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COMMENTS
| 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 (vladeta(AT)eunet.rs), Feb 14 2001
3^a(n) is the highest power of 3 dividing (3^n)! - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (benoit7848c(AT)orange.fr), 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 (pbarry(AT)wit.ie), Apr 11 2003
With leading zero, inverse binomial transform of A006095. - Paul Barry (pbarry(AT)wit.ie), 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 (deutsch(AT)duke.poly.edu), 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 (kociemba(AT)t-online.de), 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 Delarte (alonso.delarte(AT)gmail.com), 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 (alex(AT)kolmogorov.com), 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
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 28 2009: (Start)
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,...). (End)
Contribution from Nishant Shukla (n.shukla722(AT)gmail.com), Jul 11 2009: (Start)
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. (End)
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 (layman(AT)math.vt.edu), Jul 29 2009]
Subsequence of A134025; A171960(a(n)) = a(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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 R. Janjic (agnus(AT)blic.net), 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 (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 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)
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REFERENCES
| M. A. Alekseyev and T. Berger, On the expected number of random moves to solve the Tower of Hanoi puzzle, Preprint, 2008.
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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