A211519 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=2x-3y.

0, 1, 2, 3, 6, 8, 11, 15, 19, 23, 29, 34, 40, 47, 54, 61, 70, 78, 87, 97, 107, 117, 129, 140, 152, 165, 178, 191, 206, 220, 235, 251, 267, 283, 301, 318, 336, 355, 374, 393, 414, 434, 455, 477, 499, 521, 545, 568, 592, 617, 642, 667, 694, 720, 747, 775

Offset: 1

Author

Clark Kimberling, Apr 14 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211422.

Examples

			For n = 4, 1 = 2*2-3*1, 2 = 2*4-3*2 and 3 = 2*3-3*1, so (1,2,1), (2,4,2) and (3,3,1) are solutions and a(4) = 3. - _Bernard Schott_, Jan 27 2020
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1001
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Crossrefs

Cf. A211422.

Programs

Magma

[ #[:w, x, y in [1..n]|w eq 2*x-3*y]: n in [1..56]]; // Marius A. Burtea, Jan 26 2020

Magma

R:=PowerSeriesRing(Integers(), 56); [0] cat Coefficients(R!(x^2*(1+x+x^3) / ((1-x)^3*(1+x)*(1+x+x^2)))); // Marius A. Burtea, Jan 26 2020

Mathematica

t[n_] := t[n] = Flatten[Table[w - 2 x + 3 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 1, 80}]  (* A211519 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,1,0,-1,-1,1},{0,1,2,3,6,8},56] (* Ray Chandler, Aug 02 2015 *)

PARI

concat(0, Vec(x*(1 + x + x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Dec 02 2017

PARI

a(n)=(n-1)^2\4 + (n+1)\3 \\ Charles R Greathouse IV, Jun 12 2020

Formula

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

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

a(n) = floor(((n-1)^2)/4) + floor((n-2)/3) + 1. - Ridouane Oudra, Jun 12 2020

a(n) = A001399(n-2)+A001399(n-3)+A001399(n-5). - R. J. Mathar, Jun 23 2021~

Extensions

Name and offset corrected by Pontus von Brömssen, Jan 26 2020