(n-2)-98*Self(n-3)-Self(n-4)+Self(n-5): n in [1..20]]; // Vincenzo Librandi, Dec 30 2011

Crossrefs

Cf. A046181, A046183, A046190.

Sequence in context: A354752 A319036 A185211 * A092122 A285368 A003460

Adjacent sequences: A046179 A046180 A046181 * A046183 A046184 A046185

Keyword

nonn,easy

Author

Eric W. Weisstein

Extensions

More terms from Richard Choulet, Oct 03 2007

Status

approved