A001025 Powers of 16: a(n) = 16^n. (Formerly M5021 N2164)

1, 16, 256, 4096, 65536, 1048576, 16777216, 268435456, 4294967296, 68719476736, 1099511627776, 17592186044416, 281474976710656, 4503599627370496, 72057594037927936, 1152921504606846976, 18446744073709551616, 295147905179352825856, 4722366482869645213696, 75557863725914323419136, 1208925819614629174706176

Offset

0,2

Comments

Same as Pisot sequences E(1, 16), L(1, 16), P(1, 16), T(1, 16). Essentially same as Pisot sequences E(16, 256), L(16, 256), P(16, 256), T(16, 256). See A008776 for definitions of Pisot sequences.

Convolution-square (auto-convolution) of A098430. - R. J. Mathar, May 22 2009

Subsequence of A161441: A160700(a(n)) = 1. - Reinhard Zumkeller, Jun 10 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 16-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

Right-hand side of the identity ( Sum_{k = 0..n} (2*k + 1)*binomial(2*n + 1, n - k) ) * ( Sum_{k = 0..n} (-1)^k/(2*k + 1)*binomial(2*n + 1, n - k) ) = 16^n. - Peter Bala, Feb 12 2019

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

Muniru A Asiru, Table of n, a(n) for n = 0..820 (terms n = 0..100 from T. D. Noe)

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 280

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 (16).

Formula

G.f.: 1/(1-16*x).

E.g.f.: exp(16*x).

From Muniru A Asiru, Nov 07 2018: (Start)

a(n) = 16^n.

a(0) = 1, a(n) = 16*a(n-1). (End)

a(n) = 4^A005843(n) = 2^A008586(n) = A000302(n)^2 = A000079(n)*A001018(n). - Muniru A Asiru, Nov 10 2018

Maple

A001025:=-1/(-1+16*z); # Simon Plouffe in his 1992 dissertation

Mathematica

Table[4^(2*n), {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Mar 01 2009 *)

Prog

(Sage) [lucas_number1(n, 16, 0) for n in range(1, 18)] # Zerinvary Lajos, Apr 29 2009
(PARI) a(n)=1<<(4*n) \\ Charles R Greathouse IV, Feb 01 2012
(Maxima) A001025(n):=16^n$
makelist(A001025(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
(Haskell)
a001025 = (16 ^)
a001025_list = iterate (* 16) 1  -- Reinhard Zumkeller, Nov 07 2012
(GAP) List([0..20], n->16^n); # Muniru A Asiru, Nov 07 2018
(Python) print([16**n for n in range(20)]) # Stefano Spezia, Nov 10 2018

Crossrefs

Partial sums give A131865.

Cf. A000079, A000302, A001018, A005843, A008586.

Sequence in context: A220803 A229101 A220175 * A144318 A230142 A247165

Adjacent sequences: A001022 A001023 A001024 * A001026 A001027 A001028

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved