cp's OEIS Frontend

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

%I A001025 M5021 N2164 #106 Aug 27 2025 21:39:27
%S A001025 1,16,256,4096,65536,1048576,16777216,268435456,4294967296,
%T A001025 68719476736,1099511627776,17592186044416,281474976710656,
%U A001025 4503599627370496,72057594037927936,1152921504606846976,18446744073709551616,295147905179352825856,4722366482869645213696,75557863725914323419136,1208925819614629174706176
%N A001025 Powers of 16: a(n) = 16^n.
%C A001025 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 A001025 Convolution-square (auto-convolution) of A098430. - _R. J. Mathar_, May 22 2009
%C A001025 Subsequence of A161441: A160700(a(n)) = 1. - _Reinhard Zumkeller_, Jun 10 2009
%C A001025 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
%D A001025 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001025 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001025 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 A001025 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 A001025 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=280">Encyclopedia of Combinatorial Structures 280</a>
%H A001025 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A001025 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 A001025 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001025 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 A001025 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (16).
%F A001025 G.f.: 1/(1-16*x).
%F A001025 E.g.f.: exp(16*x).
%F A001025 From _Muniru A Asiru_, Nov 07 2018: (Start)
%F A001025 a(n) = 16^n.
%F A001025 a(0) = 1, a(n) = 16*a(n-1). (End)
%F A001025 a(n) = 4^A005843(n) = 2^A008586(n) = A000302(n)^2 = A000079(n)*A001018(n). - _Muniru A Asiru_, Nov 10 2018
%F A001025 a(n) = ( 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) ). - _Peter Bala_, Feb 12 2019
%F A001025 a(n) = Sum_{k = 0..2*n} A000984(k) * A000984(2*n-k). - _Peter Bala_, Aug 23 2025
%p A001025 A001025:=-1/(-1+16*z); # _Simon Plouffe_ in his 1992 dissertation
%t A001025 Table[4^(2*n), {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 01 2009 *)
%o A001025 (Sage) [lucas_number1(n,16,0) for n in range(1, 18)] # _Zerinvary Lajos_, Apr 29 2009
%o A001025 (PARI) a(n)=1<<(4*n) \\ _Charles R Greathouse IV_, Feb 01 2012
%o A001025 (Maxima) A001025(n):=16^n$
%o A001025 makelist(A001025(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o A001025 (Haskell)
%o A001025 a001025 = (16 ^)
%o A001025 a001025_list = iterate (* 16) 1  -- _Reinhard Zumkeller_, Nov 07 2012
%o A001025 (GAP) List([0..20],n->16^n); # _Muniru A Asiru_, Nov 07 2018
%o A001025 (Python) print([16**n for n in range(20)]) # _Stefano Spezia_, Nov 10 2018
%Y A001025 Partial sums give A131865.
%Y A001025 Cf. A000079, A000302, A001018, A005843, A008586.
%K A001025 nonn,easy,changed
%O A001025 0,2
%A A001025 _N. J. A. Sloane_