A048655 Generalized Pellian with second term equal to 5.
1, 5, 11, 27, 65, 157, 379, 915, 2209, 5333, 12875, 31083, 75041, 181165, 437371, 1055907, 2549185, 6154277, 14857739, 35869755, 86597249, 209064253, 504725755, 1218515763, 2941757281, 7102030325, 17145817931, 41393666187, 99933150305, 241259966797, 582453083899
Offset: 0
Links
- T. D. Noe, Table of n, a(n) for n = 0..300
- M. Bicknell, A primer on the Pell sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
- A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quarterly, Vol. 3, No. 3, 1965, pp. 161-176.
- A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434.
- A. F. Horadam, Pell identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
- Tanya Khovanova, Recursive sequences
- Index entries for linear recurrences with constant coefficients, signature (2,1)
Programs
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Magma
m:=30; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)/(1-2*x-x^2))); // G. C. Greubel, Jul 26 2018 -
Maple
with(combinat): a:=n->3*fibonacci(n, 2)+fibonacci(n+1, 2): seq(a(n), n=0..26); # Zerinvary Lajos, Apr 04 2008
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Mathematica
a[n_]:=(MatrixPower[{{1,2},{1,1}},n].{{4},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *) LinearRecurrence[{2,1},{1,5},30] (* Harvey P. Dale, Nov 05 2011 *) -
Maxima
a[0]:1$ a[1]:5$ a[n]:=2*a[n-1]+a[n-2]$ A048655(n):=a[n]$ makelist(A048655(n),n,0,30); /* Martin Ettl, Nov 03 2012 */
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PARI
a(n)=([0,1; 1,2]^n*[1;5])[1,1] \\ Charles R Greathouse IV, Feb 09 2017
Formula
a(n) = 2*a(n-1) + a(n-2); a(0)=1, a(1)=5.
a(n) = ((4+sqrt(2))(1+sqrt(2))^n - (4-sqrt(2))(1-sqrt(2))^n)/2*sqrt(2).
a(n) = P(n) - 3*P(n+1) + 2*P(n+2). - Creighton Dement, Jan 18 2005
G.f.: (1+3*x)/(1 - 2*x - x^2). - Philippe Deléham, Nov 03 2008
E.g.f.: exp(x)*(cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x)). - Vaclav Kotesovec, Feb 16 2015
a(n) = 3*Pell(n) + Pell(n+1), where Pell = A000129. - Vladimir Reshetnikov, Sep 27 2016
Comments