A211547 - OEIS

%I #32 Oct 26 2024 22:57:09

%S 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,

%T 81,81,81,100,100,100,121,121,121,144,144,144,169,169,169,196,196,196,

%U 225,225,225,256,256,256,289,289,289,324,324,324,361,361,361

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

%C Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w=3x+3y.

%C For a guide to related sequences, see A211422.

%H Colin Barker, <a href="/A211547/b211547.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).

%F a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7).

%F G.f.: x^3*(1 + x)*(1 - x + x^2) / ((1 - x)^3*(1 + x + x^2)^2). - _Colin Barker_, Dec 02 2017

%F a(n) = A075561(n-2) for n > 2. - _Georg Fischer_, Oct 23 2018

%F 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

%t t[n_] := t[n] = Flatten[Table[-2 w + 3 x + 3 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]

%t c[n_] := Count[t[n], 0]

%t t = Table[c[n], {n, 0, 60}](*A211547, squares thrice*)

%t FindLinearRecurrence[t]

%t LinearRecurrence[{1,0,2,-2,0,-1,1},{0,0,0,1,1,1,4},60] (* _Ray Chandler_, Aug 02 2015 *)

%o (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

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

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Apr 15 2012

%E 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, ... .