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
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Richard Choulet, Oct 03 2007
STATUS
approved