%I A001047 M3887 N1596
%S A001047 0,1,5,19,65,211,665,2059,6305,19171,58025,175099,527345,1586131,
%T A001047 4766585,14316139,42981185,129009091,387158345,1161737179,3485735825,
%U A001047 10458256051,31376865305,94134790219,282412759265,847255055011
%N A001047 3^n - 2^n.
%C A001047 a(n) = sum of the elements in the n-th row of triangle pertaining to 
               A036561. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 02 
               2002
%C A001047 Number of 2 X n binary arrays with a path of adjacent 1's and no path 
               of adjacent 0's from top row to bottom row. - Ron Hardin (rhhardin(AT)att.net), 
               Mar 21 2002
%C A001047 With offset 1, partial sums of A027649. - Paul Barry (pbarry(AT)wit.ie), 
               Jun 24 2003
%C A001047 Number of distinct lines through the origin in the n-dimensional lattice 
               of side length 2. A049691 has the values for the 2-dimensional lattice 
               of side length n. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), 
               Nov 19 2003
%C A001047 a(n) = A083323(n)-1 = A056182(n)/2 = (A002783(n)-1)/2 = (A003063(n+2)-A003063(n+1))/
               2. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jan 12 2004
%C A001047 a(n+1)/(n+1)=(3*3^n-2*2^n)/(n+1) is the second binomial transform of 
               the harmonic sequence 1/(n+1). - Paul Barry (pbarry(AT)wit.ie), Apr 
               19 2005
%C A001047 a(n) = A112626(n, 1). - Ross La Haye (rlahaye(AT)new.rr.com), Jan 11 
               2006
%C A001047 a(n+1) = sums of n-th row of the triangle in A036561. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), May 14 2006
%C A001047 The sequence gives the sum of the lengths of the segments in Cantor's 
               dust generating sequence up to the i-th step. Measurement unit = 
               length of the segment of i-th step. - Giorgio Balzarotti (Greenblue(AT)tiscali.it), 
               Nov 18 2006
%C A001047 Let T 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), xTy if x 
               is a proper subset of y. Then a(n) = |T|. - Ross La Haye (rlahaye(AT)new.rr.com), 
               Dec 22 2006
%C A001047 Comments from Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 04 2007: 
               (Start)
%C A001047 a(n) is prime for n = {2,3,5,17,29,31,53,59,101,277,647,1061,2381,...} 
               = A057468(n) Numbers n such that 3^n - 2^n is prime.
%C A001047 p divides a(p) - 1 for prime p.
%C A001047 Quotients (3^p - 2^p - 1)/p, where p = Prime[n], are listed in A127071(n) 
               = {2,6,42,294,15918,122010,7588770,61144062,...}.
%C A001047 Numbers n such that n divides 3^n - 2^n - 1 are listed in A127072(n) 
               = {1,2,3,4,5,7,8,9,11,13,16,17,19,23,27,29,31,32,37,41,43,45,47,49,
               53,59,61,64,67,71,73,79,81,83,89,97,...}.
%C A001047 Pseudoprimes in A127072(n) include all powers of primes {2,3,7} and some 
               composite numbers that are listed in A127073(n) = {45,245,405,561,
               637,639,833,891,...}, which includes all Carmichael numbers A002997(n) 
               = {561,1105,1729,2465,2821,6601,8911,10585,15841,29341,...}.
%C A001047 Numbers n such that n^2 divides 3^n - 2^n - 1 are listed in A127074(n) 
               = {1,2,3,4,7,49,179,619,17807,...}.
%C A001047 5 divides a(2n).
%C A001047 5^2 divides a(2*5n).
%C A001047 5^3 divides a(2*5^2n).
%C A001047 5^4 divides a(2*5^3n).
%C A001047 7 divides a(6n).
%C A001047 7^2 divides a(6*7n).
%C A001047 11 divides a(10n).
%C A001047 13 divides a(4n).
%C A001047 13^2 divides a(4*13n).
%C A001047 17 divides a(16n).
%C A001047 19 divides a(3n).
%C A001047 19^2 divides a(3*19n).
%C A001047 23^2 divides a(11n).
%C A001047 23^3 divides a(11*23n).
%C A001047 23^4 divides a(11*23^2n).
%C A001047 29 divides a(7n).
%C A001047 31 divides a(30n).
%C A001047 p divides a((p-1)n) for prime p>3.
