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A212760
Number of (w,x,y,z) with all terms in {0,...,n}, w even, and x = y + z.
5
1, 3, 12, 20, 45, 63, 112, 144, 225, 275, 396, 468, 637, 735, 960, 1088, 1377, 1539, 1900, 2100, 2541, 2783, 3312, 3600, 4225, 4563, 5292, 5684, 6525, 6975, 7936, 8448, 9537, 10115, 11340, 11988, 13357, 14079, 15600, 16400, 18081, 18963, 20812, 21780, 23805
OFFSET
0,2
COMMENTS
A signed version is A122576.
For a guide to related sequences, see A211795.
Partial sums of the positive elements of A129194. - Omar E. Pol, Dec 28 2013
FORMULA
a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7).
G.f.: ( 1+2*x+6*x^2+2*x^3+x^4 ) / ( (1+x)^3*(1-x)^4 ).
a(n) = (n+1)*(n+2)*(2*n+3+(-1)^n)/8. - Wesley Ivan Hurt, Jul 22 2014
a(n) = A260706(A001318(n+1)). - Reinhard Zumkeller, Nov 17 2015
a(n) = Sum_{i=1..n+1} numerator(i^2/2). - Wesley Ivan Hurt, Feb 26 2017
MAPLE
A212760:=n->(n+1)*(n+2)*(2*n+3+(-1)^n)/8: seq(A212760(n), n=0..50); # Wesley Ivan Hurt, Jul 22 2014
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[(Mod[w, 2] == 0) && x == y + z, s++],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
Map[t[#] &, Range[0, 50]] (* A212760 *)
Table[(n + 1) (n + 2) (2 n + 3 + (-1)^n)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Jul 22 2014 *)
CoefficientList[Series[(1 + 2 x + 6 x^2 + 2 x^3 + x^4)/((1 + x)^3 (1 - x)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 23 2014 *)
PROG
(Magma) [(n+1)*(n+2)*(2*n+3+(-1)^n)/8 : n in [0..50]]; // Wesley Ivan Hurt, Jul 22 2014
(Haskell)
a212760 = a260706 . fromInteger . a001318 . (+ 1)
-- Reinhard Zumkeller, Nov 17 2015
CROSSREFS
Cf. A211795.
Sequence in context: A374224 A063102 A122576 * A143268 A374072 A193558
KEYWORD
nonn,easy,changed
AUTHOR
Clark Kimberling, May 29 2012
STATUS
approved