%I A000302 M3518 N1428
%S A000302 1,4,16,64,256,1024,4096,16384,65536,262144,1048576,4194304,16777216,
%T A000302 67108864,268435456,1073741824,4294967296,17179869184,68719476736,274877906944,
%U A000302 1099511627776,4398046511104,17592186044416,70368744177664,281474976710656
%N A000302 Powers of 4.
%C A000302 Same as Pisot sequences E(1,4), L(1,4), P(1,4), T(1,4). See A008776 for 
               definitions of Pisot sequences.
%C A000302 The convolution square root of this sequence is A000984, the central 
               binomial coefficients: C(2n,n). - T. D. Noe (noe(AT)sspectra.com), 
               Jun 11 2002
%C A000302 a(n)=sum(k=0,n,C(2k,k)*C(2(n-k),n-k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jan 26 2003
%C A000302 With p(n) = the number of integer partitions of n, p(i) = the number 
               of parts of the i-th partition of n, d(i) = the number of different 
               parts of the i-th partition of n, m(i,j) = multiplicity of the j-th 
               part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i and 
               prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)} 
               p(i)!/(prod_{j=1}^{d(i)} m(i,j)!) * 2^(n-1) - Thomas Wieder (wieder.thomas(AT)t-online.de), 
               May 18 2005
%C A000302 Sums of rows of the triangle in A122366. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 30 2006
%C A000302 A000005(a(n)) = A005408(n+1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 04 2007
%C A000302 Number of n-permutations (n=1) of 5 objects u,v,w z x, with repetition 
               allowed, containing exactly null u (or free). Example: a(1)=4 because 
               we have v, w, z, x. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 19 2008
%C A000302 Hankel transform of A076035. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Feb 28 2009]
%C A000302 Equals the Catalan sequence: (1, 1, 2, 5, 14,...), convolved with A032443: 
               (1, 3, 11, 42,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 15 2009]
%C A000302 A160700(a(n)) = A010685(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 10 2009]
%C A000302 Numbers n such that n^3+(2n)^3+(3n)^3=36*n^3 is square [From Vincenzo 
               Librandi (vincenzo.librandi(AT)tin.it), Aug 05 2009]
%D A000302 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000302 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000302 T. D. Noe, <a href="b000302.txt">Table of n, a(n) for n = 0..100</a>
%H A000302 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000302 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=8">
               Encyclopedia of Combinatorial Structures 8</a>
%H A000302 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=269">
               Encyclopedia of Combinatorial Structures 269</a>
%H A000302 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A000302 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000302 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000302 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 A000302 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CantorDust.html">Cantor Dust</a>
%H A000302 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000302 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A000302 a(n) = 4^n; a(n) = 4a(n-1).
%F A000302 G.f.: 1/(1-4x), e.g.f.: exp(4x)
%F A000302 1 = Sum(n = 1 through infinity) 3/a(n) = 3/4 + 3/16 + 3/64 + 3/256 + 
               3/1024...; with partial sums: 3/4, 15/16, 63/64, 255/256, 1023/1024... 
               - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 16 2003
%F A000302 a(n)=A001045(2n)+A001045(2n+1). - Paul Barry (pbarry(AT)wit.ie), Apr 
               27 2004
%F A000302 a(n)=sum(2^(n-j)*binomial(n+j,j),j=0..n) - Peter C. Heinig (algorithms(AT)gmx.de), 
               Apr 06 2007
%F A000302 Hankel transform of A115967 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Jun 22 2007
%F A000302 a(n) = 6*StirlingS2(n+1,4) + 6*StirlingS2(n+1,3) + 3*StirlingS2(n+1,2) 
               + 1 = 2*StirlingS2(2^n,2^n - 1) + StirlingS2(n+1,2) + 1. - Ross La 
               Haye (rlahaye(AT)new.rr.com), Jun 26 2008
%F A000302 ((2+sqrt4)^n-(2-sqrt4)^n)/4 in Fibonacci form. Offset 1. a(3)=16. [From 
               Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008]
%e A000302 1^3+2^3+3^3=36=6^2; 4^3+8^3+12^3=2304=48^2; 16^3+32^3+48^3=147456=384^2 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 05 2009]
%p A000302 A000302 := n->4^n;
%p A000302 for n from 1 to 10 do sum(2^(n-j)*binomial(n+j,j),j=0..n); od; - Peter 
               C. Heinig (algorithms(AT)gmx.de), Apr 06 2007
%p A000302 A000302:=-1/(-1+4*z); [S. Plouffe in his 1992 dissertation.]
%p A000302 seq(binomial(n+0,0)*4^n,n=0..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 19 2008
%p A000302 with(finance):seq(mul(cashflows([1,1,2], 0 ),k=1..n),n=0..36); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jul 02 2008
%p A000302 a:=n->mul(3+mul(1, j=2..n),j=1..n):seq(a(n),n=0..26);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Dec 06 2008]
%p A000302 g:=(1+2*z)/(1-4*z): gser:=series(g, z=0, 43): seq((coeff(gser, z, n))/
               6, n=1..24);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 25 2009]
%p A000302 with(finance):seq(futurevalue(1,3,n), n=0..24);# [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Mar 25 2009]
%t A000302 Table[4^n, {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 01 2006
%t A000302 a = 1; lst = {a}; Do[a = a + 3*a; AppendTo[lst, a], {n, 0, 25}]; lst 
               [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]
%o A000302 (Other) sage: [lucas_number1(n,4,0) for n in xrange(1, 26)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%o A000302 (PARI) A000302(n)=4^n [From Michael Porter (michael_b_porter(AT)yahoo.com), 
               Nov 06 2009]
%Y A000302 Cf. A024036, A052539.
%Y A000302 a(n) = A159991(n)/A001024(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 02 2009]
%Y A000302 A032443 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]
%Y A000302 Sequence in context: A006811 A005755 A077821 this_sequence A050734 A075614 
               A083592
%Y A000302 Adjacent sequences: A000299 A000300 A000301 this_sequence A000303 A000304 
               A000305
%K A000302 easy,nonn,nice,core,new
%O A000302 0,2
%A A000302 N. J. A. Sloane (njas(AT)research.att.com).
%E A000302 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009