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A048655
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Generalized Pellian with second term equal to 5.
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25
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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
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OFFSET
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0,2
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COMMENTS
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Equals binomial transform of A143095: (1, 4, 2, 8, 4, 16, 8, 32, ...). - Gary W. Adamson, Jul 23 2008
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LINKS
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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)
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FORMULA
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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
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MAPLE
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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
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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 *)
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PROG
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(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 */
(PARI) a(n)=([0, 1; 1, 2]^n*[1; 5])[1, 1] \\ Charles R Greathouse IV, Feb 09 2017
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)/(1-2*x-x^2))); // G. C. Greubel, Jul 26 2018
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CROSSREFS
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Cf. A001333, A000129, A048654, A143095.
Sequence in context: A192300 A289775 A119503 * A181896 A041671 A215221
Adjacent sequences: A048652 A048653 A048654 * A048656 A048657 A048658
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Barry E. Williams
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STATUS
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approved
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