%I M5021 N2164 #100 Jul 12 2023 12:28:34

%S 1,16,256,4096,65536,1048576,16777216,268435456,4294967296,

%T 68719476736,1099511627776,17592186044416,281474976710656,

%U 4503599627370496,72057594037927936,1152921504606846976,18446744073709551616,295147905179352825856,4722366482869645213696,75557863725914323419136,1208925819614629174706176

%N Powers of 16: a(n) = 16^n.

%C 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.

%C Convolution-square (auto-convolution) of A098430. - _R. J. Mathar_, May 22 2009

%C Subsequence of A161441: A160700(a(n)) = 1. - _Reinhard Zumkeller_, Jun 10 2009

%C 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

%C 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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Muniru A Asiru, <a href="/A001025/b001025.txt">Table of n, a(n) for n = 0..820</a> (terms n = 0..100 from T. D. Noe)

%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=280">Encyclopedia of Combinatorial Structures 280</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (16).

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

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

%F From _Muniru A Asiru_, Nov 07 2018: (Start)

%F a(n) = 16^n.

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

%F a(n) = 4^A005843(n) = 2^A008586(n) = A000302(n)^2 = A000079(n)*A001018(n). - _Muniru A Asiru_, Nov 10 2018

%p A001025:=-1/(-1+16*z); # _Simon Plouffe_ in his 1992 dissertation

%t Table[4^(2*n), {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 01 2009 *)

%o (Sage) [lucas_number1(n,16,0) for n in range(1, 18)] # _Zerinvary Lajos_, Apr 29 2009

%o (PARI) a(n)=1<<(4*n) \\ _Charles R Greathouse IV_, Feb 01 2012

%o (Maxima) A001025(n):=16^n$

%o makelist(A001025(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */

%o (Haskell)

%o a001025 = (16 ^)

%o a001025_list = iterate (* 16) 1 -- _Reinhard Zumkeller_, Nov 07 2012

%o (GAP) List([0..20],n->16^n); # _Muniru A Asiru_, Nov 07 2018

%o (Python) print([16**n for n in range(20)]) # _Stefano Spezia_, Nov 10 2018

%Y Partial sums give A131865.

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_