A254600 Numbers of words on alphabet {0,1,...,10} with no subwords ii, for i from {0,1}.

1, 11, 119, 1289, 13961, 151211, 1637759, 17738489, 192124721, 2080893611, 22538058599, 244108628489, 2643928812281, 28636265779211, 310158017102639, 3359306563039289, 36384487784316641, 394078636910520011, 4268246759164049879, 46229175323835178889

Offset: 0

Author

Milan Janjic, Feb 02 2015

Keywords

• nonn
• easy

Comments

a(n) equals the number of sequences over the alphabet {0,1,...,9,10} such that no two consecutive terms have distance 10. - David Nacin, Jun 02 2017

Links

• Colin Barker, Table of n, a(n) for n = 0..950
• Index entries for linear recurrences with constant coefficients, signature (10,9).

Crossrefs

Cf. A015591, A055099, A126473, A126501, A126528.

Programs

Magma

[n le 1 select 11^n else 10*Self(n)+9*Self(n-1): n in [0..20]]; // Bruno Berselli, Feb 03 2015

Mathematica

RecurrenceTable[{a[0]==1, a[1]==11, a[n]== 10a[n-1] +9a[n-2]}, a[n], {n, 0, 25}]
Table[(-3 I)^(n-1)*(ChebyshevU[n-1, 5*I/3] - 3*I*ChebyshevU[n, 5*I/3]), {n,0,25}] (* G. C. Greubel, Feb 13 2021 *)

PARI

Vec((x+1) / (1-10*x-9*x^2) + O(x^30)) \\ Colin Barker, Jan 21 2017

Sage

[(-3*i)^(n-1)*( chebyshev_U(n-1, 5*i/3) -3*i*chebyshev_U(n, 5*i/3) ) for n in (0..30)] # G. C. Greubel, Feb 13 2021

Formula

G.f.: (1+x)/(1-10*x-9*x^2).

a(n) = 10*a(n-1) + 9*a(n-2) with n>1, a(0) = 1, a(1) = 11.

a(n) = ((5-sqrt(34))^n*(-6+sqrt(34)) + (5+sqrt(34))^n*(6+sqrt(34))) / (2*sqrt(34)). - Colin Barker, Jan 21 2017

a(n) = (-3*i)^(n-1) * (ChebyshevU(n-1, 5*i/3) - 3*i*ChebyshevU(n, 5*i/3)). - G. C. Greubel, Feb 13 2021