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 A001651 Numbers not divisible by 3. (Formerly M0957 N0357) 142
 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Inverse binomial transform of A084858. - Benoit Cloitre, Jun 12 2003 Earliest monotonic sequence starting with (1,2) and satisfying the condition: "a(n)+a(n-1) is not in the sequence." - Benoit Cloitre, Mar 25 2004. [The numbers of the form a(n)+a(n-1) form precisely the complement with respect to the positive integers. - David W. Wilson, Feb 18 2012] a(1) = 1; a(n) is least number which is relatively prime to the sum of all the previous terms. - Amarnath Murthy, Jun 18 2001 For n > 3, numbers having 3 as an anti-divisor. - Alexandre Wajnberg, Oct 02 2005 A011655(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009 Also numbers n such that (n+1)*(n+2)/6 = A000292(n)/n is an integer. - Ctibor O. Zizka, Oct 15 2010 Notice the property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in this case, a(n)^2-1==0 (mod 3). - Bruno Berselli, Nov 17 2010 A001651 mod 9 gives A141425. - Paul Curtz, Dec 31 2010. (Correct for the modified offset 1. - M. F. Hasler, Apr 07 2015) A215879(a(n)) > 0. - Reinhard Zumkeller, Dec 28 2012 [More precisely, A215879 is the characteristic function of A001651. - M. F. Hasler, Apr 07 2015] The set of natural numbers (1, 2, 3, ...), sequence A000027; represents the numbers of ordered compositions of n using terms in the signed set: (1, 2, -4, -5, 7, 8, -10, -11, 13, 14, ...). This follows from (1, 2, 3, ...) being the INVERT transform of A011655, signed and beginning: (1, 1, 0, -1, -1, 0, 1, 1, 0, ...). - Gary W. Adamson, Apr 28 2013 Union of A047239 and A047257. - Wesley Ivan Hurt, Dec 19 2013 Numbers whose sum of digits (and digital root) is != 0 (mod 3). - Joerg Arndt, Aug 29 2014 The number of partitions of 3*(n-1) into at most 2 parts. - Colin Barker, Apr 22 2015 a(n) is the number of partitions of 3*n into two distinct parts. - L. Edson Jeffery, Jan 14 2017 Conjectured (and like even easily proved) to be the graph bandwidth of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017 Numbers k such that Fibonacci(k) mod 4 = 1 or 3. Equivalently, sequence lists the indices of the odd Fibonacci numbers (see A014437). - Bruno Berselli, Oct 17 2017 Minimum value of n_3 such that the "rectangular spiral pattern" is the optimal solution for Ripà's n_1 X n_2 x n_3 Dots Problem, for any n_1 = n_2. For example, if n_1 = n_2 = 5, n_3 = floor((3/2)*(n_1 - 1)) + 1 = a(5). - Marco Ripà, Jul 23 2018 For n >= 54, a(n) = sat(n, P_n), the minimum number of edges in a P_n-saturated graph on n vertices, where P_n is the n-vertex path (see Dudek, Katona, and Wojda, 2003; Frick and Singleton, 2005). - Danny Rorabaugh, Nov 07 2017 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..10000 L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386. Aneta Dudek, Gyula Y. Katona, and A.Pawel Wojda, m_Path Cover Saturated Graphs, Electronic Notes in Discrete Math., 13 (April 2003), 41-44. Marietjie Frick and Joy Singleton, Lower Bound for the Size of Maximal Nontraceable Graphs, Electron. J. Combin., 12#R32 (2005), 9 pp. A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004. (See Table 5.) G. Ledin, Jr., Is Eratosthenes out?, Fib. Quart., 6 (No. 4, 1968), 261-265. G. P. Michon, Counting Polyhedra Melvyn B. Nathanson, On the fractional parts of roots of positive real numbers, Amer. Math. Monthly, 120 (2013), 409-429 [see p. 417]. M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 200), 559-564, Ex. 2.2. