A254663 Numbers of n-length words on alphabet {0,1,...,7} with no subwords ii, where i is from {0,1,...,5}.

1, 8, 58, 422, 3070, 22334, 162478, 1182014, 8599054, 62557406, 455099950, 3310814462, 24085901134, 175222936862, 1274732360302, 9273572395838, 67464471491470, 490798445231966, 3570518059606702, 25975223307710846, 188967599273189326, 1374723641527746974

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 5. - 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 (7,2).

Crossrefs

Cf. A015555, A055099, A126473, A126501, A126528, A254598, A254602.

Programs

Magma

[n le 1 select 8^n else 7*Self(n)+2*Self(n-1): n in [0..20]];

Mathematica

RecurrenceTable[{a[0] == 1, a[1] == 8, a[n] == 7 a[n - 1] + 2 a[n - 2]}, a[n], {n, 0, 20}]
LinearRecurrence[{7,2},{1,8},30] (* Harvey P. Dale, Nov 28 2023 *)

PARI

Vec((1 + x)/(1 - 7*x - 2*x^2) + O(x^30)) \\ Colin Barker, Jan 22 2017

Formula

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

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

a(n) = 2^(-1-n)*((7-r)^n*(-9+r) + (7+r)^n*(9+r)) / r, where r=sqrt(57). - Colin Barker, Jan 22 2017