A254658 Numbers of words on alphabet {0,1,...,7} with no subwords ii, where i is from {0,1,2,3}.

1, 8, 60, 452, 3404, 25636, 193068, 1454020, 10950412, 82468964, 621084396, 4677466628, 35226603980, 265296094372, 1997979076524, 15047037913156, 113321181698188, 853436423539940, 6427339691572332, 48405123535166084, 364545223512451916, 2745437058727827748

Offset: 0

Author

Milan Janjic, Feb 04 2015

Keywords

• nonn
• easy

Comments

a(n) equals the number of octonary sequences of length n such that no two consecutive terms differ by 6. - David Nacin, May 31 2017

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,4).

Crossrefs

Cf. A015561, A055099, A126473, A126501, A126528, A254598, A254602.

Programs

Magma

[n le 1 select 8^n else 7*Self(n)+4*Self(n-1): n in [0..20]];

Mathematica

RecurrenceTable[{a[0] == 1, a[1] == 8, a[n] == 7 a[n - 1] + 4 a[n - 2]}, a[n], {n, 0, 20}]
LinearRecurrence[{7,4},{1,8},30] (* Harvey P. Dale, Jan 21 2023 *)

PARI

Vec((1 + x) / (1 - 7*x -4*x^2) + O(x^30)) \\ Colin Barker, Jan 21 2017

Formula

G.f.: (1 + x)/(1 - 7*x -4*x^2).

a(n) = 7*a(n-1) + 4*a(n-2) with n>1, a(0) = 1, a(1) = 8.

a(n) = (2^(-1-n)*((7-sqrt(65))^n*(-9+sqrt(65)) + (7+sqrt(65))^n*(9+sqrt(65)))) / sqrt(65). - Colin Barker, Jan 21 2017