A139482 Binomial transform of [1, 1, 2, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ...].

1, 2, 5, 11, 20, 32, 47, 65, 86, 110, 137, 167, 200, 236, 275, 317, 362, 410, 461, 515, 572, 632, 695, 761, 830, 902, 977, 1055, 1136, 1220, 1307, 1397, 1490, 1586, 1685, 1787, 1892, 2000, 2111, 2225

Offset: 1

Author

Gary W. Adamson, Apr 23 2008

Keywords

• nonn
• easy

Comments

A007318 * [1, 1, 2, 1, -1, 1, -1, 1, ...].

The quadratic expression for a(n) follows at once by taking into account that the alternate row sums in the Pascal triangle are equal to zero (starting with the second row). - Emeric Deutsch, May 03 2008

For n > 1, 3*(8*a(n) - 13) = A016945(n-2)^2. - Vincenzo Librandi, Feb 15 2012

Examples

			a(4) = 11 = (1, 3, 3, 1) dot (1, 1, 2, 1) = (1 + 3 + 6 + 1).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Maple

1,seq((10+3*n^2-9*n)*1/2,n=2..40); # Emeric Deutsch, May 03 2008

Mathematica

Join[{1,2},FoldList[##+3&,5,3*Range@100]] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)
LinearRecurrence[{3,-3,1},{1,2,5,11},40] (* Harvey P. Dale, May 02 2015 *)

Formula

G.f.: (x^3+2*x^2-x+1)/(-x^3+3*x^2-3*x+1). - Alexander R. Povolotsky, Apr 24 2008

a(n) = (10 - 9*n + 3*n^2)/2 for n >= 2. - Emeric Deutsch, May 03 2008

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=2, a(3)=5, a(4)=11. - Harvey P. Dale, May 02 2015

Extensions

More terms from Emeric Deutsch, May 03 2008