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%I M0957 N0357 #290 Mar 05 2024 16:15:43
%S 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,
%T 38,40,41,43,44,46,47,49,50,52,53,55,56,58,59,61,62,64,65,67,68,70,71,
%U 73,74,76,77,79,80,82,83,85,86,88,89,91,92,94,95,97,98,100,101,103,104
%N Numbers not divisible by 3.
%C Inverse binomial transform of A084858. - _Benoit Cloitre_, Jun 12 2003
%C 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]
%C 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
%C For n > 3, numbers having 3 as an anti-divisor. - _Alexandre Wajnberg_, Oct 02 2005
%C Also numbers n such that (n+1)*(n+2)/6 = A000292(n)/n is an integer. - _Ctibor O. Zizka_, Oct 15 2010
%C 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
%C A001651 mod 9 gives A141425. - _Paul Curtz_, Dec 31 2010. (Correct for the modified offset 1. - _M. F. Hasler_, Apr 07 2015)
%C 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
%C Union of A047239 and A047257. - _Wesley Ivan Hurt_, Dec 19 2013
%C Numbers whose sum of digits (and digital root) is != 0 (mod 3). - _Joerg Arndt_, Aug 29 2014
%C The number of partitions of 3*(n-1) into at most 2 parts. - _Colin Barker_, Apr 22 2015
%C a(n) is the number of partitions of 3*n into two distinct parts. - _L. Edson Jeffery_, Jan 14 2017
%C 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
%C 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
%C 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
%C 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
%C From _Roger Ford_, May 09 2021: (Start)
%C a(n) is the smallest sum of arch lengths for the top arches of a semi-meander with n arches. An arch length is the number of arches covered + 1.
%C /\ The top arch has a length of 3. /\ The top arch has a length of 3.
%C / \ Both bottom arches have a //\\ The middle arch has a length of 2.
%C //\/\\ length of 1. ///\\\ The bottom arch has a length of 1.
%C Example: a(6) = 8 /\ /\
%C //\\ /\ //\\ /\ 2 + 1 + 1 + 2 + 1 + 1 = 8. (End)
%C This is the lexicographically earliest increasing sequence of positive integers such that no polynomial of degree d can be fitted to d+2 consecutive terms (equivalently, such that no iterated difference is zero). - _Pontus von Brömssen_, Dec 26 2021
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A001651/b001651.txt">Table of n, a(n) for n = 1..10000</a>
%H L. Carlitz, R. Scoville and T. Vaughan, <a href="http://www.fq.math.ca/Scanned/11-4/carlitz.pdf">Some arithmetic functions related to Fibonacci numbers</a>, Fib. Quart., Vol. 11, No. 4 (1973), pp. 337-386.
%H Aneta Dudek, Gyula Y. Katona, and A.Pawel Wojda, <a href="https://doi.org/10.1016/S1571-0653(04)00434-2">m_Path Cover Saturated Graphs</a>, Electronic Notes in Discrete Math., Vol. 13 (April 2003), pp. 41-44.
%H Marietjie Frick and Joy Singleton, <a href="https://doi.org/10.37236/1929">Lower Bound for the Size of Maximal Nontraceable Graphs</a>, Electron. J. Combin., 12#R32 (2005), 9 pp.
%H Aviezri S. Fraenkel, <a href="http://www.emis.de/journals/INTEGERS/papers/eg6/eg6.Abstract.html">New games related to old and new sequences</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004. (See Table 5.)
%H Brian Hopkins, <a href="http://ecajournal.haifa.ac.il/Volume2021/ECA2021_S1H1.pdf">Euler's Enumerations</a>, Enumerative Combinatorics and Applications, Vol. 1, No. 1 (2021), Article #S1H1.
%H G. Ledin, Jr., <a href="http://www.fq.math.ca/Scanned/6-4/ledin.pdf">Is Eratosthenes out?</a>, Fib. Quart., Vol. 6, No. 4 (1968), pp. 261-265.
%H Gerard P. Michon, <a href="http://www.numericana.com/data/polyhedra.htm">Counting Polyhedra</a>.
%H Melvyn B. Nathanson, <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.05.409">On the fractional parts of roots of positive real numbers</a>, Amer. Math. Monthly, Vol. 120, No. 5 (2013), pp. 409-429 [see p. 417].
%H M. A. Nyblom, <a href="http://www.jstor.org/stable/2695446">Some curious sequences involving floor and ceiling functions</a>, Am. Math. Monthly, Vol. 109, No. 6 (2002), pp. 559-564, Ex. 2.2.
%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 Marco Ripà, <a href="http://nntdm.net/papers/nntdm-20/NNTDM-20-1-59-71.pdf">The rectangular spiral or the n1 X n2 X ... X nk Points Problem </a>, Notes on Number Theory and Discrete Mathematics, Vol. 20, No. 1 (2014), pp. 59-71.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>, <a href="http://mathworld.wolfram.com/GraphBandwidth.html">Graph Bandwidth</a>, and <a href="http://mathworld.wolfram.com/RATSSequence.html">RATS Sequence</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(n) = 3 + a(n-2) for n > 2.
%F a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
%F a(2*n+1) = 3*n+1, a(2*n) = 3*n-1.
%F G.f.: x * (1 + x + x^2) / ((1 - x) * (1 - x^2)). - _Michael Somos_, Jun 08 2000
%F a(n) = (4-n)*a(n-1) + 2*a(n-2) + (n-3)*a(n-3) (from the Carlitz et al. article).
