A001023 Powers of 14.

1, 14, 196, 2744, 38416, 537824, 7529536, 105413504, 1475789056, 20661046784, 289254654976, 4049565169664, 56693912375296, 793714773254144, 11112006825558016, 155568095557812224, 2177953337809371136, 30491346729331195904, 426878854210636742656, 5976303958948914397184, 83668255425284801560576

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Same as Pisot sequences E(1, 14), L(1, 14), P(1, 14), T(1, 14). Essentially same as Pisot sequences E(14, 196), L(14, 196), P(14, 196), T(14, 196). See A008776 for definitions of Pisot sequences.

Number of n-permutations of 15 objects: l, m, n, o, p, q, r, s, t, u, v, w, z, x, y with repetition allowed and containing no u's, (u-free). Permutations with repetitions! If n=0 then 1 >>14^0=1 "". (no u's.) If n=1 then 13 >>14^1=14, >> l, m, n, o, p, q, r, s, t, v, w, z, x, y. (no u's.) etc. - Zerinvary Lajos, Jul 01 2009

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 14-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
• 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 278.
• 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.
• Index entries for linear recurrences with constant coefficients, signature (14).

Crossrefs

Cf. A000005, A000290, A160193.

Row 9 of A329332.

Programs

Magma

[ 14^n: n in [0..20] ]; // Vincenzo Librandi, Nov 21 2010

Magma

[ n eq 1 select 1 else 14*Self(n-1): n in [1..21] ];

Maple

A001023:=-1/(-1+14*z); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Table[14^n,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)
Denominator/@HermiteH[Range[0,20],5/28] (* Harvey P. Dale, Jul 11 2011 *)

PARI

a(n)=14^n \\ Charles R Greathouse IV, Nov 18 2011

Python

print([14**n for n in range(21)]) # Michael S. Branicky, Jan 14 2021

Sage

[lucas_number1(n,14,0) for n in range(1, 18)]# Zerinvary Lajos, Apr 29 2009

Formula

G.f.: 1/(1-14x), e.g.f.: exp(14x)

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

a(n) = 14^n; a(n) = 14*a(n-1) with a(0)=1. - Vincenzo Librandi, Nov 21 2010

Extensions

More terms from James Sellers, Sep 19 2000