A055275 First differences of 9^n (A001019).

1, 8, 72, 648, 5832, 52488, 472392, 4251528, 38263752, 344373768, 3099363912, 27894275208, 251048476872, 2259436291848, 20334926626632, 183014339639688, 1647129056757192, 14824161510814728, 133417453597332552, 1200757082375992968, 10806813741383936712, 97261323672455430408, 875351913052098873672

Offset: 0

Author

Barry E. Williams, May 28 2000

Keywords

• easy
• nonn

Comments

For n>=1, a(n) is equal to the number of functions f:{1,2...,n}->{1,2,3,4,5,6,7,8,9} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5,6,7,8,9} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007

a(n) is the number of compositions of n when there are 8 types of each natural number. - Milan Janjic, Aug 13 2010

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Index entries for linear recurrences with constant coefficients, signature (9).

Crossrefs

Cf. A001019.

Programs

Magma

[1] cat [8*9^(n-1): n in [1..25]]; // G. C. Greubel, Mar 16 2020

Maple

1, seq(8*9^(n-1), n=1..25); # G. C. Greubel, Mar 16 2020

Mathematica

q = 9; Join[{a = 1}, Table[If[n == 0, a = q*a - 1, a = q*a], {n, 0, 25}]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Join[{1},NestList[9#&,8,30]] (* Harvey P. Dale, Aug 23 2024 *)

PARI

a(n)=if(n,8*9^(n-1),1) \\ Charles R Greathouse IV, Oct 07 2015

Sage

[1]+[8*9^(n-1) for n in (1..25)] # G. C. Greubel, Mar 16 2020

Formula

G.f.: (1-x)/(1-9x).

a(n) = 8*9^(n-1); a(0)=1.

a(n) = 9a(n-1) + (-1)^n*C(1,1-n).

E.g.f.: (1 + 8*exp(9*x))/9. - G. C. Greubel, Mar 16 2020