A106514 Expansion of (1-x)/((1-2*x)*(1-2*x-x^2)).

1, 3, 9, 25, 67, 175, 449, 1137, 2851, 7095, 17553, 43225, 106051, 259423, 633089, 1541985, 3749827, 9107175, 22095249, 53559817, 129739171, 314086735, 760009793, 1838300625, 4444999651, 10745077143, 25968708369, 62749602745

Offset: 0

Author

Paul Barry, May 05 2005

Keywords

• easy
• nonn

Comments

Convolution of A000079 and A001333.

Row sums of A106513.

Equals eigensequence of a triangle with the Pell series as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-2).

Crossrefs

Cf. A000079, A001333, A002203, A094706, A106513.

Programs

Magma

I:=[1,3,9]; [n le 3 select I[n] else 4*Self(n-1) -3*Self(n-2) -2*Self(n-3): n in [1..41]]; // G. C. Greubel, Aug 05 2021

Mathematica

Table[(LucasL[n+2, 2] - 2^(n+2))/2, {n, 0, 40}] (* G. C. Greubel, Aug 05 2021 *)

Sage

[(lucas_number2(n+2, 2, -1) - 2^(n+2))/2 for n in (0..40)] # G. C. Greubel, Aug 05 2021

Formula

a(n) = Sum_{k=0..n} 2^(n-k-1)*((1-sqrt(2))^k + (1+sqrt(2))^k).

a(n) = Sum_{k=0..n} Sum_{j=0..floor((n+1)/2)} binomial(n+1, 2*j+k+1)*2^j.

a(n) = Pell(n+3) - Pell(n+2) - 2^(n+1) = A001333(n+2) - 2^(n+1). - Ralf Stephan, Jun 02 2007

From G. C. Greubel, Aug 05 2021: (Start)

a(n) = A001333(n+2) - 2^(n+1) = (A002203(n+2) - 2^(n+2))/2.

Sum_{j=0..n} a(j) = A094706(n+1). (End)