A000420 Powers of 7: a(n) = 7^n.

1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010407, 678223072849, 4747561509943, 33232930569601, 232630513987207, 1628413597910449, 11398895185373143, 79792266297612001, 558545864083284007

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Same as Pisot sequences E(1, 7), L(1, 7), P(1, 7), T(1, 7). Essentially same as Pisot sequences E(7, 49), L(7, 49), P(7, 49), T(7, 49). See A008776 for definitions of Pisot sequences.

Sum of coefficients of expansion of (1+x+x^2+x^3+x^4+x^5+x^6)^n.

a(n) is number of compositions of natural numbers into n parts < 7.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 7-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

Numbers n such that sigma(7n) = 7n + sigma(n). - Jahangeer Kholdi, Nov 23 2013

Number of ways to assign truth values to n ternary disjunctions connected by conjunctions such that the proposition is true. For example, a(2) = 49, since for the proposition '(a v b v c) & (d v e v f)' there are 49 assignments that make the proposition true. - Ori Milstein, Dec 31 2022

Equivalently, the number of length-n words over an alphabet with seven letters. - Joerg Arndt, Jan 01 2023

Examples

			a(2)=49 there are 49 compositions of natural numbers into 2 parts < 7.
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 272
• Tanya Khovanova, Recursive Sequences
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• Index entries for linear recurrences with constant coefficients, signature (7).

Crossrefs

Cf. A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A001018 (powers of 8), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48), A087752 (powers of 49).

Programs

Haskell

a000420 = (7 ^)
a000420_list = iterate (* 7) 1  -- Reinhard Zumkeller, Apr 29 2015

Magma

[7^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 27 2016

Maple

A000420:=-1/(-1+7*z); # Simon Plouffe in his 1992 dissertation. [This is actually the generating function, so convert(series(...),list) would yield the actual sequence. - M. F. Hasler, Apr 19 2015]
A000420 := n -> 7^n; # M. F. Hasler, Apr 19 2015

Mathematica

Table[7^n, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

Maxima

makelist(7^n,n,0,20); /* Martin Ettl, Dec 27 2012 */

PARI

a(n)=7^n \\ Charles R Greathouse IV, Jul 28 2015

Formula

a(n) = 7^n.

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

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

E.g.f.: exp(7*x).

4/7 - 5/7^2 + 4/7^3 - 5/7^4 + ... = 23/48. [Jolley, Summation of Series, Dover, 1961]