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A212578 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x| = 2*|x-y| - |y-z|.

%I A212578 #17 Sep 08 2022 08:46:02
%S A212578 0,1,4,13,28,55,92,147,216,309,420,561,724,923,1148,1415,1712,2057,
%T A212578 2436,2869,3340,3871,4444,5083,5768,6525,7332,8217,9156,10179,11260,
%U A212578 12431,13664,14993,16388,17885,19452,21127,22876,24739,26680,28741
%N A212578 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x| = 2*|x-y| - |y-z|.
%C A212578 For a guide to related sequences, see A211795.
%H A212578 G. C. Greubel, <a href="/A212578/b212578.txt">Table of n, a(n) for n = 0..1000</a>
%H A212578 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1, -4, 1, 2, -1).
%F A212578 a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6).
%F A212578 G.f.: x*(1 + 2*x + 4*x^2 + 2*x^3 + x^4)/(1 - 2*x - x^2 + 4*x^3 - x^4 - 2*x^5 + x^6).
%t A212578 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212578 (Do[If[2 Abs[w - x] == Abs[x - y] - Abs[y - z],
%t A212578   s = s + 1],
%t A212578 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212578 Map[t[#] &, Range[0, 45]]   (* A212578 *)
%t A212578 LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 1, 4, 13, 28, 55, 92}, 45] (* signature corrected by _Georg Fischer_, Apr 10 2019 *)
%o A212578 (PARI) my(x='x+O('x^45)); concat([0], Vec(x*(1+2*x+4*x^2+2*x^3+x^4)/(1 -2*x-x^2+4*x^3 -x^4-2*x^5+x^6))) \\ _G. C. Greubel_, Apr 10 2019
%o A212578 (Magma) R<x>:=PowerSeriesRing(Integers(), 45); [0] cat Coefficients(R!( x*(1+2*x +4*x^2+2*x^3+x^4)/(1-2*x-x^2+4*x^3-x^4-2*x^5+x^6) )); // _G. C. Greubel_, Apr 10 2019
%o A212578 (Sage) (x*(1+2*x+4*x^2+2*x^3+x^4)/(1-2*x-x^2+4*x^3-x^4-2*x^5+x^6) ).series(x, 45).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 10 2019
%Y A212578 Cf. A211795.
%K A212578 nonn,easy
%O A212578 0,3
%A A212578 _Clark Kimberling_, May 22 2012