OFFSET
1,2
COMMENTS
As n increases, this sequence is approximately geometric with common ratio r = lim_{n->infinity} a(n)/a(n-1) = (sqrt(3) + sqrt(2))^4 = 49 + 20*sqrt(6). - Ant King, Dec 27 2011
LINKS
Herman Jamke, Table of n, a(n) for n = 1..20
Eric Weisstein's World of Mathematics, Octagonal Hexagonal Number.
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).
FORMULA
G.f.: x*(-1+22*x+3*x^2) / ( (x-1)*(x^2-98*x+1) ). - R. J. Mathar, Dec 21 2011
From Ant King, Dec 27 2011: (Start)
a(n) = 98*a(n-1) - a(n-2) - 24.
a(n) = (1/24)*sqrt(3)*((1+sqrt(6))*(sqrt(3) + sqrt(2))^(4n-3) + (1-sqrt(6))*(sqrt(3) - sqrt(2))^(4n-3) + 2*sqrt(3)).
a(n) = ceiling((1/24)*sqrt(3)*(1+sqrt(6))*(sqrt(3) + sqrt(2))^(4n-3)).
(End)
MAPLE
a:=5+2*sqrt(6): b:=5-2*sqrt(6): s:=n->a^n+b^n: d:=n->sqrt(6)*(a^n-b^n):for n from 0 to 40 do x:=simplify(s(n)-1/4*d(n)): y:=simplify(1/3*d(n)-s(n)/2): if(type((1+x/2)/3, integer) and type((1+y)/4, integer)) then printf("%d, ", (1+y)/4) fi: x:=simplify(s(n+1)+1/4*d(n+1)): y:=simplify(1/3*d(n+1)+s(n+1)/2): if(type((1+x/2)/3, integer) and type((1+y)/4, integer)) then printf("%d, ", (1+y)/4) fi: od: # Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008
MATHEMATICA
LinearRecurrence[{99, -99, 1}, {1, 77, 7521}, 13] (* Ant King, Dec 27 2011 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
1 more term from Larry Reeves (larryr(AT)acm.org), May 07 2001
One more term from Lior Manor, Feb 13 2002
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008
STATUS
approved