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A000420 Powers of 7: a(n) = 7^n.
(Formerly M4431 N1874)
140
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). 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
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
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
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.
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
(Magma) [7^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 27 2016
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)