A082585 a(1)=1, a(n) = ceiling(r(5)*a(n-1)) where r(5) = (1/2)*(5 + sqrt(29)) is the positive root of X^2 = 5*X + 1.
1, 6, 32, 167, 868, 4508, 23409, 121554, 631180, 3277455, 17018456, 88369736, 458867137, 2382705422, 12372394248, 64244676663, 333595777564, 1732223564484, 8994713599985, 46705791564410, 242523671422036
Offset: 1
Links
- Index entries for linear recurrences with constant coefficients, signature (6,-4,-1).
Programs
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Maple
a:=n->sum(fibonacci(i,5), i=0..n): seq(a(n), n=1..21); # Zerinvary Lajos, Mar 20 2008
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Mathematica
Join[{a=1,b=6},Table[c=5*b+1*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011*) LinearRecurrence[{6,-4,-1},{1,6,32},30] (* Harvey P. Dale, Jan 06 2012 *)
Formula
For n > 3, a(n) = 6*a(n-1) - 4*a(n-2) - a(n-3); a(n) = floor(t(5)*r(5)^n) where t(5) = (1/10)*(1 + 7/sqrt(29)) is the positive root of 145*X^2 = 29*X + 1.
G.f.: x/((x-1)*(x^2+5*x-1)). - Colin Barker, Jan 27 2013
G.f.: 1/(1/Q(0) + 3*x^3 - 3*x) where Q(k) = 1 + k*(2*x+1) + 8*x - 2*x*(k+1)*(k+5)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Mar 15 2013