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A211523 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w+2x=5y.

%I A211523 #18 Sep 08 2022 08:46:02
%S A211523 0,0,1,2,4,5,7,10,13,17,20,24,29,34,40,45,51,58,65,73,80,88,97,106,
%T A211523 116,125,135,146,157,169,180,192,205,218,232,245,259,274,289,305,320,
%U A211523 336,353,370,388,405,423,442,461,481,500,520,541,562,584,605,627
%N A211523 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w+2x=5y.
%C A211523 For a guide to related sequences, see A211422.
%C A211523 Also, number of ordered pairs (w,x) with both terms in {1,...,n} and w+2x divisible by 5. - _Pontus von Brömssen_, Jan 17 2020
%H A211523 Colin Barker, <a href="/A211523/b211523.txt">Table of n, a(n) for n = 0..1000</a>
%H A211523 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,1,-2,1).
%F A211523 a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).
%F A211523 G.f.: x^2*(1 + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)). - _Colin Barker_, Dec 02 2017
%t A211523 t[n_] := t[n] = Flatten[Table[w + 2 x - 5 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
%t A211523 c[n_] := Count[t[n], 0]
%t A211523 t = Table[c[n], {n, 0, 70}]   (* A211523 *)
%t A211523 FindLinearRecurrence[t]
%t A211523 LinearRecurrence[{2,-1,0,0,1,-2,1},{0,0,1,2,4,5,7},57] (* _Ray Chandler_, Aug 02 2015 *)
%o A211523 (PARI) concat(vector(2), Vec(x^2*(1 + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^40))) \\ _Colin Barker_, Dec 02 2017
%o A211523 (Magma) a:=[]; for n in [0..57] do m:=0; for i,j in [1..n] do if (i+2*j) mod 5 eq 0  then m:=m+1; end if; end for; Append(~a,m); end for; a; // _Marius A. Burtea_, Jan 17 2020
%o A211523 (Magma) R<x>:=PowerSeriesRing(Integers(), 57);[0,0] cat Coefficients(R!( x^2*(1 + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4) ))); // _Marius A. Burtea_, Jan 17 2020
%Y A211523 Cf. A211422.
%K A211523 nonn,easy
%O A211523 0,4
%A A211523 _Clark Kimberling_, Apr 14 2012