A095343 Length of n-th string generated by a Kolakoski(7,1) rule starting with a(1)=1.

1, 1, 7, 7, 31, 49, 145, 289, 727, 1591, 3775, 8545, 19873, 45505, 105127, 241639, 557023, 1281937, 2953009, 6798817, 15657847, 36054295, 83027839, 191190721, 440274241, 1013846401, 2334669127, 5376208327, 12380215711, 28508840689

Offset: 1

Author

Benoit Cloitre, Jun 03 2004

Keywords

• nonn

Comments

Each string is derived from the previous string using the Kolakoski(7,1) rule and the additional condition: "string begins with 1 if previous string ends with 5 and vice versa". The strings are 1 -> 7 -> 1111111 -> 7171717 -> 11111117111111171111111711111117 -> ... and each one contains 1,1,7,7,31,... elements.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0, 4, 3).

Crossrefs

Cf. A000002, A066983, A095342, A095344.

Programs

GAP

a:=[1,1];; for n in [3..35] do a[n]:=a[n-1]-3*a[n-2]-3*(-1)^n; od; a; # G. C. Greubel, Dec 26 2019

Magma

I:=[1,1]; [n le 2 select I[n] else Self(n-1) + 3*Self(n-2) - 3*(-1)^n: n in [1..35]]; // G. C. Greubel, Dec 26 2019

Maple

seq(coeff(series(x*(1+x+3*x^2)/((1+x)*(1-x-3*x^2)), x, n+1), x, n), n = 0..35); # G. C. Greubel, Dec 26 2019

Mathematica

Table[ 3*(-1)^n + 2*Sqrt[3]^n*(Sqrt[3]*Fibonacci[n, 1/Sqrt[3]] - Fibonacci[n+1, 1/Sqrt[3]]), {n,35}]//FullSimplify (* G. C. Greubel, Dec 26 2019 *)

PARI

vector(35, n, round(3*(-1)^n + 2*(sqrt(3)/I)^n*(sqrt(3)*I* polchebyshev(n-1, 2, I/(2*sqrt(3))) - polchebyshev(n, 2, I/(2*sqrt(3)))) )) \\ G. C. Greubel, Dec 26 2019

Sage

def A095343_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x+3*x^2)/((1+x)*(1-x-3*x^2)) ).list()
a=A095343_list(35); a[1:] # G. C. Greubel, Dec 26 2019

Formula

a(1) = a(2) = 1, a(n) = a(n-1) + 3*a(n-2) - 3*(-1)^n.

G.f.: x*(1+x+3*x^2)/((1+x)*(1-x-3*x^2)). - Colin Barker, Jul 02 2012

a(n) = 3*(-1)^n + 2*(sqrt(3)/i)^n*(sqrt(3)*i*ChebyshevU(n, i/(2*sqrt(3))) - ChebyshevU(n-1, i/(2*sqrt(3)))). - G. C. Greubel, Dec 26 2019