A191903 Number of compositions of odd natural numbers into 4 parts <= n.

0, 8, 40, 128, 312, 648, 1200, 2048, 3280, 5000, 7320, 10368, 14280, 19208, 25312, 32768, 41760, 52488, 65160, 80000, 97240, 117128, 139920, 165888, 195312, 228488, 265720, 307328, 353640, 405000, 461760, 524288, 592960, 668168, 750312

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Adi Dani, Restricted compositions of natural numbers.

Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Formula

a(n) = ((n + 1)^4 - (1 + (-1)^n)/2)/2.

From R. J. Mathar, Jun 22 2011: (Start)

G.f.: 8*x*(1+x+x^2) / ( (1+x)*(1-x)^5 ).

a(n) = 8*A011863(n+1). (End)

a(n) = floor((n+1)^4/2). - Wesley Ivan Hurt, Jun 14 2013

Sum_{n>=1} 1/a(n) = 3/4 + Pi^4/720 - tanh(Pi/2)*Pi/4. - Amiram Eldar, Aug 13 2022

Example

a(1) = 8 compositions of odd numbers into 4 parts < 1.
1:(0,0,0,1),(0,0,1,1),(0,1,0,0),(1,0,0,0)
3:(0,1,1,1),(1,0,1,1),(1,1,0,1),(1,1,1,0)

Mathematica

Table[Floor[1/2*((n + 1)^4 - (1 + (-1)^n)/2)], {n, 0, 30}]

Prog

(Magma) [((n + 1)^4 - (1 + (-1)^n)/2)/2: n in [0..50]]; // Vincenzo Librandi, Jul 04 2011

Crossrefs

Cf. A011863, A171714.

Sequence in context: A105374 A162668 A227733 * A028596 A264602 A125198

Adjacent sequences: A191900 A191901 A191902 * A191904 A191905 A191906

Keyword

nonn,easy

Author

Adi Dani, Jun 19 2011

Status

approved