A212766 Number of (w,x,y,z) with all terms in {0,...,n}, w even and x odd.

0, 4, 18, 64, 150, 324, 588, 1024, 1620, 2500, 3630, 5184, 7098, 9604, 12600, 16384, 20808, 26244, 32490, 40000, 48510, 58564, 69828, 82944, 97500, 114244, 132678, 153664, 176610, 202500, 230640, 262144, 296208, 334084, 374850

Offset

0,2

Comments

Every term is even.

For a guide to related sequences, see A211795.

Links

Table of n, a(n) for n=0..34.

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Formula

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

G.f.: -2*x*(2+5*x+10*x^2+5*x^3+2*x^4) / ( (1+x)^3*(x-1)^5 ).

a(n) = (2n(n+2)-(-1)^n+1)(n+1)^2/8. - Bruno Berselli, Jun 11 2012

E.g.f.: (x*(15 + 24*x + 10*x^2 + x^3)*cosh(x) + (1 + 12*x + 25*x^2 + 10*x^3 + x^4)* sinh(x))/4. - Stefano Spezia, Sep 09 2025

a(n-1) = floor((n/2)^2)*n^2 for n>=1. - Peter Luschny, Jan 10 2026

a(n) = A210379(n)/2. - Alois P. Heinz, Jan 10 2026

From Amiram Eldar, Jan 16 2026: (Start)

Sum_{n>=1} 1/a(n) = Pi^4/360 - Pi^2/2 + 5.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^4/360 + Pi^2/2 - 5. (End)

Maple

a := n -> floor((n/2)^2)*n^2: seq(a(n+1), n = 0 ..34); # Peter Luschny, Jan 10 2026

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[(Mod[w, 2] == 0) && (Mod[x, 2] == 1), s++], {w, 0, n}, {x, 0,  n}, {y, 0, n}, {z, 0, n}]; s)]];
Map[t[#] &, Range[0, 40]]  (* A212766 *)
%/2 (* integers *)
LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 4, 18, 64, 150, 324, 588, 1024}, 40]

Crossrefs

Cf. A210379, A211795.

Sequence in context: A227162 A057414 A165910 * A100177 A083321 A362316

Adjacent sequences: A212763 A212764 A212765 * A212767 A212768 A212769

Keyword

nonn,easy

Author

Clark Kimberling, May 29 2012

Status

approved