A211533 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=3x-5y.

0, 0, 1, 1, 3, 4, 5, 8, 10, 13, 16, 19, 23, 27, 32, 36, 41, 47, 52, 59, 65, 71, 79, 86, 94, 102, 110, 119, 128, 138, 147, 157, 168, 178, 190, 201, 212, 225, 237, 250, 263, 276, 290, 304, 319, 333, 348, 364, 379, 396, 412, 428, 446, 463, 481, 499, 517, 536

Offset: 0

Author

Clark Kimberling, Apr 15 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211422.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,1,-1,0,-1,1).

Crossrefs

Cf. A211422.

Programs

Mathematica

t[n_] := t[n] = Flatten[Table[w - 3 x + 5 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211533 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,0,1,-1,1,-1,0,-1,1},{0,0,1,1,3,4,5,8,10},58] (* Ray Chandler, Aug 02 2015 *)

PARI

concat(vector(2), Vec(x^2*(1 + 2*x^2 + x^4 + x^6) / ((1 - x)^3*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)) + O(x^40))) \\ Colin Barker, Dec 02 2017

Formula

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) - a(n-6) - a(n-8) + a(n-9).

G.f.: x^2*(1 + 2*x^2 + x^4 + x^6) / ((1 - x)^3*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Dec 02 2017

a(n) ~ n^2/6. - Stefano Spezia, Apr 09 2025