A050190 T(n,5), array T as in A050186; a count of aperiodic binary words.

0, 6, 21, 56, 126, 250, 462, 792, 1287, 2002, 3000, 4368, 6188, 8568, 11628, 15500, 20349, 26334, 33649, 42504, 53125, 65780, 80730, 98280, 118755, 142500, 169911, 201376, 237336, 278256, 324625, 376992, 435897, 501942

Offset: 5

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 5..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,2,-8,12,-8,2,-1,4,-6,4, -1).

Crossrefs

Cf. A000389, A050186.

Programs

Magma

[n*floor(Binomial(n-1, 4)/5): n in [5..40]]; // G. C. Greubel, Nov 25 2017

Mathematica

Table[n*Floor[Binomial[n - 1, 4]/5], {n, 5, 50}] (* G. C. Greubel, Nov 25 2017 *)
Drop[CoefficientList[Series[(x^5*(3 + x^2 + x^3)*(2 - x + 2*x^2 + x^3 + x^4))/((1 - x)^4*(1 - x^5)^2), {x, 0, 50}], x], 4] (* G. C. Greubel, Nov 27 2017 *)

PARI

for(n=5,40, print1(n*floor(binomial(n-1, 4)/5), ", ")) \\ G. C. Greubel, Nov 25 2017

PARI

x='x+O('x^30); concat([0], Vec((x^5*(3 + x^2 + x^3)*(2 - x + 2*x^2 + x^3 + x^4))/((1 - x)^4*(1 - x^5)^2))) \\ G. C. Greubel, Nov 27 2017

Formula

a(n) = n * A051170(n).

From Ralf Stephan, Aug 18 2004: (Start)

G.f.: (x^5*(3 + x^2 + x^3)*(2 - x + 2*x^2 + x^3 + x^4))/((1 - x)^4*(1 - x^5)^2). (corrected by G. C. Greubel, Nov 27 2017)

a(n) = A000389(n) - [5 divides n]*n/5.

a(n) = n*floor(C(n-1, 4)/5). (End)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 2*a(n-5) - 8*a(n-6) + 12*a(n-7) - 8*a(n-8) + 2*a(n-9) - a(n-10) + 4*a(n-11) - 6*a(n-12) + 4*a(n-13) - a(n-14). - R. J. Mathar, May 20 2013

Extensions

More terms from Ralf Stephan, Aug 18 2004