OFFSET
0,7
COMMENTS
Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w=3x+3y.
For a guide to related sequences, see A211422.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).
FORMULA
a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7).
G.f.: x^3*(1 + x)*(1 - x + x^2) / ((1 - x)^3*(1 + x + x^2)^2). - Colin Barker, Dec 02 2017
a(n) = A075561(n-2) for n > 2. - Georg Fischer, Oct 23 2018
E.g.f.: exp(-x/2)*(exp(3*x/2)*(5 + 3*x*(x - 1)) - 5*cos(sqrt(3)*x/2) - sqrt(3)*(3 + 4*x)*sin(sqrt(3)*x/2))/27. - Stefano Spezia, Oct 17 2022
MATHEMATICA
t[n_] := t[n] = Flatten[Table[-2 w + 3 x + 3 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 60}](*A211547, squares thrice*)
FindLinearRecurrence[t]
LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {0, 0, 0, 1, 1, 1, 4}, 60] (* Ray Chandler, Aug 02 2015 *)
PROG
(PARI) concat(vector(3), Vec(x^3*(1 + x)*(1 - x + x^2) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^40))) \\\ Colin Barker, Dec 02 2017
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Clark Kimberling, Apr 15 2012
EXTENSIONS
Definition simplified by N. J. A. Sloane, Nov 17 2020. Also the old version said "squares repeated three times", which was at best ambiguous, and strictly speaking was incorrect, since "squares repeated" is 0, 0, 1, 1, 4, 4, 9, 9, ... .
STATUS
approved