A000400 Powers of 6: a(n) = 6^n.

1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 10077696, 60466176, 362797056, 2176782336, 13060694016, 78364164096, 470184984576, 2821109907456, 16926659444736, 101559956668416, 609359740010496, 3656158440062976, 21936950640377856, 131621703842267136

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

Same as Pisot sequences E(1, 6), L(1, 6), P(1, 6), T(1, 6). Essentially same as Pisot sequences E(6, 36), L(6, 36), P(6, 36), T(6, 36). See A008776 for definitions of Pisot sequences.

Central terms of the triangle in A036561. - Reinhard Zumkeller, May 14 2006

a(n) = A169604(n)/3; a(n+1) = 2*A169604(n). - Reinhard Zumkeller, May 02 2010

Number of pentagons contained within pentaflakes. - William A. Tedeschi, Sep 12 2010

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

a(n) is number of compositions of natural numbers into n parts less than 6. For example, a(2) = 36, and there are 36 compositions of natural numbers into 2 parts less than 6.

The compositions of n in which each part is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 5-colored compositions of n such that no adjacent parts have the same color.

Number of words of length n over the alphabet of six letters. - Joerg Arndt, Sep 16 2014

The number of ordered triples (x, y, z) of binary words of length n such that D(x,z) = D(x, y) + D(y, z) where D(a, b) is the Hamming distance from a to b. - Geoffrey Critzer, Mar 06 2017

a(n) is the area of a triangle with vertices at (2^n, 3^n), (2^(n+1), 3^(n+1)), and (2^(n+2), 3^(n+2)); a(n) is also one fifth the area of a triangle with vertices at (2^n, 3^(n+2)), (2^(n+1), 3^(n+1)), and (2^(n+2), 3^n). - J. M. Bergot, May 07 2018

a(n) is the number of possible outcomes of n distinguishable 6-sided dice. - Stefano Spezia, Jul 06 2024

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 86.
• 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
• C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
• Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 271
• 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
• Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• Eric Weisstein's World of Mathematics, Pentaflake
• Index entries for linear recurrences with constant coefficients, signature (6).

Crossrefs

Column 3 of A225816.

Row 6 of A003992.

Row 3 of A329332.

Cf. A000005, A000290, A011577, A036561, A159991, A169604,

Programs

Haskell

a000400 = (6 ^)
a000400_list = iterate (* 6) 1  -- Reinhard Zumkeller, Nov 21 2013

Mathematica

6^Range[0, 40] (* Harvey P. Dale, Jan 24 2013 *)

Maxima

A000400(n):=6^n$
makelist(A000400(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI

a(n)=6^n \\ Charles R Greathouse IV, Jun 16 2011

Scala

(List.fill(50)(6: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, May 31 2019

Formula

a(n) = 6^n.

a(0) = 1; a(n) = 6*a(n-1).

G.f.: 1/(1-6*x). - Simon Plouffe in his 1992 dissertation.

E.g.f.: exp(6*x).

A000005(a(n)) = A000290(n+1). - Reinhard Zumkeller, Mar 04 2007

a(n) = A159991(n)/A011577(n). - Reinhard Zumkeller, May 02 2009

a(n) = det(|s(i+3,j)|, 1 <= i,j <= n), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 04 2013