A257285 - OEIS

%I #16 Nov 15 2023 19:07:10

%S 1,8,52,308,1732,9428,50212,263348,1365892,7026068,35916772,182729588,

%T 926230852,4681485908,23608756132,118849087028,597466660612,

%U 3000218204948,15052630632292,75469311591668,378171191679172,1894154493279188,9483966605929252

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

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

%C 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.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-20).

%F From _Vincenzo Librandi_, May 04 2015: (Start)

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

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

%F E.g.f.: exp(4*x)*(4*exp(x) - 3). - _Stefano Spezia_, Nov 15 2023

%t Table[4 5^n - 3 4^n, {n, 0, 30}] (* _Vincenzo Librandi_, May 04 2015 *)

%o (PARI) a(n)=4*5^n-3*4^n

%o (Magma) [4*5^n-3*4^n: n in [0..30]]; // _Vincenzo Librandi_, May 04 2015

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

%K nonn,easy

%O 0,2

%A _M. F. Hasler_, May 03 2015