A257287 a(n) = 6*7^n - 5*6^n.

1, 12, 114, 978, 7926, 61962, 472614, 3541578, 26190726, 191733162, 1392520614, 10049975178, 72163811526, 516030592362, 3677517616614, 26134444136778, 185292033880326, 1311149786699562, 9262681804120614, 65346572412186378

Offset: 0

Author

M. F. Hasler, May 03 2015

Keywords

• nonn
• easy

Comments

First differences of 7^n - 6^n = A016169.

a(n-1) is the number of numbers with n digits having the largest digit equal to 6. Note that this is independent of the base b > 6.

Equivalently, number of n-letter words over a 7-letter alphabet {a,b,c,d,e,f,g}, which must not start with the first letter of the alphabet, and in which the last letter of the alphabet must appear.

Links

• Index entries for linear recurrences with constant coefficients, signature (13,-42).

Crossrefs

Cf. A016169.

See also A000225, A027649, A255463, A257285, A257286, A257287, A257288, A257289, and A088924.

Programs

Magma

[6*7^n-5*6^n: n in [0..30]]; // Vincenzo Librandi, May 04 2015

Mathematica

Table[6 7^n - 5 6^n, {n, 0, 30}] (* Vincenzo Librandi, May 04 2015 *)
LinearRecurrence[{13,-42},{1,12},20] (* Harvey P. Dale, Dec 10 2023 *)

PARI

a(n)=6*7^n-5*6^n

Formula

From Vincenzo Librandi, May 04 2015: (Start)

G.f.: (1-x)/((1-6*x)*(1-7*x)).

a(n) = 13*a(n-1) - 42*a(n-2). (End)

E.g.f.: exp(6*x)*(6*exp(x) - 5). - Stefano Spezia, Nov 15 2023