A084634 Binomial transform of 1, 1, 1, 2, 2, 2, 2, 2, ...

1, 2, 4, 9, 21, 48, 106, 227, 475, 978, 1992, 4029, 8113, 16292, 32662, 65415, 130935, 261990, 524116, 1048385, 2096941, 4194072, 8388354, 16776939, 33554131, 67108538, 134217376, 268435077, 536870505, 1073741388, 2147483182, 4294966799, 8589934063

Offset

0,2

Comments

Partial sums of A000325.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Formula

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

a(n) = 1 + n + n*(n-1)/2 + 2*Sum_{k=3..n} C(n, k).

O.g.f.: (1-3*x+3*x^2)/((1-2*x)*(1-x)^3). - R. J. Mathar, Apr 07 2008

a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4). - R. J. Mathar, Apr 07 2008

a(n) = Sum_{i=0..n} (2^i - i). - Ctibor O. Zizka, Oct 15 2010

a(n) = A000225(n+1) - binomial(n+1,2). - G. C. Greubel, Mar 18 2023

Maple

A084634:=n->2^(n+1) - (n^2 +n +2)/2; seq(A084634(n), n=0..50); # Wesley Ivan Hurt, Jan 31 2014

Mathematica

LinearRecurrence[{5, -9, 7, -2}, {1, 2, 4, 9}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)

Prog

(Sage) [2^(n+1)-1-binomial(n+1, 2) for n in range(52)] # Zerinvary Lajos, May 29 2009
(Magma) [2^(n+1)-1-Binomial(n+1, 2): n in [0..50]]; // G. C. Greubel, Mar 18 2023

Crossrefs

Cf. A000225, A000325,

Sequence in context: A351644 A027711 A307548 * A137256 A051164 A182904

Adjacent sequences: A084631 A084632 A084633 * A084635 A084636 A084637

Keyword

nonn,easy

Author

Paul Barry, Jun 06 2003

Status

approved