A001020 Powers of 11: a(n) = 11^n.

1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 214358881, 2357947691, 25937424601, 285311670611, 3138428376721, 34522712143931, 379749833583241, 4177248169415651, 45949729863572161, 505447028499293771, 5559917313492231481, 61159090448414546291

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Same as Pisot sequences E(1, 11), L(1, 11), P(1, 11), T(1, 11). Essentially same as Pisot sequences E(11, 121), L(11, 121), P(11, 121), T(11, 121). See A008776 for definitions of Pisot sequences.

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

a(n), for n <= 4, gives the n-th row of Pascal's triangle (A007318); a(n), n >= 5 "sort of" gives the n-th row of Pascal's triangle, but now the binomial coefficients with more than one digit overlap. - Daniel Forgues, Aug 12 2012

Numbers k such that sigma(11*k) = 11*k + sigma(k). - Jahangeer Kholdi, Nov 13 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 275.
• 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 (11).

Crossrefs

Cf. A096884, A097659, A007318.

Programs

Magma

[11^n: n in [0..100]]; // Vincenzo Librandi, Apr 24 2011

Maple

A001020:=-1/(-1+11*z); # Simon Plouffe in his 1992 dissertation

Mathematica

Table[11^n,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

Maxima

makelist(11*n,n,0,20); /* Martin Ettl, Dec 17 2012 */

PARI

a(n)=n^11 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: 1/(1-11*x).

E.g.f.: exp(11*x).

a(n) = 11*a(n-1), n > 0; a(0)=1. - Philippe Deléham, Nov 23 2008