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A000420 Powers of 7.
(Formerly M4431 N1874)
84
1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010407, 678223072849, 4747561509943, 33232930569601, 232630513987207, 1628413597910449, 11398895185373143, 79792266297612001, 558545864083284007 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Same as Pisot sequences E(1,7), L(1,7), P(1,7), T(1,7). 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

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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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).

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]

EXAMPLE

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

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 *)

PROG

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

(Haskell)

a000420 = (7 ^)

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

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

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).

Sequence in context: A045578 A126627 A206453 * A050737 A195908 A033143

Adjacent sequences:  A000417 A000418 A000419 * A000421 A000422 A000423

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 29 17:16 EDT 2016. Contains 273510 sequences.