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%I #45 May 09 2020 05:14:00
%S 1,6,42,294,2058,14406,100842,705894,4941258,34588806,242121642,
%T 1694851494,11863960458,83047723206,581334062442,4069338437094,
%U 28485369059658,199397583417606,1395783083923242,9770481587462694
%N First differences of 7^n (A000420).
%C Partial sum of A055270.
%C Conjecture in "Introduction à la théorie des nombres" by J. M. Deconinck and Armel Mercier: this is the period length of the fraction 1/7^n. For example 1/7^2=0.0204081632653061224489795918367346938775510204....with a period of 42 digits =6*7=a(2). The period of 1/7^3 has exactly 294=a(3) digits. - _Benoit Cloitre_, Feb 02 2002
%C Also phi(7^n), where phi is Euler's totient function. - _Alonso del Arte_, May 08 2006
%C For n>=1, a(n) is equal to the number of functions f:{1,2...,n}->{1,2,3,4,5,6,7} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5,6,7} 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 6 types of each part. - _Milan Janjic_, Aug 13 2010
%C Apart from the first term, number of monic squarefree polynomials over F_7 of degree n. - _Charles R Greathouse IV_, Feb 07 2012
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%D Jean-Marie De Koninck and Armel Mercier, Introduction à la théorie des nombres, Collection Universitaire de Mathématiques, Modulo, 1994.
%H G. C. Greubel, <a href="/A055272/b055272.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 (7).
%F G.f.: (1-x)/(1-7*x).
%F G.f.: 1/( 1 - 6*Sum(k>=1, x^k) ).
%F a(n) = 6*7^(n-1), a(0)=1.
%F E.g.f.: (1 + 6*exp(7*x))/7. - _G. C. Greubel_, Mar 16 2020
%p 1, seq(6*7^(n-1), n=1..20); # _G. C. Greubel_, Mar 16 2020
%t Table[EulerPhi[7^n], {n, 0, 19}] (* _Alonso del Arte_, May 08 2006 *)
%o (PARI) a(n)=round(7^n*6/7) \\ _Charles R Greathouse IV_, Feb 07 2012
%o (Sage) [1]+[6*7^(n-1) for n in (1..20)] # _G. C. Greubel_, Mar 16 2020
%Y Cf. A000420, A055270.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, May 28 2000
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