A046181 Indices of octagonal numbers which are also triangular.

1, 3, 63, 261, 6141, 25543, 601723, 2502921, 58962681, 245260683, 5777740983, 24033043981, 566159653621, 2354993049423, 55477868313843, 230765285799441, 5436264935102961, 22612643015295763, 532698485771776303, 2215808250213185301, 52199015340698974701

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

Colin Barker, Table of n, a(n) for n = 1..1000

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) - 32; for n even, a(n+1) = 49*a(n) - 16 + 10*sqrt(24*a(n)^2 - 16*a(n) + 1). - 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) - 32.

a(n) = (1/24)*sqrt(2)((sqrt(6) - (-1)^n)*(sqrt(3) + sqrt(2))^(2*n - 1) + (sqrt(6) + (-1)^n)*(sqrt(3) - sqrt(2))^(2*n - 1) + 4*sqrt(2)).

a(n) = ceiling((1/24)*sqrt(2)*(sqrt(6) - (-1)^n)*(sqrt(3) + sqrt(2))^(2*n - 1)).

G.f.: x*(1 + 2*x - 38*x^2 + 2*x^3 + x^4)/((1 - x)*(1 - 10*x + x^2)*(1 + 10*x + x^2)).

(End)

Mathematica

LinearRecurrence[{1, 98, -98, -1, 1}, {1, 3, 63, 261, 6141}, 18] (* Ant King, Nov 01 2011 *)

Prog

(PARI) Vec(-x*(x^4+2*x^3-38*x^2+2*x+1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^50)) \\ Colin Barker, Jun 23 2015

Crossrefs

Cf. A046182, A046183.

Sequence in context: A087886 A123754 A048354 * A151993 A120053 A139293

Adjacent sequences: A046178 A046179 A046180 * A046182 A046183 A046184

Keyword

nonn,easy

Author

Eric W. Weisstein

Extensions

More terms from Richard Choulet, Oct 03 2007

Status

approved