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Marco Ripà, The rectangular spiral or the n1 X n2 X ... X nk Points Problem , Notes on Number Theory and Discrete Mathematics, 2014, 20(1), 59-71. Eric Weisstein's World of Mathematics, Complete Bipartite Graph, Graph Bandwidth, and RATS Sequence. Index entries for linear recurrences with constant coefficients, signature (1,1,-1). FORMULA a(n) = 3 + a(n-2) for n > 2. a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3. a(2*n+1) = 3*n+1, a(2*n) = 3*n-1. G.f.: x * (1 + x + x^2) / ((1 - x) * (1 - x^2)). - Michael Somos, Jun 08 2000 a(n) = (4-n)*a(n-1) + 2*a(n-2) + (n-3)*a(n-3) (from the Carlitz et al. article). a(n) = floor((3*n-1)/2). [Corrected by Gary Detlefs] a(1) = 1, a(n) = 2*a(n-1) - 3*floor(a(n-1)/3). - Benoit Cloitre, Aug 17 2002 a(n+1) = 1 + n - n mod 2 + (n + n mod 2)/2. - Reinhard Zumkeller, Dec 17 2002 a(1) = 1, a(n+1) = a(n) + (a(n) mod 3). - Reinhard Zumkeller, Mar 23 2003 a(1) = 1, a(n) = 3*(n-1) - a(n-1). - Benoit Cloitre, Apr 12 2003 a(n) = 3*(2*n-1)/4 - (-1)^n/4. - Benoit Cloitre, Jun 12 2003 Nearest integer to (Sum_{k>=n} 1/k^3)/(Sum_{k>=n} 1/k^4). - Benoit Cloitre, Jun 12 2003 Partial sums of A040001. a(n) = A032766(n-1)+1. - Paul Barry, Sep 02 2003 a(n) = T(n, 1) = T(n, n-1), where T is the array in A026386. - Emeric Deutsch, Feb 18 2004 a(n) = sqrt(3*A001082(n)+1). - Zak Seidov, Dec 12 2007 a(n) = A077043(n) - A077043(n-1). - Reinhard Zumkeller, Dec 28 2007 a(n) = A001477(n-1) + A008619(n-1). - Yosu Yurramendi, Aug 10 2008 Euler transform of length 3 sequence [2, 1, -1]. - Michael Somos, Sep 06 2008 a(n) = n - 1 + ceiling(n/2). - Michael Somos, Jan 15 2011 a(n) = 3*A000217(n)+1 - 2*Sum_{i=1..n-1} a(i), for n>1. - Bruno Berselli, Nov 17 2010 a(n) = 3*floor(n/2) + (-1)^(n+1). - Gary Detlefs, Dec 29 2011 a(n) = 2n - 1 - floor(n/2). - Wesley Ivan Hurt, Oct 25 2013 a(n) = (3n - 2 + (n mod 2)) / 2. - Wesley Ivan Hurt, Mar 31 2014 a(n) = A000217(n) - A000982(n-1). - Bui Quang Tuan, Mar 28 2015 1/1^3 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + ... = 4 Pi^3/(3 sqrt(3)). - M. F. Hasler, Mar 29 2015 E.g.f.: (4 + sinh(x) - cosh(x) + 3*(2*x - 1)*exp(x))/4. - Ilya Gutkovskiy, May 24 2016 a(n) = a(n+k-1) + a(n-k) - a(n-1) for n > k >= 0. - Bob Selcoe, Feb 03 2017 a(n) = -a(1-n) for all n in Z. - Michael Somos, Jul 31 2018 EXAMPLE G.f.: x + 2*x^2 + 4*x^3 + 5*x^4 + 7*x^5 + 8*x^6 + 10*x^7 + 11*x^8 + 13*x^9 + ... MAPLE A001651 := n -> 3*floor(n/2) - (-1)^n; # Corrected by M. F. Hasler, Apr 07 2015 A001651:=(1+z+z**2)/(z+1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation a[1]:=1:a[2]:=2:for n from 3 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=1..69); # Zerinvary Lajos, Mar 16 2008, offset corrected by M. F. Hasler, Apr 07 2015 MATHEMATICA Select[Table[n, {n, 200}], Mod[#, 3]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *) Drop[Range[200 + 1], {1, -1, 3}] - 1 (* József Konczer, May 24 2016 *) Floor[(3 Range[70] - 1)/2] (* Eric W. Weisstein, Apr 24 2017 *) CoefficientList[Series[(x^2 + x + 1)/((x - 1)^2 (x + 1)), {x, 0, 70}],   x] (* or *) LinearRecurrence[{1, 1, -1}, {1, 2, 4}, 70] (* Robert G. Wilson v, Jul 25 2018 *) PROG (PARI) {a(n) = n + (n-1)\2}; /* Michael Somos, Jan 15 2011 */ (MAGMA) [3*(2*n-1)/4-(-1)^n/4: n in [1..80]]; // Vincenzo Librandi, Jun 07 2011 (Haskell) a001651 = (`div` 2) . (subtract 1) . (* 3) a001651_list = filter ((/= 0) . (`mod` 3)) [1..] -- Reinhard Zumkeller, Jul 07 2012, Aug 23 2011 (PARI) x='x+O('x^100); Vec(x*(1+x+x^2)/((1-x)*(1-x^2))) \\ Altug Alkan, Oct 22 2015 (GAP) Filtered([0..110], n->n mod 3<>0); # Muniru A Asiru, Jul 24 2018 CROSSREFS Differs from A059564 after 35 = a(24) = A059564(24). Cf. A000726, A001082, A003105, A005408 (n=1 or 3 mod 4), A007494, A008585 (complement), A011655, A026386, A032766, A191967, A225227. Sequence in context: A127450 A292640 A059564 * A224999 A274384 A195175 Adjacent sequences:  A001648 A001649 A001650 * A001652 A001653 A001654 KEYWORD nonn,easy AUTHOR EXTENSIONS This is a list, so the offset should be 1. I corrected this and adjusted some of the comments and formulas. Other lines probably also need to be adjusted. - N. J. A. Sloane, Jan 01 2011 Offset of pre-2011 formulas verified or corrected by M. F. Hasler, Apr 07-18 2015 and by Danny Rorabaugh, Oct 23 2015 STATUS approved

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Last modified September 23 12:15 EDT 2018. Contains 315274 sequences. (Running on oeis4.)