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%I M3463 #390 Jan 31 2024 09:10:20
%S 0,1,4,13,40,121,364,1093,3280,9841,29524,88573,265720,797161,2391484,
%T 7174453,21523360,64570081,193710244,581130733,1743392200,5230176601,
%U 15690529804,47071589413,141214768240,423644304721,1270932914164
%N a(n) = (3^n - 1)/2.
%C Partial sums of A000244. Values of base 3 strings of 1's.
%C 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 Also number of 3-block bicoverings of an n-set (if offset is 1, cf. A059443). - _Vladeta Jovovic_, Feb 14 2001
%C 3^a(n) is the highest power of 3 dividing (3^n)!. - _Benoit Cloitre_, Feb 04 2002
%C Apart from the a(0) and a(1) terms, maximum number of coins among which a lighter or heavier counterfeit coin can be identified (but not necessarily labeled as heavier or lighter) by n weighings. - _Tom Verhoeff_, Jun 22 2002, updated Mar 23 2017
%C n such that A001764(n) is not divisible by 3. - _Benoit Cloitre_, Jan 14 2003
%C 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 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_, Mar 22 2003
%C Binomial transform of A000079 (with leading zero). - _Paul Barry_, Apr 11 2003
%C With leading zero, inverse binomial transform of A006095. - _Paul Barry_, Aug 19 2003
%C 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
%C The number of triangles of all sizes (not counting holes) in Sierpiński's triangle after n inscriptions. - Lee Reeves (leereeves(AT)fastmail.fm), May 10 2004
%C 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
%C 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 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
%C 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_, Jul 28 2006
%C 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
%C Let P(A) be the power set of an n-element set A. Then a(n) = the number of [unordered] pairs of elements {x,y} of P(A) for which x and y are disjoint [and both nonempty]. Wieder calls these "disjoint usual 2-combinations". - _Ross La Haye_, Jan 10 2008 [This is because each of the elements of {1, 2, ..., n} can be in the first subset, in the second or in neither. Because there are three options for each, the total number of options is 3^n. However, since the sets being empty is not an option we subtract 1 and since the subsets are unordered we then divide by 2! (The number of ways two objects can be arranged.) Thus we obtain (3^n-1)/2 = a(n). - _Chayim Lowen_, Mar 03 2015]
%C Also, still with P(A) being the power set of a n-element set A, a(n) is the number of 2-element subsets {x,y} of P(A) such that the union of x and y is equal to A. Cf. A341590. - _Fabio Visonà_, Feb 20 2021
%C 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, ...). - _Gary W. Adamson_, May 28 2009
%C 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. - Nishant Shukla (n.shukla722(AT)gmail.com), Jul 11 2009
%C 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. - _John W. Layman_, Jul 29 2009
%C Subsequence of A134025; A171960(a(n)) = a(n). - _Reinhard Zumkeller_, Jan 20 2010
%C 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). - _Milan Janjic_, Jan 27 2010
%C 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 Gary Detlefs, and treated as A(a, b; c, d; k) in the Wolfdieter Lang link given below. - _Wolfdieter Lang_, Oct 18 2010
%C 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). - _Gary Detlefs_, Nov 16 2010
%C This sequence also describes the total number of moves to solve the [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Towers of Hanoi puzzle (cf. A183111 - A183125).
%C From _Adi Dani_, Jun 08 2011: (Start)
%C a(n) is number of compositions of odd numbers into n parts less than 3. For example, a(3) = 13 and there are 13 compositions odd numbers into 3 parts < 3:
%C 1: (0, 0, 1), (0, 1, 0), (1, 0, 0);
%C 3: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0), (1, 1, 1);
%C 5: (1, 2, 2), (2, 1, 2), (2, 2, 1).
