A001023 Powers of 14. (Formerly M4949 N2120)

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

Offset

0,2

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).

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

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 *)

Prog

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

Crossrefs

Cf. A000005, A000290, A160193.

Row 9 of A329332.

Sequence in context: A207743 A207720 A171288 * A278476 A067221 A072533

Adjacent sequences: A001020 A001021 A001022 * A001024 A001025 A001026

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Sep 19 2000

Status

approved