A211521 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w + 2x = 4y.

0, 0, 1, 2, 4, 5, 9, 11, 16, 18, 25, 28, 36, 39, 49, 53, 64, 68, 81, 86, 100, 105, 121, 127, 144, 150, 169, 176, 196, 203, 225, 233, 256, 264, 289, 298, 324, 333, 361, 371, 400, 410, 441, 452, 484, 495, 529, 541, 576, 588, 625, 638, 676, 689, 729, 743

Offset: 0

Author

Clark Kimberling, Apr 14 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211422.

Also, number of ordered pairs (w,x) with both terms in {1,...,n} and w+2x divisible by 4. - Pontus von Brömssen, Jan 19 2020

Links

• Colin Barker, Table of n, a(n) for n = 0..999
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

Crossrefs

Cf. A211422.

Programs

Magma

a:=[0]; for n in [1..55] do m:=0; for i, j in [1..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

Magma

R:=PowerSeriesRing(Integers(), 57); [0,0] cat Coefficients(R!( x^3*(1 + x + x^2 + x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)))); // Marius A. Burtea, Jan 19 2020

Mathematica

t[n_] := t[n] = Flatten[Table[w + 2 x - 4 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211521 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,1,-1,1,-1,-1,1},{0,0,1,2,4,5,9},56] (* Ray Chandler, Aug 02 2015 *)

PARI

concat(vector(2), Vec(x^2*(1 + x + x^2 + x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)) + O(x^40))) \\ Colin Barker, Dec 02 2017

Formula

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7).

a(n) = (2*n^2-n+1+(n-1)*(-1)^n+(-1)^((2*n+1-(-1)^n)/4)-(-1)^((6*n+1-(-1)^n)/4))/8. - Luce ETIENNE, Dec 31 2015

G.f.: x^2*(1 + x + x^2 + x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)). - Colin Barker, Dec 02 2017

Extensions

Offset corrected by Pontus von Brömssen, Jan 19 2020