0,2

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

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

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]

A160700(a(n)) = A010685(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 10 2009]

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]

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).

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

a(n) = 4^n; a(n) = 4a(n-1).

G.f.: 1/(1-4x), e.g.f.: exp(4x)

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

a(n)=A001045(2n)+A001045(2n+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+sqrt4)^n-(2-sqrt4)^n)/4 in Fibonacci form. Offset 1. a(3)=16. [From Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008]

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]

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.]

seq(binomial(n+0, 0)*4^n, n=0..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 19 2008

with(finance):seq(mul(cashflows([1, 1, 2], 0 ), k=1..n), n=0..36); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 02 2008

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]

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]

with(finance):seq(futurevalue(1, 3, n), n=0..24); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2009]

Table[4^n, {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006

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]

(Other) sage: [lucas_number1(n, 4, 0) for n in xrange(1, 26)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

(PARI) A000302(n)=4^n [From Michael Porter (michael_b_porter(AT)yahoo.com), Nov 06 2009]

Cf. A024036, A052539.

a(n) = A159991(n)/A001024(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 02 2009]

A032443 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]

Sequence in context: A006811 A005755 A077821 this_sequence A050734 A075614 A083592

Adjacent sequences: A000299 A000300 A000301 this_sequence A000303 A000304 A000305

easy,nonn,nice,core,new

N. J. A. Sloane (njas(AT)research.att.com).

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009