A213041 Number of triples (w,x,y) with all terms in {0..n} and 2*|w-x| = max(w,x,y) - min(w,x,y).

1, 2, 7, 12, 21, 30, 43, 56, 73, 90, 111, 132, 157, 182, 211, 240, 273, 306, 343, 380, 421, 462, 507, 552, 601, 650, 703, 756, 813, 870, 931, 992, 1057, 1122, 1191, 1260, 1333, 1406, 1483, 1560, 1641, 1722, 1807, 1892, 1981, 2070, 2163, 2256

Offset: 0

Author

Clark Kimberling, Jun 10 2012

Keywords

• nonn
• easy

Comments

See A212959 for a guide to related sequences.

For n > 3, a(n-2) is the number of distinct values of the magic constant in a perimeter-magic (n-1)-gon of order n (see A342819). - Stefano Spezia, Mar 23 2021

Links

• Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10-13).
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A002620, A004526, A058331, A212959, A168277 (first differences), A342819.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Max[w, x, y] - Min[w, x, y] == 2 Abs[w - x],
s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A213041 *)

PARI

Vec((1+3*x^2)/((1-x)^3*(1+x)) + O(x^99)) \\ Altug Alkan, May 06 2016

Formula

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3.

G.f.: (1 + 3*x^2)/((1 - x)^3 * (1 + x)).

a(n) = (n+1)^2 - 2*A004526(n-1) - 2. - Wesley Ivan Hurt, Jul 15 2013

a(n) = A002620(n+2) + 3*A002620(n). - R. J. Mathar, Jul 15 2013

a(n)+a(n+1) = A058331(n+1). - R. J. Mathar, Jul 15 2013

a(n) = n*(n+1) + (1+(-1)^n)/2. - Wesley Ivan Hurt, May 06 2016

E.g.f.: x*(x + 2)*exp(x) + cosh(x). - Ilya Gutkovskiy, May 06 2016

a(n) = A000384(n+1) - A137932(n+2). - Federico Provvedi, Aug 17 2023