0,3

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

Number of walks of length 2n+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,....,2n+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]

Except for the first term, a(n)=3*a(n-1)+1 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 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]

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.

T. D. Noe, Table of n, a(n) for n=0..200

Arcytech, The Sierpinski Triangle Fractal

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 372

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, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Mephisto Waltz Sequence

Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).

G. Xiao, Sigma Server, Operate on "3^n"

Index entries for sequences related to sorting

Index entries for sequences related to linear recurrences with constant coefficients

G.f.: x/((1-x)*(1-3*x)). a(n)=4*a(n-1)-3*a(n-2), n>1. a(0)=0, a(1)=1.

a(n)=3a(n-1)+1, a(0)=0

E.g.f. (exp(3x)-exp(x))/2 - 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

a(n+1)=sum{k=0..n, binomial(n+1, k+1)2^k} - Paul Barry (pbarry(AT)wit.ie), Aug 20 2004

a(0)=0, a(n)=Sum_{i = 0 to n-1} 3^i for n>0.

a(n) = A125118(n,2) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 21 2006

a(n) = StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye (rlahaye(AT)new.rr.com), Jan 10 2008

a(n)=Sum_{k, 0<=k<=n} A106566(n,k)*A106233(k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2008]

There are 4 3-block bicoverings of a 3-set: {{1, 2, 3}, {1, 2}, {3}}, {{1, 2, 3}, {1, 3}, {2}}, {{1, 2, 3}, {1}, {2, 3}} and {{1, 2}, {1, 3}, {2, 3}}.

Ternary......decimal (comment from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 14 2007):

0.................0

1.................1

11................4

111..............13

1111.............40

11111...........121

111111..........364

1111111........1093

11111111.......3280

111111111......9841

1111111111....29524

etc...........etc.

A003462 := n-> (3^n - 1)/2;

a:=n->sum(3^(n-j), j=1..n): seq(a(n), n=0..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 04 2007

with(combinat):seq(stirling2(n, 2)+stirling2(n, 3), n=1..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 04 2007

A003462:=1/(3*z-1)/(z-1); [S. Plouffe in his 1992 dissertation.]

with(finance):seq(add(futurevalue( 1, 2, k), k=0..n), n=-1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2008

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]+2 od: seq(a[n], n=0..33); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2008]

lst={}; Do[p=(3^n-1)/2; AppendTo[lst, p], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 29 2008]

a=0; lst={a}; Do[a=a*3+1; AppendTo[lst, a], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 25 2008]

(PARI) a(n)=(3^n-1)/2

(Other) sage: [lucas_number1(n, 4, 3) for n in xrange(0, 27)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

(Other) sage: [gaussian_binomial(n, 1, 3) for n in xrange(0, 27)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]

(Other) sage: [(3^n - 1)/2 for n in xrange(0, 27)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2009]

Sequences used for Shell sort: A003462, A033622, A036562, A036564, A036569, A055875.

Cf. A002718, A059443, A059945-A059951.

Cf. A064099 (minimal number of weighings to detect lighter or heavier coin among n coins).

Cf. A028491, A014753, A097933.

Cf. A000225, A000392.

Cf. A004125, A120444 [From John W. Layman (layman(AT)math.vt.edu), Jul 29 2009]

Sequence in context: A027121 A025567 A076040 this_sequence A091141 A098183 A171556

Adjacent sequences: A003459 A003460 A003461 this_sequence A003463 A003464 A003465

nonn,easy,nice

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

More terms from Michael Somos

Corrected my comment of Jan 10 2008. - Ross La Haye (rlahaye(AT)new.rr.com), Oct 29 2008