A212759 Number of (w,x,y,z) with all terms in {0,...,n} and w, x, and y even.

1, 2, 24, 32, 135, 162, 448, 512, 1125, 1250, 2376, 2592, 4459, 4802, 7680, 8192, 12393, 13122, 19000, 20000, 27951, 29282, 39744, 41472, 54925, 57122, 74088, 76832, 97875, 101250, 126976, 131072, 162129, 167042, 204120, 209952

Offset: 0

Author

Clark Kimberling, May 29 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}},
Module[{s = 0}, (Do[If[(Mod[w, 2] == 0) && (Mod[x, 2] == 0) && (Mod[y, 2] == 0),
s++], {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
Map[t[#] &, Range[0, 50]]  (* A212759 *)
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {1, 2, 24, 32, 135, 162, 448, 512, 1125}, 45]

Formula

a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6) -4*a(n-7) -a(n-8) +a(n-9).

G.f.: (1+x+18*x^2+21*x^4+x^5+2*x^6+4*x^3 ) / ( (1+x)^4*(1-x)^5 ).

a(n) = (n+1)*(2*n^3+9*n^2+15*n+9+(3*n^2+9*n+7)*(-1)^n)/16. - Luce ETIENNE, Sep 23 2015