%C (End)
%C 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
%C a(n) is the total number of holes (triangles removed) after the n-th step of a Sierpiński triangle production. - _Ivan N. Ianakiev_, Oct 29 2013
%C a(n) solves Sum_{j = a(n) + 1 .. a(n+1)} j = k^2 for some integer k, given a(0) = 0 and requiring smallest a(n+1) > a(n). Corresponding k = 3^n. - _Richard R. Forberg_, Mar 11 2015
%C a(n+1) equals the number of words of length n over {0, 1, 2, 3} avoiding 01, 02 and 03. - _Milan Janjic_, Dec 17 2015
%C For n >= 1, a(n) is also the total number of words of length n, over an alphabet of three letters, such that one of the letters appears an odd number of times (See A006516 for 4 letter words, and the Balakrishnan reference there). - _Wolfdieter Lang_, Jul 16 2017
%C Also, the number of maximal cliques, maximum cliques, and cliques of size 4 in the n-Apollonian network. - _Andrew Howroyd_, Sep 02 2017
%C For n > 1, the number of triangles (cliques of size 3) in the (n-1)-Apollonian network. - _Andrew Howroyd_, Sep 02 2017
%C a(n) is the largest number that can be represented with n trits in balanced ternary. Correspondingly, -a(n) is the smallest number that can be represented with n trits in balanced ternary. - _Thomas König_, Apr 26 2020
%C These form Sierpinski nesting-stars, which alternate pattern on 3^n+1/2 star numbers A003154, based on the square configurations of 9^n. The partial sums of 3^n are delineated according to the geometry of a hexagram, see illustrations in links. (3*a(n-1) + 1) create Sierpinski-anti-triangles, representing the number of holes in a (n+1) Sierpinski triangle (see illustrations). - _John Elias_, Oct 18 2021
%C For n > 1, a(n) is the number of iterations necessary to calculate the hyperbolic functions with CORDIC. - _Mathias Zechmeister_, Jul 26 2022
%C a(n) is the least number k such that A065363(k) = n. - _Amiram Eldar_, Sep 03 2022
%C For all n >= 0, Sum_{k=a(n)+1..a(n+1)} 1/k < Sum_{j=a(n+1)+1..a(n+2)} 1/j. These are the minimal points which partition the infinite harmonic series into a monotonically increasing sequence. Each partition approximates log(3) from below as n tends to infinity. - _Joseph Wheat_, Apr 15 2023
%C a(n) is also the number of 3-cycles in the n-Dorogovtsev-Goltsev-Mendes graph (using the convention the 0-Dorogovtsev-Goltsev-Mendes graph is P_2). - _Eric W. Weisstein_, Dec 06 2023
%D 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 Paulo Ribenboim, The Book of Prime Number Records, Springer-Verlag, NY, 2nd ed., 1989, p. 60.
%D Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 53.
%D Amir Sapir, The Tower of Hanoi with Forbidden Moves, The Computer J. 47 (1) (2004) 20, case three-in-a row, sequence a(n).
%D Robert Sedgewick, Algorithms, 1992, pp. 109.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Paolo Xausa, <a href="/A003462/b003462.txt">Table of n, a(n) for n = 0..2000</a> (terms 0..200 from T. D. Noe)
%H A. Abdurrahman, <a href="https://arxiv.org/abs/1909.10889">CM Method and Expansion of Numbers</a>, arXiv:1909.10889 [math.NT], 2019.
%H Max A. Alekseyev and Toby Berger, <a href="http://arxiv.org/abs/1304.3780">Solving the Tower of Hanoi with Random Moves</a>. In: J. Beineke, J. Rosenhouse (eds.) The Mathematics of Various Entertaining Subjects: Research in Recreational Math, Princeton University Press, 2016, pp. 65-79. ISBN 978-0-691-16403-8
%H Arcytech, <a href="http://web.archive.org/web/20100418132731/http://www.arcytech.org/java/fractals/sierpinski.shtml">The Sierpinski Triangle Fractal</a>.
%H Jean-Luc Baril and Helmut Prodinger, <a href="https://arxiv.org/abs/2205.01383">Enumeration of partial Lukasiewicz paths</a>, arXiv:2205.01383 [math.CO], 2022.
%H Beáta Bényi and Toshiki Matsusaka, <a href="https://arxiv.org/abs/2106.05585">Extensions of the combinatorics of poly-Bernoulli numbers</a>, arXiv:2106.05585 [math.CO], 2021.
%H Göksal Bilgici and Tuncay Deniz Şentürk, <a href="https://doi.org/10.2478/amsil-2019-0005">Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers</a>, Annales Mathematicae Silesianae (2019) Vol. 33, 55-65.
%H Sung-Hyuk Cha, <a href="http://naun.org/multimedia/UPress/ami/16-125.pdf">On Complete and Size Balanced k-ary Tree Integer Sequences</a>, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75. - From _N. J. A. Sloane_, Dec 24 2012
%H Nachum Dershowitz, <a href="https://arxiv.org/abs/2006.06516">Between Broadway and the Hudson: A Bijection of Corridor Paths</a>, arXiv:2006.06516 [math.CO], 2020.