%C A001047 p divides a((p-1)/2)) for prime p = {5,19,23,29,43,47,53,...} = A097936(n) 
               Primes p such that p divides 3^((p-1)/2) - 2^((p-1)/2). Also primes 
               p such that 6 is a square mod p, except {2,3}, A038876(n).
%C A001047 p^(k+1) divides a(p^k*(p-1)/2*n) for prime p = {5,19,23,29,43,47,53,...} 
               = A097936(n).
%C A001047 p^(k+1) divides a(p^k*(p-1)*n) for prime p>3.
%C A001047 Note the exception that for p = 23, p^(k+2) divides a(p^k*(p-1)/2*n).
%C A001047 There are no more such exceptions for primes p up to 600000. (End)
%C A001047 Final digits of terms follow sequence 1,5,9,5. - Enoch Haga (Enokh(AT)comcast.net), 
               Nov 26 2007
%D A001047 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001047 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001047 Archimedeans Problems Drive, Eureka, 24 (1961), 20.
%D A001047 G. Kreweras, Inversion des polynomes de Bell bidimensionnels et application 
               au denombrement des relations binaires connexes. C. R. Acad. Sci. 
               Paris Ser. A-B 268 1969 A577-A579.
%D A001047 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]
%H A001047 T. D. Noe, <a href="b001047.txt">Table of n, a(n) for n=0..200</a>
%H A001047 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001047 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=397">
               Encyclopedia of Combinatorial Structures 397</a>
%H A001047 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/collatz/
               collatz.htm">Relation to Collatz problem</a>
%H A001047 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 A001047 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.
%F A001047 G.f.: x/(1-2x)(1-3x). a(n) = 5 a(n-1) - 6 a(n-2).
%F A001047 a(n) = 3*a(n-1) + 2^(n-1). - Jon Perry, Aug 23, 2002
%F A001047 Starting 0, 0, 1, 5, 19, .. this is 3^n/3-2^n/2+0^n/6, the binomial transform 
               of A086218. - Paul Barry (pbarry(AT)wit.ie), Aug 18 2003
%F A001047 Binomial transform of A000225. - Ross La Haye (rlahaye(AT)new.rr.com), 
               Feb 07 2005
%F A001047 a(n) = Sum[C(n, k)2^k, {k, 0, n-1}] - Ross La Haye (rlahaye(AT)new.rr.com), 
               Aug 20 2005
%F A001047 a(n) = 2^(2n) - A083324(n). - Ross La Haye (rlahaye(AT)new.rr.com), Sep 
               10 2005
%F A001047 E.g.f.: e^(3*x)-e^(2*x). [From Mohammad K. Azarian (azarian(AT)evansville.edu), 
               Jan 14 2009]
%p A001047 a(n)=seq(sum(2^i*3^(n-i),i=0..n),n=0..40); - Giorgio Balzarotti (Greenblue(AT)tiscali.it), 
               Nov 18 2006
%p A001047 with(combinat):a:=n->stirling2(n,3)-stirling2(n-1,3): seq(a(n), n=2..27); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
%p A001047 A001047:=1/(3*z-1)/(2*z-1); [S. Plouffe in his 1992 dissertation, dropping 
               the initial zero.]
%p A001047 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=5*a[n-1]-6*a[n-2]od: seq(a[n], 
               n=0..33);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 
               14 2008]
%p A001047 seq(add(2^(n-k)*binomial(n,k),k=1..n),n=0..25);# [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Apr 18 2009]
%t A001047 Table[ 3^n - 2^n, {n, 0, 25} ]
%o A001047 (Python) [3^n - 2^n for n in range(25)] - Ross La Haye (rlahaye(AT)new.rr.com), 
               Aug 19 2005
%o A001047 (Other) sage: [lucas_number1(n,5,6) for n in xrange(0, 26)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%Y A001047 Cf. A000225, A016189, A036561.
%Y A001047 a(n) = row sums of A091913, row 2 of A047969, column 1 of A090888 and 
               column 1 of A038719.
%Y A001047 Cf. A097936, A038876, A127071, A127072, A127073, A127074, A002997, A057468.
%Y A001047 Sequence in context: A049612 A001870 A025568 this_sequence A099448 A124806 
               A059509
%Y A001047 Adjacent sequences: A001044 A001045 A001046 this_sequence A001048 A001049 
               A001050
%K A001047 nonn,easy,nice
%O A001047 0,3
%A A001047 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A001047 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000
%E A001047 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009