This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A211433 #19 Dec 27 2024 18:08:07
%S A211433 1,1,7,11,23,27,45,53,77,85,115,127,163,175,217,233,281,297,351,371,
%T A211433 431,451,517,541,613,637,715,743,827,855,945,977,1073,1105,1207,1243,
%U A211433 1351,1387,1501,1541,1661,1701,1827,1871,2003,2047,2185,2233
%N A211433 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+2x+4y=0.
%C A211433 For a guide to related sequences, see A211422.
%C A211433 Also, a(n) is the number of ordered pairs (w,x) with both terms in {-n,...,0,...,n} and w+2x divisible by 4. If (w,x) is such a pair it is easy to see that (-w,x), (-w,-x), and (w,-x) also are such pairs. The number of pairs with both w and x positive is given by A211521(n). If w=0, x must be even, and if x=0, w must be divisible by 4. This means that a(n) = 4*A211521(n) + 2*floor(n/2) + 2*floor(n/4) + 1. Since the sequences A211521(n), floor(n/2), floor(n/4), 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 19 2020
%H A211433 Pontus von Brömssen, <a href="/A211433/b211433.txt">Table of n, a(n) for n = 0..1024</a>
%H A211433 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,-1,1).
%F A211433 Conjectures from _Colin Barker_, May 15 2017: (Start)
%F A211433 G.f.: (1 + 5*x^2 + 4*x^3 + 5*x^4 + x^6) / ((1 - x)^3*(1 + x)^2*(1 + x^2)).
%F A211433 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7) for n>6.
%F A211433 (End)
%t A211433 t[n_] := t[n] = Flatten[Table[w + 2 x + 4 y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
%t A211433 c[n_] := Count[t[n], 0]
%t A211433 t = Table[c[n], {n, 0, 70}] (* A211433 *)
%t A211433 (t - 1)/2 (* integers *)
%o A211433 (Magma) a:=[]; for n in [0..50] do m:=0; for i, j in [-n..n] do if (i+2*j) mod 4 eq 0 then m:=m+1; end if; end for; Append(~a, m); end for; a; // _Marius A. Burtea_, Jan 19 2020
%o A211433 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1 + 5*x^2 + 4*x^3 + 5*x^4 + x^6) / ((1 - x)^3*(1 + x)^2*(1 + x^2)))); // _Marius A. Burtea_, Jan 19 2020
%Y A211433 Cf. A211422, A211521.
%K A211433 nonn,easy
%O A211433 0,3
%A A211433 _Clark Kimberling_, Apr 10 2012