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A211519 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=2x-3y.

%I A211519 #42 Sep 08 2022 08:46:02
%S A211519 0,1,2,3,6,8,11,15,19,23,29,34,40,47,54,61,70,78,87,97,107,117,129,
%T A211519 140,152,165,178,191,206,220,235,251,267,283,301,318,336,355,374,393,
%U A211519 414,434,455,477,499,521,545,568,592,617,642,667,694,720,747,775
%N A211519 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=2x-3y.
%C A211519 For a guide to related sequences, see A211422.
%H A211519 Colin Barker, <a href="/A211519/b211519.txt">Table of n, a(n) for n = 1..1001</a>
%H A211519 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A211519 a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).
%F A211519 G.f.: x^2*(1 + x + x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)). - _Colin Barker_, Dec 02 2017
%F A211519 a(n) = floor(((n-1)^2)/4) + floor((n-2)/3) + 1. - _Ridouane Oudra_, Jun 12 2020
%F A211519 a(n) = A001399(n-2)+A001399(n-3)+A001399(n-5). - _R. J. Mathar_, Jun 23 2021~
%e A211519 For n = 4, 1 = 2*2-3*1, 2 = 2*4-3*2 and 3 = 2*3-3*1, so (1,2,1), (2,4,2) and (3,3,1) are solutions and a(4) = 3. - _Bernard Schott_, Jan 27 2020
%t A211519 t[n_] := t[n] = Flatten[Table[w - 2 x + 3 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
%t A211519 c[n_] := Count[t[n], 0]
%t A211519 t = Table[c[n], {n, 1, 80}]  (* A211519 *)
%t A211519 FindLinearRecurrence[t]
%t A211519 LinearRecurrence[{1,1,0,-1,-1,1},{0,1,2,3,6,8},56] (* _Ray Chandler_, Aug 02 2015 *)
%o A211519 (PARI) concat(0, Vec(x*(1 + x + x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)) + O(x^60))) \\ _Colin Barker_, Dec 02 2017
%o A211519 (PARI) a(n)=(n-1)^2\4 + (n+1)\3 \\ _Charles R Greathouse IV_, Jun 12 2020
%o A211519 (Magma) [ #[<w, x, y>:w, x, y in [1..n]|w eq 2*x-3*y]: n in [1..56]]; // _Marius A. Burtea_, Jan 26 2020
%o A211519 (Magma) R<x>:=PowerSeriesRing(Integers(), 56); [0] cat Coefficients(R!(x^2*(1+x+x^3) / ((1-x)^3*(1+x)*(1+x+x^2)))); // _Marius A. Burtea_, Jan 26 2020
%Y A211519 Cf. A211422.
%K A211519 nonn,easy
%O A211519 1,3
%A A211519 _Clark Kimberling_, Apr 14 2012
%E A211519 Name and offset corrected by _Pontus von Brömssen_, Jan 26 2020