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

0,3

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).

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

Mathematica

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

Prog

(Magma) [2^n -n*(n^2-3*n+8)/6: n in [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

Crossrefs

Cf. A000225, A000325, A084634.

Sequence in context: A214999 A348756 A348755 * A294529 A154222 A114199

Adjacent sequences: A084632 A084633 A084634 * A084636 A084637 A084638

Keyword

easy,nonn

Author

Paul Barry, Jun 06 2003

Status

approved