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A001025
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Powers of 16: a(n) = 16^n.
(Formerly M5021 N2164)
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56
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1, 16, 256, 4096, 65536, 1048576, 16777216, 268435456, 4294967296, 68719476736, 1099511627776, 17592186044416, 281474976710656, 4503599627370496, 72057594037927936, 1152921504606846976, 18446744073709551616, 295147905179352825856, 4722366482869645213696, 75557863725914323419136, 1208925819614629174706176
(list;
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refs;
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history;
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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).
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FORMULA
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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
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MAPLE
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A001025:=-1/(-1+16*z); # Simon Plouffe in his 1992 dissertation
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MATHEMATICA
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Table[4^(2*n), {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Mar 01 2009 *)
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PROG
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(Sage) [lucas_number1(n, 16, 0) for n in xrange(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) for n in range(0, 20): print(16**n, end=', ') # Stefano Spezia, Nov 10 2018
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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