A211523 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w+2x=5y.

0, 0, 1, 2, 4, 5, 7, 10, 13, 17, 20, 24, 29, 34, 40, 45, 51, 58, 65, 73, 80, 88, 97, 106, 116, 125, 135, 146, 157, 169, 180, 192, 205, 218, 232, 245, 259, 274, 289, 305, 320, 336, 353, 370, 388, 405, 423, 442, 461, 481, 500, 520, 541, 562, 584, 605, 627

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 5. - Pontus von Brömssen, Jan 17 2020

Links

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

Crossrefs

Cf. A211422.

Programs

Magma

a:=[]; for n in [0..57] do m:=0; for i,j in [1..n] do if (i+2*j) mod 5 eq 0  then m:=m+1; end if; end for; Append(~a,m); end for; a; // Marius A. Burtea, Jan 17 2020

Magma

R:=PowerSeriesRing(Integers(), 57);[0,0] cat Coefficients(R!( x^2*(1 + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4) ))); // Marius A. Burtea, Jan 17 2020

Mathematica

t[n_] := t[n] = Flatten[Table[w + 2 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}]   (* A211523 *)
FindLinearRecurrence[t]
LinearRecurrence[{2,-1,0,0,1,-2,1},{0,0,1,2,4,5,7},57] (* Ray Chandler, Aug 02 2015 *)

PARI

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

Formula

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

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