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%I #318 Apr 17 2024 11:13:31
%S 1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,
%T 73,76,79,82,85,88,91,94,97,100,103,106,109,112,115,118,121,124,127,
%U 130,133,136,139,142,145,148,151,154,157,160,163,166,169,172,175,178,181,184,187
%N a(n) = 3*n + 1.
%C Numbers k such that the concatenation of the first k natural numbers is not divisible by 3. E.g., 16 is in the sequence because we have 123456789101111213141516 == 1 (mod 3).
%C Ignoring the first term, this sequence represents the number of bonds in a hydrocarbon: a(#of carbon atoms) = number of bonds. - Nathan Savir (thoobik(AT)yahoo.com), Jul 03 2003
%C n such that Sum_{k=0..n} (binomial(n+k,n-k) mod 2) is even (cf. A007306). - _Benoit Cloitre_, May 09 2004
%C Hilbert series for twisted cubic curve. - _Paul Barry_, Aug 11 2006
%C If Y is a 3-subset of an n-set X then, for n >= 3, a(n-3) is the number of 3-subsets of X having at least two elements in common with Y. - _Milan Janjic_, Nov 23 2007
%C a(n) = A144390 (1, 9, 23, 43, 69, ...) - A045944 (0, 5, 16, 33, 56, ...). From successive spectra of hydrogen atom. - _Paul Curtz_, Oct 05 2008
%C Number of monomials in the n-th power of polynomial x^3+x^2+x+1. - _Artur Jasinski_, Oct 06 2008
%C A145389(a(n)) = 1. - _Reinhard Zumkeller_, Oct 10 2008
%C Union of A035504, A165333 and A165336. - _Reinhard Zumkeller_, Sep 17 2009
%C Hankel transform of A076025. - _Paul Barry_, Sep 23 2009
%C From _Jaroslav Krizek_, May 28 2010: (Start)
%C a(n) = numbers k such that the antiharmonic mean of the first k positive integers is an integer.
%C A169609(a(n-1)) = 1. See A146535 and A169609. Complement of A007494.
%C See A005408 (odd positive integers) for corresponding values A146535(a(n)). (End)
%C Apart from the initial term, A180080 is a subsequence; cf. A180076. - _Reinhard Zumkeller_, Aug 14 2010
%C Also the maximum number of triangles that n + 2 noncoplanar points can determine in 3D space. - _Carmine Suriano_, Oct 08 2010
%C A089911(4*a(n)) = 3. - _Reinhard Zumkeller_, Jul 05 2013
%C The number of partitions of 6*n into at most 2 parts. - _Colin Barker_, Mar 31 2015
%C For n >= 1, a(n)/2 is the proportion of oxygen for the stoichiometric combustion reaction of hydrocarbon CnH2n+2, e.g., one part propane (C3H8) requires 5 parts oxygen to complete its combustion. - _Kival Ngaokrajang_, Jul 21 2015
%C Exponents n > 0 for which 1 + x^2 + x^n is reducible. - _Ron Knott_, Oct 13 2016
%C Also the number of independent vertex sets in the n-cocktail party graph. - _Eric W. Weisstein_, Sep 21 2017
%C Also the number of (not necessarily maximal) cliques in the n-ladder rung graph. - _Eric W. Weisstein_, Nov 29 2017
%C Also the number of maximal and maximum cliques in the n-book graph. - _Eric W. Weisstein_, Dec 01 2017
%C For n>=1, a(n) is the size of any snake-polyomino with n cells. - _Christian Barrientos_ and _Sarah Minion_, Feb 27 2018
%C The sum of two distinct terms of this sequence is never a square. See Lagarias et al. p. 167. - _Michel Marcus_, May 20 2018
%C It seems that, for any n >= 1, there exists no positive integer z such that digit_sum(a(n)*z) = digit_sum(a(n)+z). - _Max Lacoma_, Sep 18 2019
%C For n > 2, a(n-2) is the number of distinct values of the magic constant in a normal magic triangle of order n (see formula 5 in Trotter). - _Stefano Spezia_, Feb 18 2021
%C Number of 3-permutations of n elements avoiding the patterns 132, 231, 312. See Bonichon and Sun. - _Michel Marcus_, Aug 20 2022
%C Erdős & Sárközy conjecture that a set of n positive integers with property P must have some element at least a(n-1) = 3n - 2. Property P states that, for x, y, and z in the set and z < x, y, z does not divide x+y. An example of such a set is {2n-1, 2n, ..., 3n-2}. Bedert proves this for large enough n. (This is an upper bound, and is exact for all known n; I have verified it for n up to 12.) - _Charles R Greathouse IV_, Feb 06 2023
%C a(n-1) = 3*n-2 is the dimension of the vector space of all n X n tridiagonal matrices, equals the number of nonzero coefficients: n + 2*(n-1) (see Wikipedia link). - _Bernard Schott_, Mar 03 2023
%D W. Decker, C. Lossen, Computing in Algebraic Geometry, Springer, 2006, p. 22
%D Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269.
