A274583 Expansion of (1 + x + x^2 - x^3 - x^4 + x^6)/((1 - x)^3*(1 + x + x^2)^2).

1, 2, 3, 4, 5, 7, 9, 10, 13, 16, 17, 21, 25, 26, 31, 36, 37, 43, 49, 50, 57, 64, 65, 73, 81, 82, 91, 100, 101, 111, 121, 122, 133, 144, 145, 157, 169, 170, 183, 196, 197, 211, 225, 226, 241, 256, 257, 273, 289, 290, 307, 324, 325, 343, 361, 362, 381, 400, 401, 421, 441, 442, 463, 484, 485

Offset

0,2

Comments

Interleaving of A002522, A002061 and A000290.

Links

Table of n, a(n) for n=0..64.

Ilya Gutkovskiy, Illustration

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

Formula

G.f.: (1 + x + x^2 - x^3 - x^4 + x^6)/((1 - x)^3*(1 + x + x^2)^2).

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7).

a(n) = 1 + 10*n/9 - n^2/9 + (n/3 - 8/9)*floor(n/3) + (n/3 - 4/9)*floor((n+1)/3). - Vaclav Kotesovec, Jun 29 2016

Example

Illustration of initial terms:
-------------------------------------------------------------------
                                          o       o         o o o o
                    o     o       o o o   o o o   o o o o   o o o o
    o   o     o o   o o   o o o   o o o   o o o   o o o o   o o o o
o   o   o o   o o   o o   o o o   o o o   o o o   o o o o   o o o o
-------------------------------------------------------------------
-------------------------------------------------------------------
0   1    2     3     4      5       6       7        8         9

Mathematica

LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 2, 3, 4, 5, 7, 9}, 65]
Table[1 + 10*n/9 - n^2/9 + (n/3 - 8/9)*Floor[n/3] + (n/3 - 4/9)*Floor[(n+1)/3], {n, 0, 100}] (* Vaclav Kotesovec, Jun 29 2016 *)

Prog

(PARI) Vec((1+x+x^2-x^3-x^4+x^6)/((1-x)^3*(1+x+x^2)^2) + O(x^99)) \\ Altug Alkan, Jul 05 2016

Crossrefs

Cf. A000290, A002061, A002522.

Sequence in context: A301599 A036408 A307002 * A055600 A139528 A178434

Adjacent sequences: A274580 A274581 A274582 * A274584 A274585 A274586

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Jun 29 2016

Status

approved