A267322 Expansion of (1 + x + x^2 + x^4 + 2*x^5)/(1 - x^3)^3.

1, 1, 1, 3, 4, 5, 6, 9, 12, 10, 16, 22, 15, 25, 35, 21, 36, 51, 28, 49, 70, 36, 64, 92, 45, 81, 117, 55, 100, 145, 66, 121, 176, 78, 144, 210, 91, 169, 247, 105, 196, 287, 120, 225, 330, 136, 256, 376, 153, 289, 425, 171, 324, 477, 190, 361, 532, 210, 400, 590, 231, 441, 651

Offset

0,4

Comments

Triangular numbers alternating with squares and pentagonal numbers.

Links

Table of n, a(n) for n=0..62.

Ilya Gutkovskiy, Extended illustration of initial terms

Eric Weisstein's World of Mathematics, Triangular Number

Eric Weisstein's World of Mathematics, Square Number

Eric Weisstein's World of Mathematics, Pentagonal Number

Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1)

Formula

G.f.: (1 + x + x^2 + x^4 + 2*x^5)/(1 - x^3)^3.

a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9).

a(3k) = A000217(k+1), a(3k+1) = A000290(k+1), a(3k+2) = A000326(k+1).

Sum_{n>=0} 1/a(n) = 2 - Pi/sqrt(3) + Pi^2/6 + 3*log(3) = 5.1269715686...

a(n) = (floor(n/3) + 1)*((n+1)*floor(n/3) - 3*floor(n/3)^2 + 2)/2. - Bruno Berselli, Apr 08 2016

Example

Illustration of initial terms:
==========================================================
n:    0   1   2     3     4     5       6       7       8
----------------------------------------------------------
                                                        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
==========================================================
      1   1   1     3     4     5       6       9      12
----------------------------------------------------------

Mathematica

LinearRecurrence[{0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 1, 1, 3, 4, 5, 6, 9, 12}, 70]
Table[(Floor[n/3] + 1) ((n + 1) Floor[n/3] - 3 Floor[n/3]^2 + 2)/2, {n, 0, 70}] (* Bruno Berselli, Apr 08 2016 *)
CoefficientList[Series[(1+x+x^2+x^4+2x^5)/(1-x^3)^3, {x, 0, 70}], x] (* Harvey P. Dale, Dec 31 2023 *)

Prog

(PARI) x='x+O('x^99); Vec((1+x+x^2+x^4+2*x^5)/(1-x^3)^3) \\ Altug Alkan, Apr 07 2016

Crossrefs

Cf. A000217, A000290, A000326, A123596, A124093, A271391.

Sequence in context: A047427 A228235 A228895 * A218929 A088875 A022884

Adjacent sequences: A267319 A267320 A267321 * A267323 A267324 A267325

Keyword

nonn,easy,changed

Author

Ilya Gutkovskiy, Apr 07 2016

Status

approved