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COMMENTS
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Partial sums of A000244. Values of base 3 strings of 1's.
a(n) = (3^n-1)/2 is also the number of different nonparallel lines determined by pair of vertices in the n dimensional hypercube. Example: when n = 2 the square has 4 vertices and then the relevant lines are: x = 0, y = 0, x = 1, y = 1, y = x, y = 1-x and when we identify parallel lines only 4 remain: x = 0, y = 0, y = x, y = 1-x so a(2) = 4 - Noam Katz (noamkj(AT)hotmail.com), Feb 11 2001
Also number of 3-block bicoverings of an n-set (if offset is 1, cf. A059443) - Vladeta Jovovic, Feb 14 2001
3^a(n) is the highest power of 3 dividing (3^n)! - Benoit Cloitre, Feb 04 2002
Apart from a(0) term, maximum number of coins among which a lighter or heavier counterfeit coin can be detected by n weighings. - Tom Verhoeff (Tom.Verhoeff(AT)acm.org), Jun 22 2002
n such that A001764(n) is not divisible by 3 - Benoit Cloitre, Jan 14 2003
Consider the mapping f(a/b) = (a + 2b)/(2a + b). Taking a = 1 b = 2 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/2,4/5,13/14,40/41,... converging to 1. Sequence contains the numerators = (3^n-1)/2. The same mapping for N i.e. f(a/b) = (a + Nb)/(a+b) gives fractions converging to N^(1/2). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
Binomial transform of A000079 (with leading zero). - Paul Barry, Apr 11 2003
With leading zero, inverse binomial transform of A006095. - Paul Barry, Aug 19 2003
Number of walks of length 2*n+2 in the path graph P_5 from one end to the other one. Example: a(2)=4 because in the path ABCDE we have ABABCDE, ABCBCDE, ABCDCDE and ABCDEDE. - Emeric Deutsch, Apr 02 2004
The number of triangles of all sizes (not counting holes) in Sierpinski's triangle after n inscriptions. - Lee Reeves (leereeves(AT)fastmail.fm), May 10 2004
Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2*n+1, s(0) = 1, s(2n+1) = 4. - Herbert Kociemba, Jun 10 2004
Number of non-degenerate right-angled incongruent integer-edged Heron triangles whose circumdiameter is the product of n distinct primes of shape 4k + 1. - Alex Fink and R. K. Guy, Aug 18 2005
Also numerator of the sum of the reciprocals of the first n powers of 3, with A000244 being the sequence of denominators. With the exception of n < 2, the base 10 digital root of a(n) is always 4. In base 3 the digital root of a(n) is the same as the digital root of n. - Alonso del Arte, Jan 24 2006
The sequence 3*a(n), n>=1, gives the number of edges of the Hanoi graph H_3^{n} with 3 pegs and n>=1 discs. - Daniele Parisse (daniele.parisse(AT)eads.com), Jul 28 2006
Numbers n such that a(n) is prime are listed in A028491 = {3,7,13,71,103,541,1091,...}. 2^(m+1) divides a(2^m*k) for m>0. 5 divides a(4k). 5^2 divides a(20k). 7 divides a(6k). 7^2 divides a(42k). 11^2 divides a(5k). 13 divides a(3k). 17 divides a(16k). 19 divides a(18k). 1093 divides a(7k). 41 divides a(8k). p divides a((p-1)/5) for prime p = {41,431,491,661,761,1021,1051,1091,1171,...}. p divides a((p-1)/4) for prime p = {13,109,181,193,229,277,313,421,433,541,...}. p divides a((p-1)/3) for prime p = {61,67,73,103,151,193,271,307,367,...} = A014753, 3 and -3 are both cubes (one implies other) mod these primes p=1 mod 6. p divides a((p-1)/2) for prime p = {11,13,23,37,47,59,61,71,73,83,97,...} = A097933(n). p divides a(p-1) for prime p>7. p^2 divides a(p*(p-1)k) for all prime p except p = 3. p^3 divides a(p*(p-1)*(p-2)k) for prime p = 11. - Alexander Adamchuk, Jan 22 2007
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which x and y are disjoint. Wieder calls these "disjoint usual 2-combinations". - Ross La Haye (rlahaye(AT)new.rr.com), Jan 10 2008
Starting with offset 1 = binomial transform of A003945: (1, 3, 6, 12, 24,...) and double bt of (1, 2, 1, 2, 1, 2,...); equals polcoeff inverse of (1, -4, 3, 0, 0, 0,...). [From Gary W. Adamson, May 28 2009]
Also the constant of the polynomials C(x)=3x+1 that form a sequence by performing this operation repeatedly and taking the result at each step as the input at the next. [From Nishant Shukla (n.shukla722(AT)gmail.com), Jul 11 2009]
It appears that this is A120444(3^n-1) = A004125(3^n) - A004125(3^n-1), where A004125 is the sum of remainders of n mod k for k=1,2,3,...,n. [From John W. Layman, Jul 29 2009]
Subsequence of A134025; A171960(a(n)) = a(n). [From Reinhard Zumkeller, Jan 20 2010]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=3, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). [From Milan Janjic, Jan 27 2010]
This is the sequence A(0,1;2,3;2) = A(0,1;4,-3;0) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. [From Wolfdieter Lang, Oct 18 2010]
It appears that if s(n) is a first order rational sequence of the form s(0)=0, s(n)= (2*s(n-1)+1)/(s(n-1)+2), n>0, then s(n)= a(n)/(a(n)+1). [From Gary Detlefs, Nov 16 2010]
This sequence also describes the total number of moves solving the [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle (Cf. A183111 - A183125).
From Adi Dani, Jun 8 2011: (Start)
a(n) is number of compositions of odd numbers into n parts <3.
For example, a(3)=13 and there are 13 compositions odd numbers into 3 parts <3:
1: (0,0,1),(0,1,0),(1,0,0);
3: (0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1),(2,1,0),(1,1,1);
5: (1,2,2),(2,1,2),(2,2,1). (End)
It appears that a(n) is the number of unordered pairs of disjoint subsets of {1,2,...,n}. [John W. Layman, Mar 07 2012]
For n > 1: A008344(a(n)) = 0. [Reinhard Zumkeller, May 09 2012]
Pisano period lengths: 1, 2, 1, 2, 4, 2, 6, 4, 1, 4, 5, 2, 3, 6, 4, 8, 16, 2, 18, 4,... - R. J. Mathar, Aug 10 2012
A085059(a(n)) = 1 for n > 0. - Reinhard Zumkeller, Jan 31 2013
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REFERENCES
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Sung-Hyuk Cha, On Complete and Size Balanced k-ary Tree Integer Sequences, INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND INFORMATICS, Issue 2, Volume 6, 2012, pp. 67-75; http://naun.org/multimedia/UPress/ami/16-125.pdf. - From N. J. A. Sloane, Dec 24 2012
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
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]
J. G. Mauldon, Strong solutions for the counterfeit coin problem. IBM Research Report RC 7476 (#31437) 9/15/78, IBM Thomas J. Watson Research Center, P. O. Box 218, Yorktown Heights, N. Y. 10598
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 60.
P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 53.
R. Sedgewick, Algorithms, 1992, pp. 109.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
K. Zsigmondy, Zur Theorie der Potenreste, Monatsh. Math., 3 (1892), 265-284.
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