A048776 First partial sums of A048739; second partial sums of A000129.

1, 4, 12, 32, 81, 200, 488, 1184, 2865, 6924, 16724, 40384, 97505, 235408, 568336, 1372096, 3312545, 7997204, 19306972, 46611168, 112529329, 271669848, 655869048, 1583407968, 3822685009, 9228778012, 22280241060, 53789260160, 129858761409, 313506783008

Offset

0,2

Links

Table of n, a(n) for n=0..29.

Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Formula

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

a(n) = (((7/2 + (5/2)*sqrt(2))*(1+sqrt(2))^n - (7/2 - (5/2)*sqrt(2))*(1-sqrt(2))^n)/2*sqrt(2)) - (n+3)/2.

a(n) = (A000129(n+3) - (n+3))/2 = Sum_{j} A047662(n-j+1, j+1). - Henry Bottomley, Jul 09 2001

From R. J. Mathar, Feb 06 2010: (Start)

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).

G.f.: -1/((x^2+2*x-1) * (x-1)^2). (End)

Define an array with m(n,1)=1 and m(1,k) = k*(k+1)/2 for n=1,2,3,... The interior terms are m(n,k) = m(n,k-1) + m(n-1,k-1) + m(n-1,k). The sum of the terms in each antidiagonal=a(n). - J. M. Bergot, Dec 01 2012 [This is A154948 without the first column. The diagonal is m(n,n) = A161731(n-1). R. J. Mathar, Dec 06 2012]

E.g.f.: exp(x)*(10*cosh(sqrt(2)*x) + 7*sqrt(2)*sinh(sqrt(2)*x) - 2*(3 + x))/4. - Stefano Spezia, May 13 2023

Maple

with(combinat):seq((fibonacci(n+3, 2)-n-3)/2, n=0..25); # Zerinvary Lajos, Jun 02 2008

Mathematica

a=b=0; Table[c=2*b+a+n; a=b; b=c, {n, 1, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011*)
LinearRecurrence[{4, -4, 0, 1}, {1, 4, 12, 32}, 30] (* Harvey P. Dale, Aug 27 2014 *)

Crossrefs

Cf. A001333, A000129, A048739.

Sequence in context: A118442 A097067 A038592 * A135248 A205976 A291038

Adjacent sequences: A048773 A048774 A048775 * A048777 A048778 A048779

Keyword

easy,nonn

Author

Barry E. Williams

Extensions

More terms from Harvey P. Dale, Aug 27 2014

Status

approved