%H Roger B. Eggleton, <a href="http://dx.doi.org/10.1155/2015/216475">Maximal Midpoint-Free Subsets of Integers</a>, International Journal of Combinatorics Volume 2015, Article ID 216475, 14 pages.
%H John Elias, <a href="/A003462/a003462.png">Sierpinski Nesting Stars</a>, <a href="/A003462/a003462_1.png">Stars of 3*9^n-1/2</a>, <a href="/A003462/a003462_2.png">Stars of 9^n-1/2</a>, <a href="/A003462/a003462.jpg">Sierpinski Anti-Triangles</a>.
%H Graham Everest, Shaun Stevens, Duncan Tamsett and Tom Ward, <a href="http://www.jstor.org/stable/27642221">Primes generated by recurrence sequences</a>, Amer. Math. Monthly, Vol. 114, No. 5 (2007), pp. 417-431.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=372">Encyclopedia of Combinatorial Structures 372</a>.
%H Takao Komatsu, <a href="https://doi.org/10.22436/jnsa.012.12.05">Some recurrence relations of poly-Cauchy numbers</a>, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
%H G. Kreweras, <a href="/A000108/a000108_1.pdf">Sur les éventails de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
%H H. V. Krishna and N. J. A. Sloane, <a href="/A003462/a003462_2.pdf">Correspondence, 1975</a>.
%H Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H Wolfdieter Lang, <a href="/A003462/a003462.pdf">Notes on certain inhomogeneous three term recurrences</a>.
%H László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Nemeth/nemeth6.html">The trinomial transform triangle</a>, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also <a href="https://arxiv.org/abs/1807.07109">arXiv:1807.07109</a> [math.NT], 2018.
%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Yash Puri and Thomas Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A. G. Shannon, <a href="/A003462/a003462_1.pdf">Letter to N. J. A. Sloane, Dec 06 1974</a>.
%H Morgan Ward, <a href="http://dx.doi.org/10.1090/S0002-9904-1936-06435-9">Note on divisibility sequences</a>, Bull. Amer. Math. Soc., 42 (1936), 843-845.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ApollonianNetwork.html">Apollonian Network</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Dorogovtsev-Goltsev-MendesGraph.html">Dorogovtsev-Goltsev-Mendes Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximumClique.html">Maximum Clique</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MephistoWaltzSequence.html">Mephisto Waltz Sequence</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Weighing.html">Weighing</a>.
%H Thomas Wieder, The number of certain k-combinations of an n-set, <a href="http://www.math.nthu.edu.tw/~amen/2008/070301.pdf">Applied Mathematics Electronic Notes</a>, Vol. 8 (2008).
%H K. Zsigmondy, <a href="https://doi.org/10.1007%2FBF01692444">Zur Theorie der Potenzreste</a>, Monatsh. Math., Vol. 3 (1892), 265-284.
%H <a href="/index/So#sorting">Index entries for sequences related to sorting</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3)
%F G.f.: x/((1-x)*(1-3*x)).
%F a(n) = 4*a(n-1) - 3*a(n-2), n > 1. a(0) = 0, a(1) = 1.
%F a(n) = 3*a(n-1) + 1, a(0) = 0.
%F E.g.f.: (exp(3*x) - exp(x))/2. - _Paul Barry_, Apr 11 2003
%F a(n+1) = Sum_{k = 0..n} binomial(n+1, k+1)*2^k. - _Paul Barry_, Aug 20 2004
%F a(n) = Sum_{i = 0..n-1} 3^i, for n > 0; a(0) = 0.
