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%I #33 Sep 08 2022 08:45:01
%S 1,7,56,448,3584,28672,229376,1835008,14680064,117440512,939524096,
%T 7516192768,60129542144,481036337152,3848290697216,30786325577728,
%U 246290604621824,1970324836974592,15762598695796736,126100789566373888,1008806316530991104
%N First differences of 8^n (A001018).
%C 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
%C a(n) is the number of compositions of n when there are 7 types of each natural number. - _Milan Janjic_, Aug 13 2010
%C For n>0, a(n) is not the sum of two nonnegative cubes. - _Bruno Berselli_, Mar 20 2012
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%D 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).
%H Vincenzo Librandi, <a href="/A055274/b055274.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (8).
%F G.f.: (1-x)/(1-8*x).
%F G.f.: 1/( 1 - 7*Sum_{k>=1} x^k ).
%F a(n) = 7*8^(n-1); a(0)=1.
%F a(n) = 8*a(n-1) + (-1)^n * C(1, 1-n).
%F a(n) = 7*Sum_{k=0..n-1} a(k), for n>=1. - _Adi Dani_, Jun 24 2011
%F E.g.f.: (1 + 7*exp(8*x))/8. - _G. C. Greubel_, Mar 16 2020
%p 1, seq(7*8^(n-1), n=1..20); # _G. C. Greubel_, Mar 16 2020
%t 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 *)
%o (PARI) my(x='x+O('x^66)); Vec((1-x)/(1-8*x)) /* _Joerg Arndt_, Jun 25 2011 */
%o (Magma) [1] cat [7*8^(n-1): n in [1..20]]; // _G. C. Greubel_, Mar 16 2020
%o (Sage) [1]+[7*8^(n-1) for n in (1..20)] # _G. C. Greubel_, Mar 16 2020
%Y Cf. A001018.
%K nonn,easy
%O 0,2
%A _Barry E. Williams_, May 28 2000
%E More terms from _James A. Sellers_, May 29 2000
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