A211547 The squares n^2, n >= 0, each one written three times.

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.

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

Cf. A075561, A211422, A211435 (triply repeated triangular numbers).

Sequence in context: A288797 A200600 A048761 * A075561 A256796 A117405

Adjacent sequences: A211544 A211545 A211546 * A211548 A211549 A211550

Keyword

nonn,easy

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