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
Examples
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)
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).
Programs
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Magma
[((n + 1)^4 - (1 + (-1)^n)/2)/2: n in [0..50]]; // Vincenzo Librandi, Jul 04 2011
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Mathematica
Table[Floor[1/2*((n + 1)^4 - (1 + (-1)^n)/2)], {n, 0, 30}]
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