A055272 First differences of 7^n (A000420).

1, 6, 42, 294, 2058, 14406, 100842, 705894, 4941258, 34588806, 242121642, 1694851494, 11863960458, 83047723206, 581334062442, 4069338437094, 28485369059658, 199397583417606, 1395783083923242, 9770481587462694

Offset: 0

Author

Barry E. Williams, May 28 2000

Keywords

• easy
• nonn

Comments

Partial sum of A055270.

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

Also phi(7^n), where phi is Euler's totient function. - Alonso del Arte, May 08 2006

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

a(n) is the number of compositions of n when there are 6 types of each part. - Milan Janjic, Aug 13 2010

Apart from the first term, number of monic squarefree polynomials over F_7 of degree n. - Charles R Greathouse IV, Feb 07 2012

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
• Jean-Marie De Koninck and Armel Mercier, Introduction à la théorie des nombres, Collection Universitaire de Mathématiques, Modulo, 1994.

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 (7).

Crossrefs

Cf. A000420, A055270.

Programs

Maple

1, seq(6*7^(n-1), n=1..20); # G. C. Greubel, Mar 16 2020

Mathematica

Table[EulerPhi[7^n], {n, 0, 19}] (* Alonso del Arte, May 08 2006 *)

PARI

a(n)=round(7^n*6/7) \\ Charles R Greathouse IV, Feb 07 2012

Sage

[1]+[6*7^(n-1) for n in (1..20)] # G. C. Greubel, Mar 16 2020

Formula

G.f.: (1-x)/(1-7*x).

G.f.: 1/( 1 - 6*Sum(k>=1, x^k) ).

a(n) = 6*7^(n-1), a(0)=1.

E.g.f.: (1 + 6*exp(7*x))/7. - G. C. Greubel, Mar 16 2020