%F a(n) = floor((3*n-1)/2). [Corrected by _Gary Detlefs_]
%F a(1) = 1, a(n) = 2*a(n-1) - 3*floor(a(n-1)/3). - _Benoit Cloitre_, Aug 17 2002
%F a(n+1) = 1 + n - n mod 2 + (n + n mod 2)/2. - _Reinhard Zumkeller_, Dec 17 2002
%F a(1) = 1, a(n+1) = a(n) + (a(n) mod 3). - _Reinhard Zumkeller_, Mar 23 2003
%F a(1) = 1, a(n) = 3*(n-1) - a(n-1). - _Benoit Cloitre_, Apr 12 2003
%F a(n) = 3*(2*n-1)/4 - (-1)^n/4. - _Benoit Cloitre_, Jun 12 2003
%F Nearest integer to (Sum_{k>=n} 1/k^3)/(Sum_{k>=n} 1/k^4). - _Benoit Cloitre_, Jun 12 2003
%F Partial sums of A040001. a(n) = A032766(n-1)+1. - _Paul Barry_, Sep 02 2003
%F a(n) = T(n, 1) = T(n, n-1), where T is the array in A026386. - _Emeric Deutsch_, Feb 18 2004
%F a(n) = sqrt(3*A001082(n)+1). - _Zak Seidov_, Dec 12 2007
%F a(n) = A077043(n) - A077043(n-1). - _Reinhard Zumkeller_, Dec 28 2007
%F a(n) = A001477(n-1) + A008619(n-1). - _Yosu Yurramendi_, Aug 10 2008
%F Euler transform of length 3 sequence [2, 1, -1]. - _Michael Somos_, Sep 06 2008
%F A011655(a(n)) = 1. - _Reinhard Zumkeller_, Nov 30 2009
%F a(n) = n - 1 + ceiling(n/2). - _Michael Somos_, Jan 15 2011
%F a(n) = 3*A000217(n)+1 - 2*Sum_{i=1..n-1} a(i), for n>1. - _Bruno Berselli_, Nov 17 2010
%F a(n) = 3*floor(n/2) + (-1)^(n+1). - _Gary Detlefs_, Dec 29 2011
%F A215879(a(n)) > 0. - _Reinhard Zumkeller_, Dec 28 2012 [More precisely, A215879 is the characteristic function of A001651. - _M. F. Hasler_, Apr 07 2015]
%F a(n) = 2n - 1 - floor(n/2). - _Wesley Ivan Hurt_, Oct 25 2013
%F a(n) = (3n - 2 + (n mod 2)) / 2. - _Wesley Ivan Hurt_, Mar 31 2014
%F a(n) = A000217(n) - A000982(n-1). - _Bui Quang Tuan_, Mar 28 2015
%F 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
%F E.g.f.: (4 + sinh(x) - cosh(x) + 3*(2*x - 1)*exp(x))/4. - _Ilya Gutkovskiy_, May 24 2016
%F a(n) = a(n+k-1) + a(n-k) - a(n-1) for n > k >= 0. - _Bob Selcoe_, Feb 03 2017
%F a(n) = -a(1-n) for all n in Z. - _Michael Somos_, Jul 31 2018
%F a(n) = n + A004526(n-1). - _David James Sycamore_, Sep 06 2021
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - _Amiram Eldar_, Dec 04 2021
%e 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 + ...
%p A001651 := n -> 3*floor(n/2) - (-1)^n; # Corrected by _M. F. Hasler_, Apr 07 2015
%p A001651:=(1+z+z**2)/(z+1)/(z-1)**2; # _Simon Plouffe_ in his 1992 dissertation
%p 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
%t Select[Table[n,{n,200}],Mod[#,3]!=0&] (* _Vladimir Joseph Stephan Orlovsky_, Feb 18 2011 *)
%t Drop[Range[200 + 1], {1, -1, 3}] - 1 (* _József Konczer_, May 24 2016 *)
%t Floor[(3 Range[70] - 1)/2] (* _Eric W. Weisstein_, Apr 24 2017 *)
%t CoefficientList[Series[(x^2 + x + 1)/((x - 1)^2 (x + 1)), {x, 0, 70}],
%t x] (* or *)
%t LinearRecurrence[{1, 1, -1}, {1, 2, 4}, 70] (* _Robert G. Wilson v_, Jul 25 2018 *)
%o (PARI) {a(n) = n + (n-1)\2}; /* _Michael Somos_, Jan 15 2011 */
%o (PARI) x='x+O('x^100); Vec(x*(1+x+x^2)/((1-x)*(1-x^2))) \\ _Altug Alkan_, Oct 22 2015
%o (Magma) [3*(2*n-1)/4-(-1)^n/4: n in [1..80]]; // _Vincenzo Librandi_, Jun 07 2011
%o (Haskell)
%o a001651 = (`div` 2) . (subtract 1) . (* 3)
%o a001651_list = filter ((/= 0) . (`mod` 3)) [1..]
%o -- _Reinhard Zumkeller_, Jul 07 2012, Aug 23 2011
%o (GAP) Filtered([0..110],n->n mod 3<>0); # _Muniru A Asiru_, Jul 24 2018
%o (Python)
%o print([k for k in range(1, 105) if k%3]) # _Michael S. Branicky_, Sep 06 2021
%o (Python)
%o def A001651(n): return (n<<1)-(n>>1)-1 # _Chai Wah Wu_, Mar 05 2024
%Y Differs from A059564 after 35 = a(24) = A059564(24).
%Y Cf. A000726, A001082, A003105, A005408 (n=1 or 3 mod 4), A007494, A008585 (complement), A011655, A026386, A032766, A073010, A191967, A225227, A004526.
%Y Cf. A000027, A000217, A000292, A000982, A001477, A008619, A014437, A040001, A047239, A047257, A077043, A084858, A113801, A141425, A215879.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E 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
%E Offset of pre-2011 formulas verified or corrected by _M. F. Hasler_, Apr 07-18 2015 and by _Danny Rorabaugh_, Oct 23 2015
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