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

0, 1, 4, 13, 28, 55, 92, 147, 216, 309, 420, 561, 724, 923, 1148, 1415, 1712, 2057, 2436, 2869, 3340, 3871, 4444, 5083, 5768, 6525, 7332, 8217, 9156, 10179, 11260, 12431, 13664, 14993, 16388, 17885, 19452, 21127, 22876, 24739, 26680, 28741

Offset: 0

Author

Clark Kimberling, May 22 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Crossrefs

Cf. A211795.

Programs

Magma

R:=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

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 Abs[w - x] == Abs[x - y] - Abs[y - z],
  s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 45]]   (* A212578 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 1, 4, 13, 28, 55, 92}, 45] (* signature corrected by Georg Fischer, Apr 10 2019 *)

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

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

Formula

a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6).

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).