A055274 First differences of 8^n (A001018).

1, 7, 56, 448, 3584, 28672, 229376, 1835008, 14680064, 117440512, 939524096, 7516192768, 60129542144, 481036337152, 3848290697216, 30786325577728, 246290604621824, 1970324836974592, 15762598695796736, 126100789566373888, 1008806316530991104

Offset

0,2

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} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5,6,7,8} 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 7 types of each natural number. - Milan Janjic, Aug 13 2010

For n>0, a(n) is not the sum of two nonnegative cubes. - Bruno Berselli, Mar 20 2012

References

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

F. Conti, R. Dvornicich, T. Franzoni and S. Mortola, Il Fibonacci N. 0 (included in Il Fibonacci, Unione Matematica Italiana, 2011), 1990, Problem 0.12.4 (see Berselli's comment).

Links

Vincenzo Librandi, 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 (8).

Formula

G.f.: (1-x)/(1-8*x).

G.f.: 1/( 1 - 7*Sum_{k>=1} x^k ).

a(n) = 7*8^(n-1); a(0)=1.

a(n) = 8*a(n-1) + (-1)^n * C(1, 1-n).

a(n) = 7*Sum_{k=0..n-1} a(k), for n>=1. - Adi Dani, Jun 24 2011

E.g.f.: (1 + 7*exp(8*x))/8. - G. C. Greubel, Mar 16 2020

Maple

1, seq(7*8^(n-1), n=1..20); # G. C. Greubel, Mar 16 2020

Mathematica

q = 8; Join[{a = 1}, Table[If[n == 0, a = q*a - 1, a = q*a], {n, 0, 25}]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)

Prog

(PARI) my(x='x+O('x^66)); Vec((1-x)/(1-8*x)) /* Joerg Arndt, Jun 25 2011 */
(Magma) [1] cat [7*8^(n-1): n in [1..20]]; // G. C. Greubel, Mar 16 2020
(Sage) [1]+[7*8^(n-1) for n in (1..20)] # G. C. Greubel, Mar 16 2020

Crossrefs

Cf. A001018.

Sequence in context: A242159 A057090 A156362 * A152776 A155197 A147839

Adjacent sequences: A055271 A055272 A055273 * A055275 A055276 A055277

Keyword

nonn,easy

Author

Barry E. Williams, May 28 2000

Extensions

More terms from James A. Sellers, May 29 2000

Status

approved