A009968 Powers of 24: a(n) = 24^n.

1, 24, 576, 13824, 331776, 7962624, 191102976, 4586471424, 110075314176, 2641807540224, 63403380965376, 1521681143169024, 36520347436056576, 876488338465357824, 21035720123168587776, 504857282956046106624, 12116574790945106558976, 290797794982682557415424, 6979147079584381377970176, 167499529910025153071284224, 4019988717840603673710821376

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Same as Pisot sequences E(1, 24), L(1, 24), P(1, 24), T(1, 24). Essentially same as Pisot sequences E(24, 576), L(24, 576), P(24, 576), T(24, 576). See A008776 for definitions of Pisot sequences.

If X_1, X_2, ..., X_n is a partition of the set {1, 2, ..., 2*n} into blocks of size 2 then, for n >= 1, a(n) is equal to the number of functions f : {1, 2, ..., 2*n} -> {1, 2, 3, 4, 5} such that for fixed y_1, y_2, ..., y_n in {1, 2, 3, 4, 5} we have f(X_i) <> {y_i}, (i = 1, 2, ..., n). - Milan Janjic, May 24 2007

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

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (24).

Crossrefs

Column k = 4 of A225816.

Programs

Magma

[24^n: n in [0..100]]; // Vincenzo Librandi, Nov 21 2010

Mathematica

NestList[24#&, 1, 20] (* Harvey P. Dale, Feb 04 2017 *)

PARI

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

Python

[24**n for n in range(21)] # Michael S. Branicky, Jan 24 2021

SageMath

[lucas_number1(n,24,0) for n in range(1, 17)] # Zerinvary Lajos, Apr 29 2009

Scala

LazyList.iterate(1: BigInt)( * 24).take(24).toList // _Alonso del Arte, Apr 24 2020

Formula

G.f.: 1/(1 - 24*x). - Philippe Deléham, Nov 23 2008

E.g.f.: exp(24x). - Zerinvary Lajos, Apr 29 2009

a(n) = 24^n; a(n) = 24*a(n-1) for n > 0, a(0) = 1. - Vincenzo Librandi, Nov 21 2010

a(n) = det(|s(i + 4, j)|, 1 <= i, j <= n), where s(n, k) are Stirling numbers of the first kind. - Mircea Merca, Apr 04 2013