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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
OFFSET
1,2
COMMENTS
From Ant King, Nov 01 2011: (Start)
lim_{n -> oo} a(2n+1)/a(2n) = (1/5)*(59 + 24*sqrt(6)).
lim_{n -> oo} a(2n)/a(2n-1) = (1/5)*(11 + 4*sqrt(6)).
(End)
LINKS
Eric Weisstein's World of Mathematics, Octagonal Triangular Number
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
Sequence in context: A123754 A377733 A048354 * A151993 A120053 A139293
KEYWORD
nonn,easy
EXTENSIONS
More terms from Richard Choulet, Oct 03 2007
STATUS
approved