A048875 Generalized Pellian with second term of 6.

1, 6, 25, 106, 449, 1902, 8057, 34130, 144577, 612438, 2594329, 10989754, 46553345, 197203134, 835365881, 3538666658, 14990032513, 63498796710, 268985219353, 1139439674122, 4826743915841, 20446415337486, 86612405265785, 366896036400626, 1554196550868289

Offset

0,2

Links

T. D. Noe, Table of n, a(n) for n = 0..200

M. Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.

L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Pellian Representations, Fib. Quart. Vol. 10, No. 5, (1972), pp. 449-488.

Tanya Khovanova, Recursive Sequences

A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.

Index entries for linear recurrences with constant coefficients, signature (4,1).

Formula

a(n) = ((4+sqrt(5))*(2+sqrt(5))^n - (4-sqrt(5))*(2-sqrt(5))^n)*sqrt(5)/2.

a(n) = 4a(n-1) + a(n-2); a(0)=1, a(1)=6.

Binomial transform of A134418: (1, 5, 14, 48, 152, ...). - Gary W. Adamson, Nov 23 2007

G.f.: (1+2*x)/(1-4*x-x^2). - Philippe Deléham, Nov 03 2008

a(-1 - n) = (-1)^n * A097924(n) for all n in Z. - Michael Somos, Feb 23 2014

a(n) = A001076(n+1) + 2*A001076(n). - R. J. Mathar, Sep 11 2019

Example

G.f. = 1 + 6*x + 25*x^2 + 106*x^3 + 449*x^4 + 1902*x^5 + 8057*x^6 + 34130*x^7 + ...

Maple

with(combinat): a:=n->2*fibonacci(n-1, 4)+fibonacci(n, 4): seq(a(n), n=1..17); # Zerinvary Lajos, Apr 04 2008

Mathematica

LinearRecurrence[{4, 1}, {1, 6}, 40] (* Harvey P. Dale, Nov 30 2011 *)
a[ n_] := (4 I ChebyshevT[ n + 1, -2 I] - 3 ChebyshevT[ n, -2 I]) I^n / 5; (* Michael Somos, Feb 23 2014 *)
a[ n_] := If[ n < 0, SeriesCoefficient[ (1 + 6 x) / (1 + 4 x - x^2), {x, 0, -n}], SeriesCoefficient[ (1 + 2 x) / (1 - 4 x - x^2), {x, 0, n}]]; (* Michael Somos, Feb 23 2014 *)

Prog

(Maxima) a[0]:1$ a[1]:6$ a[n]:=4*a[n-1]+a[n-2]$ makelist(a[n], n, 0, 30); /* Martin Ettl, Nov 03 2012 */
(PARI) {a(n) = ( 4*I*polchebyshev( n+1, 1, -2*I) - 3*polchebyshev( n, 1, -2*I)) * I^n / 5}; /* Michael Somos, Feb 23 2014 */
(PARI) {a(n) = if( n<0, polcoeff( (1 + 6*x) / (1 + 4*x - x^2) + x * O(x^-n), -n), polcoeff( (1 + 2*x) / (1 - 4*x - x^2) + x * O(x^n), n))}; \\ Michael Somos, Feb 23 2014

Crossrefs

Cf. A015448, A001076, A001077, A033887.

Cf. A134418.

Cf. A097924.

Sequence in context: A188178 A147543 A212258 * A295202 A346894 A094669

Adjacent sequences: A048872 A048873 A048874 * A048876 A048877 A048878

Keyword

easy,nice,nonn

Author

Barry E. Williams

Extensions

Corrected by T. D. Noe, Nov 07 2006

Status

approved