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A000302
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Powers of 4.
(Formerly M3518 N1428)
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184
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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 (noe(AT)sspectra.com), Jun 11 2002
a(n)=sum(k=0,n,C(2k,k)*C(2(n-k),n-k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (wieder.thomas(AT)t-online.de), May 18 2005
Sums of rows of the triangle in A122366. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 30 2006
A000005(a(n)) = A005408(n+1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 04 2007
Hankel transform of A076035. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 28 2009]
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]
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]
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REFERENCES
| 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
| 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
S. 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.
S. 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 entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| 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 (qntmpkt(AT)yahoo.com), Jun 16 2003
a(n)=A001045(2*n)+A001045(2*n+1). - Paul Barry (pbarry(AT)wit.ie), 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 (kolotoko(AT)wanadoo.fr), 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 (rlahaye(AT)new.rr.com), 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]
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MAPLE
| 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); [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[4^n, {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006
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PROG
| (PARI) A000302(n)=4^n [From Michael Porter (michael_b_porter(AT)yahoo.com), Nov 06 2009]
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CROSSREFS
| Cf. A024036, A052539, A032443.
a(n) = A159991(n)/A001024(n) = A047653(n) + A181765(n). A160700(a(n)) = A010685(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com)]
Sequence in context: A005755 A077821 A206450 * A050734 A075614 A083592
Adjacent sequences: A000299 A000300 A000301 * A000303 A000304 A000305
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KEYWORD
| easy,nonn,nice,core
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009
Partially edited by Joerg Arndt (arndt(AT)jjj.de), Mar 11 2010
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