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A211434 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+2x+5y=0.

%I A211434 #29 Dec 27 2024 18:08:02
%S A211434 1,1,5,9,17,25,33,45,57,73,89,105,125,145,169,193,217,245,273,305,337,
%T A211434 369,405,441,481,521,561,605,649,697,745,793,845,897,953,1009,1065,
%U A211434 1125,1185,1249,1313,1377,1445,1513,1585,1657,1729,1805,1881
%N A211434 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+2x+5y=0.
%C A211434 For a guide to related sequences, see A211422.
%C A211434 Also, a(n) is the number of ordered pairs (w,x) with both terms in {-n,...,0,...,n} and w+2x divisible by 5. If (w,x) is such a pair it is easy to see that (-x,w), (-w,-x), and (x,-w) also are such pairs. If both w and x are nonzero these four pairs lie one in each quadrant. If one of w or x is zero, the other must be a multiple of 5. This means that a(n) equals 4*A211523(n) (the nonzero pairs) plus 4*floor(n/5) + 1 (pairs with w or x equal to zero). Since the sequences A211523(n), floor(n/5), and the constant sequence all satisfy the recurrence conjectured in the formula section, a(n) must also satisfy the recurrence, so this proves the conjecture. - _Pontus von Brömssen_, Jan 17 2020
%H A211434 Pontus von Brömssen, <a href="/A211434/b211434.txt">Table of n, a(n) for n = 0..1024</a>
%H A211434 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,1,-2,1).
%F A211434 Conjectures from _Colin Barker_, May 15 2017: (Start)
%F A211434 G.f.: (1 - x + 4*x^2 + 4*x^4 - x^5 + x^6) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
%F A211434 a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) for n>6.
%F A211434 (End)
%F A211434 a(n) = (4*n*(n+1) + c(n))/5, where c(n) is 5 if n is 0 or 4 (mod 5), -3 if n is 1 or 3 (mod 5), and 1 if n is 2 (mod 5). - _Pontus von Brömssen_, Jan 17 2020
%t A211434 t[n_] := t[n] = Flatten[Table[w + 2 x + 5 y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
%t A211434 c[n_] := Count[t[n], 0]
%t A211434 t = Table[c[n], {n, 0, 70}]  (* A211434 *)
%t A211434 (t - 1)/4                    (* A011858 *)
%o A211434 (Magma) a:=[]; for n in [0..50] do m:=0; for i, j in [-n..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 19 2020
%o A211434 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1 - x + 4*x^2 + 4*x^4 - x^5 + x^6) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)))); // _Marius A. Burtea_, Jan 19 2020
%Y A211434 Cf. A211422, A211523.
%K A211434 nonn,easy
%O A211434 0,3
%A A211434 _Clark Kimberling_, Apr 11 2012