%F a(n) = A125118(n, 2) for n > 1. - _Reinhard Zumkeller_, Nov 21 2006
%F a(n) = StirlingS2(n+1, 3) + StirlingS2(n+1, 2). - _Ross La Haye_, Jan 10 2008
%F a(n) = Sum_{k = 0..n} A106566(n, k)*A106233(k). - _Philippe Deléham_, Oct 30 2008
%F a(n) = 2*a(n-1) + 3*a(n-2) + 2, n > 1. - _Gary Detlefs_, Jun 21 2010
%F a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3), a(0) = 0, a(1) = 1, a(2) = 4. Observation by G. Detlefs. See the W. Lang comment and link. - _Wolfdieter Lang_, Oct 18 2010
%F A008344(a(n)) = 0, for n > 1. - _Reinhard Zumkeller_, May 09 2012
%F A085059(a(n)) = 1 for n > 0. - _Reinhard Zumkeller_, Jan 31 2013
%F G.f.: Q(0)/2 where Q(k) = 1 - 1/(9^k - 3*x*81^k/(3*x*9^k - 1/(1 - 1/(3*9^k - 27*x*81^k/(9*x*9^k - 1/Q(k+1)))))); (continued fraction ). - _Sergei N. Gladkovskii_, Apr 12 2013
%F a(n) = A001065(3^n) where A001065(m) is the sum of the proper divisors of m for positive integer m. - _Chayim Lowen_, Mar 03 2015
%F a(n) = A000244(n) - A007051(n) = A007051(n)-1. - _Yuchun Ji_, Oct 23 2018
%F Sum_{n>=1} 1/a(n) = A321872. - _Amiram Eldar_, Nov 18 2020
%e 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 Ternary........Decimal
%e 0.................0
%e 1.................1
%e 11................4
%e 111..............13
%e 1111.............40 etc. - _Zerinvary Lajos_, Jan 14 2007
%e There are altogether a(3) = 13 three letter words over {A,B,C} with say, A, appearing an odd number of times: AAA; ABC, ACB, ABB, ACC; BAC, CAB, BAB, CAC; BCA, CBA, BBA, CCA. - _Wolfdieter Lang_, Jul 16 2017
%p A003462 := n-> (3^n - 1)/2: seq(A003462(n), n=0..30);
%p A003462:=1/(3*z-1)/(z-1); # _Simon Plouffe_ in his 1992 dissertation
%t (3^Range[0, 30] - 1)/2 (* _Harvey P. Dale_, Jul 13 2011 *)
%t LinearRecurrence[{4, -3}, {0, 1}, 30] (* _Harvey P. Dale_, Jul 13 2011 *)
%t Accumulate[3^Range[0, 30]] (* _Alonso del Arte_, Sep 10 2017 *)
%t CoefficientList[Series[x/(1 - 4x + 3x^2), {x, 0, 30}], x] (* _Eric W. Weisstein_, Sep 28 2017 *)
%t Table[FromDigits[PadRight[{},n,1],3],{n,0,30}] (* _Harvey P. Dale_, Jun 01 2022 *)
%o (PARI) a(n)=(3^n-1)/2
%o (Sage) [(3^n - 1)/2 for n in range(0,30)] # _Zerinvary Lajos_, Jun 05 2009
%o (Haskell)
%o a003462 = (`div` 2) . (subtract 1) . (3 ^)
%o a003462_list = iterate ((+ 1) . (* 3)) 0 -- _Reinhard Zumkeller_, May 09 2012
%o (Maxima) A003462(n):=(3^n - 1)/2$
%o makelist(A003462(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o (Magma) [(3^n-1)/2: n in [0..30]]; // _Vincenzo Librandi_, Feb 21 2015
%o (PARI) concat(0, Vec(x/((1-x)*(1-3*x)) + O(x^30))) \\ _Altug Alkan_, Nov 01 2015
%o (GAP)
%o A003462:=List([0..30],n->(3^n-1)/2); # _Muniru A Asiru_, Sep 27 2017
%Y Sequences used for Shell sort: A033622, A003462, A036562, A036564, A036569, A055875.
%Y Cf. A179526 (repeats), A113047 (characteristic function).
%Y Cf. A000225, A000392, A004125, A014753, A028491 (indices of primes), A059443 (column k = 3), A065363, A097933, A120444, A321872 (sum reciprocals).
%Y Cf. A064099 (minimal number of weightings to detect lighter or heavier coin among n coins).
%Y Cf. A039755 (column k = 1).
%Y Cf. A006516 (binomial transform, and special 4 letter words).
%Y Cf. A341590.
%Y Cf. A003462(n) (3-cycles), A367967(n) (5-cycles), A367968(n) (6-cycles).
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Michael Somos_
%E Corrected my comment of Jan 10 2008. - _Ross La Haye_, Oct 29 2008
%E Removed comment that duplicated a formula. - _Joerg Arndt_, Mar 11 2010
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