A048878 Generalized Pellian with second term of 9.
1, 9, 37, 157, 665, 2817, 11933, 50549, 214129, 907065, 3842389, 16276621, 68948873, 292072113, 1237237325, 5241021413, 22201322977, 94046313321, 398386576261, 1687592618365, 7148757049721, 30282620817249, 128279240318717, 543399582092117, 2301877568687185
Offset: 0
Examples
a(n) = 4a(n-1) + a(n-2); a(0)=1, a(1)=9.
Links
- Harry J. Smith, Table of n, a(n) for n = 0..1584
- Tanya Khovanova, Recursive Sequences.
- Index entries for linear recurrences with constant coefficients, signature (4,1).
Programs
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Maple
with(combinat): a:=n->5*fibonacci(n-1,4)+fibonacci(n,4): seq(a(n), n=1..16); # Zerinvary Lajos, Apr 04 2008
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Mathematica
LinearRecurrence[{4,1},{1,9},31] (* or *) CoefficientList[ Series[ (1+5x)/(1-4x-x^2),{x,0,30}],x] (* Harvey P. Dale, Jul 12 2011 *) -
PARI
{ default(realprecision, 2000); for (n=0, 2000, a=round(((7+sqrt(5))*(2+sqrt(5))^n - (7-sqrt(5))*(2-sqrt(5))^n )/10*sqrt(5)); if (a > 10^(10^3 - 6), break); write("b048878.txt", n, " ", a); ); } \\ Harry J. Smith, May 31 2009
Formula
a(n) = ( (7+sqrt(5))(2+sqrt(5))^n - (7-sqrt(5))(2-sqrt(5))^n )/2*sqrt(5).
G.f.: (1+5*x)/(1-4*x-x^2). - Philippe Deléham, Nov 03 2008
a(n) = F(3*n+3) + F(3*n-2); F = A000045. - Yomna Bakr and Greg Dresden, May 25 2024