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A000420 Powers of 7.
(Formerly M4431 N1874)
43
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; 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

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

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Index entries for sequences related to linear recurrences with constant coefficients

FORMULA

a(n) = 7^n; 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); [S. Plouffe in his 1992 dissertation.]

MATHEMATICA

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

CROSSREFS

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 (njas(AT)research.att.com).

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Last modified February 14 11:10 EST 2012. Contains 205615 sequences.