A226122 Expansion of (1+2*x+x^2+x^3+2*x^4+x^5)/(1-2*x^3+x^6).

1, 2, 1, 3, 6, 3, 5, 10, 5, 7, 14, 7, 9, 18, 9, 11, 22, 11, 13, 26, 13, 15, 30, 15, 17, 34, 17, 19, 38, 19, 21, 42, 21, 23, 46, 23, 25, 50, 25, 27, 54, 27, 29, 58, 29, 31, 62, 31, 33, 66, 33, 35, 70, 35, 37, 74, 37, 39, 78, 39

Offset: 0

Author

Paul Curtz, May 27 2013

Keywords

• nonn
• easy

Comments

A226023 (starting from A226023(-2)=0) and successive differences:

0, -1, 0, 2, 3, 6, 12, 15, 20, 30,...

-1, 1, 2, 1, 3, 6, 3, 5, 10, 5,... = a(n-1)

2, 1, -1, 2, 3, -3, 2, 5, -5, 2,...

-1, -2, 3, 1, -6, 5, 3, -10, 7, 5,...

-1, 5, -2, -7, 11, -2, -13, 17, -2, -19,...

6, -7, -5, 18, -13, -11, 30, -19, -17, 42,...

-13, 2, 23, -31, 2, 41, -49, 2, 59, 67,...

15, 21, -54, 33, 39, -90, 51, 57, -126, 69,... multiples of 3

6, -75, 87, 6, -129, 141, 6, -183, 195, 6,... multiples of 3

-81, 162, -81, -135, 270, -135, -189, 378, -189, -243,... multiples of 27

The last line is -27*a(n+3)*A131561(n+1).

The recurrences in the Formula field hold for the array.

Examples

			Given A130823 = 1,1,1,3,3,3,5,5,5,7,7,7,... and A131534 = 1,2,1,1,2,1,1,2,1,1,2,1,..., then a(0)=1*1=1, a(1)=1*2=2, a(2)=1*1=1, a(3)=3*1=3, a(4)=3*2=6, etc.
Given A226023(n) from A226023(-1)=-1, then a(0)=0-(-1)=1, a(1)=2-0=2, a(2)=3-2=1, a(3)=6-3=3, a(4)=12-6=6, etc.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Crossrefs

Cf. A005408, A016754, A130823, A131534, A226023.

Programs

Mathematica

repeat=20; Table[{1, 2, 1}, {repeat}]*(2*Range[repeat]-1) // Flatten
(* or *) Table[Floor[(2*n+1)/3]*Floor[(2*n+5)/3], {n, -1, 59}] // Differences (* Jean-François Alcover, May 29 2013 *)

Formula

a(n) = A130823(n-1) * A131534(n).

a(n) = A226023(n) - A226023(n-1) with A226023(-1)=-1.

a(n) = 3*a(n-3) -3*a(n-6) +a(n-9) = a(n-1) +2*a(n-3) -2*a(n-4) -a(n-6) +a(n-7). [Ralf Stephan]

From Bruno Berselli, May 29 2013: (Start)

G.f.: (1+x)^3*(1-x+x^2)/((1-x)^2*(1+x+x^2)^2).

a(n) = 2*a(n-3)-a(n-6).

a(3n)*a(3n-1)-a(3n-2) = A016754(n-1), n>0. (End)