A141775 Binomial transform of (1, 2, 0, 1, 2, 0, 1, 2, 0, ...).

1, 3, 5, 8, 15, 31, 64, 129, 257, 512, 1023, 2047, 4096, 8193, 16385, 32768, 65535, 131071, 262144, 524289, 1048577, 2097152, 4194303, 8388607, 16777216, 33554433, 67108865, 134217728, 268435455, 536870911, 1073741824, 2147483649, 4294967297, 8589934592, 17179869183

Offset: 0

Author

Gary W. Adamson, Jul 03 2008

Keywords

• nonn
• easy

Comments

From Paul Curtz, Jun 15 2011: (Start)

A square array of a(n) and its higher order differences is defined by T(0,k) = a(k) and T(n,k) = T(n-1,k+1)-T(n-1,k):

1, 3, 5, 8, 15, 31,

2, 2, 3, 7, 16, 33,

0, 1, 4, 9, 17, 32, see A130785(n).

1, 3, 5, 8, 15, 31,

2, 2, 3, 7, 16, 33,

a(n) is identical to its third differences: T(n+3,k) = T(n,k).

The main diagonal is T(n,n) = 2^n. Subdiagonals are T(n,n-1) = A014551(n) and T(n,n-2) = A062510(n).

(End)

Examples

			a(4) = 8 = (1, 2, 0, 1) dot (1, 3, 3, 1) = (1 + 6 + 0 + 1).
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Crossrefs

Cf. A000079, A057079.

Programs

Magma

I:=[1,3,5]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018

Mathematica

LinearRecurrence[{3,-3,2},{1,3,5},40] (* Harvey P. Dale, May 29 2012 *)

PARI

x='x+O('x^30); Vec((x-1)*(1+x)/((2*x-1)*(x^2-x+1))) \\ G. C. Greubel, Jan 15 2018

Formula

From Paul Curtz, Jun 15 2011: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).

a(n) = 2^n - A128834(n).

a(n) - 2a(n-1)= A057079(n+1).

a(n) + a(n+3) = 9*2^n.

a(n+6) - a(n) = 63*2^n.

a(n) = A130785(n) - A130785(n-1). (End)

G.f.: (x-1)*(1+x) / ( (2*x-1)*(x^2-x+1) ). - R. J. Mathar, Jun 22 2011

a(n) = 2^n + (2*sin((Pi*n)/3))/sqrt(3). - Colin Barker, Feb 10 2017