1,3

Arithmetic derivative of 3^n: a(n) = A003415(A000244(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 26 2002

Binomial transform of A053220(n+1) is a(n+2). Binomial transform of A001787 is a(n+1). Binomial transform of A045883(n-1). - Michael Somos, Jul 10 2003

If X_1,X_2,...,X_n are 3-blocks of a (3n+1)-set X then, for n>=1, a(n+2) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). > - Milan R. Janjic (agnus(AT)blic.net), Nov 18 2007

Let S be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xSy if x is a subset of y. Then a(n+1) = the sum of the differences in size (i.e. |y|-|x|) for all (x, y) of S. - Ross La Haye (rlahaye(AT)new.rr.com), Nov 19 2007

With a different offset, number of n-permutations of 4 objects u,v,w,z, with repetition allowed, containing exactly one u. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 27 2007

Sum(n>=2,1/a(n))=3*log(3/2) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Sep 19 2009]

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]

Index entries for sequences related to linear recurrences with constant coefficients

Milan Janjic, Two Enumerative Functions

F. Ellermann, Illustration of binomial transforms

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 715

G.f.: (x/(1-3*x))^2. E.g.f.: (1+(3x-1)exp(3x))/9. a(n) = 3^(n-2)*(n-1); (convolution of A000244, powers of 3, with itself) - from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de).

a(n)=6a(n-1)-9a(n-2); n>2; a(1)=0, a(2)=1. - Barry E. Williams, Jan 13 2000

A027471(n)=A036290(n)/3 - Paul Barry (pbarry(AT)wit.ie), Feb 06 2004

a(n)=sum{k=0..n, 3^(n-k)binomial(n-k+1, k)binomial(1, (k+1)/2)(1-(-1)^k)/2}

a(n)=sum{k=0..2n, T(n, k)*k}/3, where T(n, k) is given by A027907; a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)(j+k)}}/3; a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)(j-k)}}; a(n+1)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)(j+k+1)}}. - Paul Barry (pbarry(AT)wit.ie), Feb 15 2005

Numerators of sequence a[ 2, n ] in (b^2)[ i, j ] where b[ i, j ] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.

a:=n->sum(3^(n-2), j=2..n): seq(a(n), n=1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2008

with(finance):seq(add(futurevalue(1, 2, n), k=0..n), n=-1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008

with(finance):seq(add(futurevalue( 3, 2, n), k=0..n)/3, n=-1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008

(PARI) a(n)=if(n<1, 0, (n-1)*3^(n-2))

(Other) sage: [lucas_number1(n, 6, 9) for n in xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

Second column of A027465.

Cf. A006234.

Partial sums of A081038.

Sequence in context: A005325 A099623 A119852 this_sequence A037695 A094829 A055145

Adjacent sequences: A027468 A027469 A027470 this_sequence A027472 A027473 A027474

nonn

Olivier Gerard (olivier.gerard(AT)gmail.com)

Edited by Michael Somos, Jul 10, 2003