A048695 -id:A048695 - cp's OEIS frontend

A048771 Partial sums of A048695.

1, 9, 26, 68, 169, 413, 1002, 2424, 5857, 14145, 34154, 82460, 199081, 480629, 1160346, 2801328, 6763009, 16327353, 39417722, 95162804, 229743337, 554649485, 1339042314, 3232734120, 7804510561, 18841755249, 45488021066, 109817797388, 265123615849

Offset: 0

Barry E. Williams

			a(n)=[ {(8+(9/2)*sqrt(2))(1+sqrt(2))^n -(8-(9/2)*sqrt(2))(1-sqrt(2))^n}/ 2*sqrt(2) ]-7/2.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1)

Cf. A001333, A000129, A048694, A048695.

Table[6*Fibonacci[n, 2] + Fibonacci[n+1, 2], {n, 0, 22}] // Accumulate (* Jean-François Alcover, Mar 25 2013 *)
Accumulate[LinearRecurrence[{2,1},{1,8},40]] (* or *) LinearRecurrence[ {3,-1,-1},{1,9,26},40] (* Harvey P. Dale, May 01 2013 *)

a(n)=2*a(n-1)+a(n-1)+7; a(0)=1, a(1)=9.
G.f. ( 1+6*x ) / ( (x-1)*(x^2+2*x-1) ). a(n)=A048739(n)+6*A048739(n-1). - R. J. Mathar, Nov 08 2012
a(0)=1, a(1)=9, a(2)=26, a(n)=3*a(n-1)-a(n-2)-a(n-3). - Harvey P. Dale, May 01 2013

More terms from Harvey P. Dale, May 01 2013

A051959 Expansion of (1+6x)/((1-2x-x^2)*(1-x)^2).

Original entry on oeis.org

1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728, 339809, 820438, 1980784, 4782112, 11545121, 27872474, 67290196, 162453000, 392196337, 946845822, 2285888136, 5518622256, 13323132817, 32164888066, 77652909132, 187470706520, 452594322369, 1092659351462, 2637913025504, 6368485402688

Offset: 0

Barry E. Williams, Jan 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.
Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Cf. A000129, A048771, A048695.

I:=[1,10,36,104]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2) +Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012

LinearRecurrence[{4,-4,0,1},{1,10,36,104},40] (* Vincenzo Librandi, Jun 22 2012 *)

def A051959(n):
    @CachedFunction
    def a(n):
        if n<4: return (1,10,36,104)[n]
        else: return 4*a(n-1) -4*a(n-2) +a(n-4)
    return a(n)
[A051959(n) for n in range(41)] # G. C. Greubel, Nov 11 2024

a(n) = 2*a(n-1) + a(n-2) + (7*n+1), with a(0)=1, a(1)=10.
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).
a(n) = ( (25 + 17*sqrt(2))*(1+sqrt(2))^n - (25 - 17*sqrt(2))*(1-sqrt(2))^n )/(4*sqrt(2)) - (7*n + 15)/2.
a(n) = (1/2)*(4*Pell(n+2) - 3*Pell(n) - 7*n - 15), with Pell(n) = A000129(n). - Ralf Stephan, May 15 2007
E.g.f.: (1/4)*exp(x)*(-30 - 14*x + 25*sqrt(2)*sinh(sqrt(2)*x) + 34*cosh(sqrt(2)*x)). - G. C. Greubel, Nov 11 2024

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A048771 Partial sums of A048695.

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A051959 Expansion of (1+6x)/((1-2x-x^2)*(1-x)^2).

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cp's OEIS Frontend

A048771 Partial sums of A048695.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A051959 Expansion of (1+6*x)/((1-2*x-x^2)*(1-x)^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A051959 Expansion of (1+6x)/((1-2x-x^2)*(1-x)^2).