A048778 First partial sums of A048745; second partial sums of A048654.

1, 6, 20, 56, 145, 362, 888, 2160, 5233, 12654, 30572, 73832, 178273, 430418, 1039152, 2508768, 6056737, 14622294, 35301380, 85225112, 205751665, 496728506, 1199208744, 2895146064, 6989500945, 16874148030, 40737797084, 98349742280, 237437281729, 573224305826, 1383885893472

Offset: 0

Author

Barry E. Williams

Keywords

• easy
• nonn

Comments

Define a triangle T by T(n,0) = n*(n+1) + 1, T(n,n) = (n+1)*(n+2)/2, and T(r,c) = T(r-1,c) + T(r-1,c-1) + T(r-2,c-1). Then a(n) is the sum of row n. - J. M. Bergot, Mar 06 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Crossrefs

Cf. A000129, A001333, A048745, A048776, A048654.

Programs

Magma

I:=[1, 6, 20, 56]; [n le 4 select I[n] else 4*Self(n-1) - 4*Self(n-2) + Self(n-4): n in [1..41]]; // G. C. Greubel, Aug 09 2022

Mathematica

Table[(Fibonacci[n+3,2] +2*Fibonacci[n+2,2] -(3*n+7))/2, {n, 0, 40}] (* G. C. Greubel, Aug 09 2022 *)

PARI

N=66;  x='x+O('x^N);
gf= ( -1-2*x ) / ( (x^2+2*x-1)*(x-1)^2 );  Vec(Ser(gf))
/* Joerg Arndt, Mar 07 2013 */

SageMath

[(lucas_number1(n+3, 2, -1) + 2*lucas_number1(n+2, 2, -1) -3*n-7)/2 for n in (0..40)] # G. C. Greubel, Aug 09 2022

Formula

a(n) = 2*a(n-1) + a(n-2) + 3*n + 1, with a(0)=1, a(1)=6.

a(n) = ( ((13 + 9*sqrt(2))/2)*(1 + sqrt(2))^n - ((13 - 9*sqrt(2))/2)*(1 -sqrt(2))^n )/2*sqrt(2) - (3*n + 7)/2.

From R. J. Mathar, Nov 08 2012: (Start)

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

a(n) = A048776(n) + 2*A048776(n-1). (End)

a(n) = (Pell(n+3) + 2*Pell(n+2) - 3*n - 7)/2, where Pell(n) = A000129(n). - G. C. Greubel, Aug 09 2022

Extensions

Corrected by T. D. Noe, Nov 08 2006