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A001025 Powers of 16: a(n) = 16^n.
(Formerly M5021 N2164)
65
1, 16, 256, 4096, 65536, 1048576, 16777216, 268435456, 4294967296, 68719476736, 1099511627776, 17592186044416, 281474976710656, 4503599627370496, 72057594037927936, 1152921504606846976, 18446744073709551616, 295147905179352825856, 4722366482869645213696, 75557863725914323419136, 1208925819614629174706176 (list; graph; refs; listen; history; text; internal format)
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.
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.
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.
Sequence in context: A220803 A229101 A220175 * A144318 A230142 A247165
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)