Search: id:A003462 Results 1-1 of 1 results found. %I A003462 M3463 %S A003462 0,1,4,13,40,121,364,1093,3280,9841,29524,88573,265720,797161,2391484, %T A003462 7174453,21523360,64570081,193710244,581130733,1743392200,5230176601, %U A003462 15690529804,47071589413,141214768240,423644304721,1270932914164 %N A003462 (3^n - 1)/2. %C A003462 Partial sums of A000244. Values of base 3 strings of 1's. %C A003462 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 %C A003462 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 %C A003462 3^a(n) is the highest power of 3 dividing (3^n)! - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 04 2002 %C A003462 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 %C A003462 n such that A001764(n) is not divisible by 3 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 14 2003 %C A003462 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 %C A003462 Binomial transform of A000079 (with leading zero). - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003 %C A003462 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 %C A003462 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 %C A003462 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 %C A003462 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 %C A003462 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 %C A003462 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 %C A003462 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 %C A003462 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 %C A003462 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 28 2009: (Start) %C A003462 Starting with offset 1 = binomial transform of A003945: (1, 3, 6, 12, 24,...) %C A003462 and double bt of (1, 2, 1, 2, 1, 2,...). %C A003462 Equals polcoeff inverse of (1, -4, 3, 0, 0, 0,...). (End) %C A003462 Contribution from Nishant Shukla (n.shukla722(AT)gmail.com), Jul 11 2009: (Start) %C A003462 Also the constant of the polynomials C(x)=3x+1 that form a sequence by performing %C A003462 this operation repeatedly and taking the result at each step as the input at the next. (End) %C A003462 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] %C A003462 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] %D A003462 M. A. Alekseyev and T. Berger, On the expected number of random moves to solve the Tower of Hanoi puzzle, Preprint, 2008. %D A003462 G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431. %D A003462 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] %D A003462 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 %D A003462 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 60. %D A003462 P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 53. %D A003462 R. Sedgewick, Algorithms, 1992, pp. 109. %D A003462 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A003462 K. Zsigmondy, Zur Theorie der Potenreste, Monatsh. Math., 3 (1892), 265-284. %H A003462 T. D. Noe, Table of n, a(n) for n=0..200 %H A003462 Arcytech, The Sierpinski Triangle Fractal %H A003462 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 372 %H A003462 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. %H A003462 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A003462 Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. %H A003462 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A003462 Eric Weisstein's World of Mathematics, Mephisto Waltz Sequence %H A003462 Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008). %H A003462 G. Xiao, Sigma Server, Operate on "3^n" %H A003462 Index entries for sequences related to sorting %H A003462 Index entries for sequences related to linear recurrences with constant coefficients %F A003462 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. %F A003462 a(n)=3a(n-1)+1, a(0)=0 %F A003462 E.g.f. (exp(3x)-exp(x))/2 - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003 %F A003462 With leading zero, inverse binomial transform of A006095. - Paul Barry (pbarry(AT)wit.ie), Aug 19 2003 %F A003462 a(n+1)=sum{k=0..n, binomial(n+1, k+1)2^k} - Paul Barry (pbarry(AT)wit.ie), Aug 20 2004 %F A003462 a(0)=0, a(n)=Sum_{i = 0 to n-1} 3^i for n>0. %F A003462 a(n) = A125118(n,2) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 21 2006 %F A003462 a(n) = StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye (rlahaye(AT)new.rr.com), Jan 10 2008 %F A003462 a(n)=Sum_{k, 0<=k<=n} A106566(n,k)*A106233(k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2008] %e A003462 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}}. %e A003462 Ternary......decimal (comment from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 14 2007): %e A003462 0.................0 %e A003462 1.................1 %e A003462 11................4 %e A003462 111..............13 %e A003462 1111.............40 %e A003462 11111...........121 %e A003462 111111..........364 %e A003462 1111111........1093 %e A003462 11111111.......3280 %e A003462 111111111......9841 %e A003462 1111111111....29524 %e A003462 etc...........etc. %p A003462 A003462 := n-> (3^n - 1)/2; %p A003462 a:=n->sum(3^(n-j),j=1..n): seq(a(n), n=0..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 04 2007 %p A003462 with(combinat):seq(stirling2(n,2)+stirling2(n,3), n=1..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 04 2007 %p A003462 A003462:=1/(3*z-1)/(z-1); [S. Plouffe in his 1992 dissertation.] %p A003462 with(finance):seq(add(futurevalue( 1, 2, k),k=0..n),n=-1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2008 %p A003462 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] %t A003462 lst={};Do[p=(3^n-1)/2;AppendTo[lst, p], {n, 0, 5!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 29 2008] %t A003462 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] %o A003462 (PARI) a(n)=(3^n-1)/2 %o A003462 (Other) sage: [lucas_number1(n,4,3) for n in xrange(0, 27)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009] %o A003462 (Other) sage: [gaussian_binomial(n,1,3) for n in xrange(0,27)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009] %o A003462 (Other) sage: [(3^n - 1)/2 for n in xrange(0,27)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2009] %Y A003462 Sequences used for Shell sort: A003462, A033622, A036562, A036564, A036569, A055875. %Y A003462 Cf. A002718, A059443, A059945-A059951. %Y A003462 Cf. A064099 (minimal number of weighings to detect lighter or heavier coin among n coins). %Y A003462 Cf. A028491, A014753, A097933. %Y A003462 Cf. A000225, A000392. %Y A003462 Cf. A004125, A120444 [From John W. Layman (layman(AT)math.vt.edu), Jul 29 2009] %Y A003462 Sequence in context: A027121 A025567 A076040 this_sequence A091141 A098183 A094628 %Y A003462 Adjacent sequences: A003459 A003460 A003461 this_sequence A003463 A003464 A003465 %K A003462 nonn,easy,nice,new %O A003462 0,3 %A A003462 N. J. A. Sloane (njas(AT)research.att.com). %E A003462 More terms from Michael Somos %E A003462 Corrected my comment of Jan 10 2008. - Ross La Haye (rlahaye(AT)new.rr.com), Oct 29 2008 Search completed in 0.003 seconds