A084635 Binomial transform of 1,0,1,0,1,1,1,...

1, 1, 2, 4, 8, 17, 38, 86, 192, 419, 894, 1872, 3864, 7893, 16006, 32298, 64960, 130375, 261310, 523300, 1047416, 2095801, 4192742, 8386814, 16775168, 33552107, 67106238, 134214776, 268432152, 536867229, 1073737734, 2147479122, 4294962304, 8589929103

Offset: 0

Author

Paul Barry, Jun 06 2003

Keywords

• easy
• nonn

Comments

Without its first term, it is the binomial transform of 1,1,1,1,2,2,2,2,2...

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Crossrefs

Cf. A000225, A000325, A084634.

Programs

Magma

[2^n -n*(n^2-3*n+8)/6: n in [0..50]]; // G. C. Greubel, Mar 19 2023

Mathematica

Table[2^n -n -Binomial[n,3], {n,0,50}] (* G. C. Greubel, Mar 19 2023 *)

SageMath

[2^n -n*(n^2-3*n+8)/6 for n in range(51)] # G. C. Greubel, Mar 19 2023

Formula

a(n) = 2^n - n*(n^2 - 3*n + 8)/6.

a(n) = 1 + C(n, 2) + Sum_{k=4..n} C(n, k).

O.g.f.: (1-5*x+10*x^2-10*x^3+5*x^4)/((1-x)^4*(1-2*x)). - R. J. Mathar, Apr 02 2008

a(n) = A000225(n) - (n-1) - binomial(n, 3). - G. C. Greubel, Mar 19 2023