A001021 Powers of 12.

1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072, 1283918464548864, 15407021574586368, 184884258895036416, 2218611106740436992

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Same as Pisot sequences E(1, 12), L(1, 12), P(1, 12), T(1, 12). Essentially same as Pisot sequences E(12, 144), L(12, 144), P(12, 144), T(12, 144). See A008776 for definitions of Pisot sequences.

Central terms of the triangle in A100851. - Reinhard Zumkeller, Mar 04 2006

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 12-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

Starting with 12, this sequence appears in the film "Vollmond" (1998, dir. Fredi Murer), when the children write it on the sidewalk at night. - Alonso del Arte, Dec 21 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 276.
• 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 (12).

Crossrefs

Cf. A024140 and A178248, 12^n +/- 1; A016125.

Cf. A000005, A000244, A000302, A000351, A000384, A001222, A008585, A008776, A100851, A159991.

Programs

Haskell

a001021 = (12 ^)
a001021_list = iterate (* 12) 1  -- Reinhard Zumkeller, Mar 31 2012

Maple

A001021:=-1/(-1+12*z); # Simon Plouffe in his 1992 dissertation

Mathematica

Table[12^n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

PARI

a(n)=12^n \\ Charles R Greathouse IV, Dec 21 2011

Formula

G.f.: 1/(1-12*x).

E.g.f.: exp(12x).

a(n) = 12*a(n-1). - Zerinvary Lajos, Apr 27 2009

a(n) = A159991(n)/A000351(n). - Reinhard Zumkeller, May 02 2009

From Reinhard Zumkeller, Mar 31 2012: (Start)

a(n) = A000302(n) * A000244(n). - Reinhard Zumkeller, Mar 31 2012

A001222(a(n)) = A008585(n); A000005(a(n)) = A000384(a(n)). (End)

a(n) = det(|ps(i+2, j)|, 1 <= i, j <= n), where ps(n, k) are Legendre-Stirling numbers of the first kind. - Mircea Merca, Apr 04 2013