A257285 a(n) = 4*5^n - 3*4^n.

1, 8, 52, 308, 1732, 9428, 50212, 263348, 1365892, 7026068, 35916772, 182729588, 926230852, 4681485908, 23608756132, 118849087028, 597466660612, 3000218204948, 15052630632292, 75469311591668, 378171191679172, 1894154493279188, 9483966605929252

Offset: 0

Author

M. F. Hasler, May 03 2015

Keywords

• nonn
• easy

Comments

First differences of 5^n - 4^n = A005060.

a(n-1) is the number of numbers with n digits having the largest digit equal to 4. Note that this is independent of the base b>4. Equivalently, number of n-letter words over a 5-letter alphabet {a,b,c,d,e}, 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 (9,-20).

Crossrefs

Cf. A005060. See also A000225, A027649, A255463, A257286 - A257289 and A088924.

Programs

Magma

[4*5^n-3*4^n: n in [0..30]]; // Vincenzo Librandi, May 04 2015

Mathematica

Table[4 5^n - 3 4^n, {n, 0, 30}] (* Vincenzo Librandi, May 04 2015 *)

PARI

a(n)=4*5^n-3*4^n

Formula

From Vincenzo Librandi, May 04 2015: (Start)

G.f.: (1-x)/((1-4*x)*(1-5*x)).

a(n) = 9*a(n-1) - 20*a(n-2). - (End)

E.g.f.: exp(4*x)*(4*exp(x) - 3). - Stefano Spezia, Nov 15 2023