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A211547
The squares n^2, n >= 0, each one written three times.
7
0, 0, 0, 1, 1, 1, 4, 4, 4, 9, 9, 9, 16, 16, 16, 25, 25, 25, 36, 36, 36, 49, 49, 49, 64, 64, 64, 81, 81, 81, 100, 100, 100, 121, 121, 121, 144, 144, 144, 169, 169, 169, 196, 196, 196, 225, 225, 225, 256, 256, 256, 289, 289, 289, 324, 324, 324, 361, 361, 361
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.
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
Cf. A075561, A211422, A211435 (triply repeated triangular numbers).
Sequence in context: A288797 A200600 A048761 * A075561 A376073 A256796
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