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A046181 Indices of octagonal numbers which are also triangular. 3
1, 3, 63, 261, 6141, 25543, 601723, 2502921, 58962681, 245260683, 5777740983, 24033043981, 566159653621, 2354993049423, 55477868313843, 230765285799441, 5436264935102961, 22612643015295763, 532698485771776303, 2215808250213185301, 52199015340698974701 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Ant King, Nov 01 2011: (Start)

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

limit 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*(24*a(n)^2 - 16*a(n) + 1)^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) - 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

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Last modified September 26 10:59 EDT 2017. Contains 292518 sequences.