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A001025
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Powers of 16: a(n) = 16^n.
(Formerly M5021 N2164)
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63
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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
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OFFSET
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0,2
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COMMENTS
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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.
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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FORMULA
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G.f.: 1/(1-16*x).
E.g.f.: exp(16*x).
a(n) = 16^n.
a(0) = 1, a(n) = 16*a(n-1). (End)
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MAPLE
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MATHEMATICA
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PROG
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(Sage) [lucas_number1(n, 16, 0) for n in range(1, 18)] # Zerinvary Lajos, Apr 29 2009
(Haskell)
a001025 = (16 ^)
(Python) print([16**n for n in range(20)]) # Stefano Spezia, Nov 10 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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