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A000302
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Powers of 4.
(Formerly M3518 N1428)
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223
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1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, 68719476736, 274877906944, 1099511627776, 4398046511104, 17592186044416, 70368744177664, 281474976710656
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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(1,4), L(1,4), P(1,4), T(1,4). See A008776 for definitions of Pisot sequences.
The convolution square root of this sequence is A000984, the central binomial coefficients: C(2n,n). - T. D. Noe, Jun 11 2002
a(n)=sum(k=0,n,C(2k,k)*C(2(n-k),n-k)). - Benoit Cloitre, Jan 26 2003
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, May 18 2005
Sums of rows of the triangle in A122366. - Reinhard Zumkeller, Aug 30 2006
A000005(a(n)) = A005408(n+1). - Reinhard Zumkeller, Mar 04 2007
Hankel transform of A076035. [From Philippe DELEHAM, Feb 28 2009]
Equals the Catalan sequence: (1, 1, 2, 5, 14,...), convolved with A032443: (1, 3, 11, 42,...). [From Gary W. Adamson, May 15 2009]
a(n) = A188915(A006127(n)). [Reinhard Zumkeller, Apr 14 2011]
Sum of coefficients of expansion of (1+x+x^2+x^3)^n.
a(n) is number of compositions of natural numbers into n parts <4.
a(2)=16 there are 16 compositions of natural numbers into 2 parts <4.
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 4-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Squares in A002984. [Reinhard Zumkeller, Dec 28 2011]
a(n) is the minimum number whose arithmetic derivative is n times the number itself: 1’=0=0*1; 4’=4=1*4; 16’=32=2*16; 64’=192=3*64, etc. - Paolo P. Lava, Feb 21 2012
Row sums of Pascal's triangle using the rule that going left increases the value by a factor of k = 3. For example, the first three rows are {1}, {3, 1}, and {9, 6, 1}. Using this rule gives row sums as (k+1)^n. - Jon Perry, Oct 11 2012
sum(k=0,n,binomial(2*k+l,k)*binomial(2*(n-k)-l,n-k)) for every real number l - Rui Duarte and António Guedes de Oliveira, Feb 16 2013
First differences of A002450. - Omar E. Pol, Feb 20 2013
Sum of all peak heights in Dyck paths of semilength n+1. - David Scambler, Apr 22 2013
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REFERENCES
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R. Duarte and A. G. de Oliveira, Short note on the convolution of binomial coefficients, arXiv preprint arXiv:1302.2100, 2013
D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
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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T. D. Noe, Table of n, a(n) for n = 0..100
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 8
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 269
Tanya Khovanova, Recursive Sequences
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Cantor Dust
Index entries for "core" sequences
Index to divisibility sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (4).
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FORMULA
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a(n) = 4^n.
a(n) = 4*a(n-1).
G.f.: 1/(1-4*x).
E.g.f.: exp(4*x).
1 = sum(n>=1, 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, Jun 16 2003
a(n)=A001045(2*n)+A001045(2*n+1). - Paul Barry, Apr 27 2004
a(n)=sum(2^(n-j)*binomial(n+j,j),j=0..n) - Peter C. Heinig (algorithms(AT)gmx.de), Apr 06 2007
Hankel transform of A115967 . - Philippe DELEHAM, Jun 22 2007
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, Jun 26 2008
((2+sqrt(4))^n-(2-sqrt(4))^n)/4. Offset 1. a(3)=16. [From Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008]
a(n) = sum(k=0..n,C(2*n+1,k)). [Mircea Merca, Jun 25 2011]
sum_{n>=1} mobius(n)/a(n) = 0.1710822479183... - R. J. Mathar, Aug 12 2012
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MAPLE
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A000302 := n->4^n;
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
A000302:=-1/(-1+4*z); [Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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Table[4^n, {n, 0, 30}] - Stefan Steinerberger, Apr 01 2006
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PROG
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(PARI) A000302(n)=4^n [From Michael B. Porter, Nov 06 2009]
(Haskell)
a000302 = (4 ^)
a000302_list = iterate (* 4) 1 -- Reinhard Zumkeller, Apr 04 2012
(Maxima) A000302(n):=4^n$ makelist(A000302(n), n, 0, 30); [Martin Ettl, Oct 24 2012]
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CROSSREFS
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Cf. A024036, A052539, A032443, A000351 (Binomial transform).
a(n) = A159991(n)/A001024(n) = A047653(n) + A181765(n). A160700(a(n)) = A010685(n). [From Reinhard Zumkeller]
Sequence in context: A077821 A215877 A206450 * A050734 A075614 A083592
Adjacent sequences: A000299 A000300 A000301 * A000303 A000304 A000305
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KEYWORD
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easy,nonn,nice,core
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Partially edited by Joerg Arndt, Mar 11 2010
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STATUS
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approved
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