A212761 Number of (w,x,y,z) with all terms in {0,...,n}, w odd, x and y even.

0, 2, 12, 32, 90, 162, 336, 512, 900, 1250, 1980, 2592, 3822, 4802, 6720, 8192, 11016, 13122, 17100, 20000, 25410, 29282, 36432, 41472, 50700, 57122, 68796, 76832, 91350, 101250, 119040, 131072, 152592, 167042, 192780, 209952, 240426

Offset: 0

Author

Clark Kimberling, May 29 2012

Keywords

• nonn
• easy

Comments

Every term is even.

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] == 1) && (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]]  (* A212761 *)
%/2 (* integers *)
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 2, 12, 32, 90, 162, 336, 512, 900}, 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.: 2*x*(1+5*x+6*x^2+9*x^3+2*x^4+x^5) / ( (1+x)^4*(1-x)^5 ).

a(n) = (n+1)*(2*n^3+7*n^2+7*n+1+(n^2+n-1)*(-1)^n)/16. - Luce ETIENNE, Oct 10 2015