%H N. J. A. Sloane, <a href="/A016777/b016777.txt">Table of n, a(n) for n = 0..10000</a>
%H Hacène Belbachir, Toufik Djellal, and Jean-Gabriel Luque, <a href="https://arxiv.org/abs/1703.00323">On the self-convolution of generalized Fibonacci numbers</a>, arXiv:1703.00323 [math.CO], 2017.
%H Benjamin Bedert, <a href="https://arxiv.org/abs/2301.07065">On a problem of Erdős and Sárközy about sequences with no term dividing the sum of two larger terms</a>, arXiv preprint, arXiv:2301.07065 [math.NT], 2023.
%H Nicolas Bonichon and Pierre-Jean Morel, <a href="https://arxiv.org/abs/2202.12677">Baxter d-permutations and other pattern avoiding classes</a>, arXiv:2202.12677 [math.CO], 2022.
%H Paul Erdős and András Sárközy, <a href="https://users.renyi.hu/~p_erdos/1970-13.pdf">On the divisibility properties of sequences of integers</a>, Proc. London Math. Soc. (3), 21 (1970), pp. 97-101.
%H Leonhard Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">Observatio de summis divisorum</a> p. 9.
%H Leonhard Euler, <a href="https://arxiv.org/abs/math/0411587">An observation on the sums of divisors</a>, arXiv:math/0411587 [math.HO], 2004-2009, see p. 9.
%H L. B. W. Jolley, <a href="https://archive.org/details/summationofserie00joll">Summation of Series</a>, Dover, 1961, pp. 16, 38.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Konrad Knopp, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ACM1954.0001.001">Theorie und Anwendung der unendlichen Reihen</a>, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")
%H J. C. Lagarias, A. M. Odlyzko, and J. B. Shearer, <a href="http://dx.doi.org/10.1016/0097-3165(82)90005-X">On the density of sequences of integers the sum of no two of which is a square. I. Arithmetic progressions</a>, Journal of Combinatorial Theory. Series A, 33 (1982), pp. 167-185.
%H T. Mansour, <a href="https://arxiv.org/abs/math/9909019">Permutations avoiding a set of patterns from S_3 and a pattern from S_4</a>, arXiv:math/9909019 [math.CO], 1999.
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
%H Nathan Sun, <a href="https://arxiv.org/abs/2208.08506">On d-permutations and Pattern Avoidance Classes</a>, arXiv:2208.08506 [math.CO], 2022.
%H Terrel Trotter, <a href="http://www.teherba.org/trottermath.net/simpleops/magictri.html">Normal Magic Triangles of Order n</a>, Journal of Recreational Mathematics Vol. 5, No. 1, 1972, pp. 28-32.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BookGraph.html">Book Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Clique.html">Clique</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LadderRungGraph.html">Ladder Rung Graph</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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tridiagonal_matrix">Tridiagonal matrix</a>.
%H Chengcheng Yang, <a href="https://arxiv.org/abs/2011.15010">A Problem of Erdös Concerning Lattice Cubes</a>, arXiv:2011.15010 [math.CO], 2020. See Table p. 27.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F G.f.: (1+2*x)/(1-x)^2.
%F a(n) = A016789(n) - 1.
%F a(n) = 3 + a(n-1).
%F Sum_{n>=1} (-1)^n/a(n) = (1/3)*(Pi/sqrt(3) + log(2)). [Jolley, p. 16, (79)] - _Benoit Cloitre_, Apr 05 2002
%F (1 + 4*x + 7*x^2 + 10*x^3 + ...) = (1 + 2*x + 3*x^2 + ...)/(1 - 2*x + 4*x^2 - 8*x^3 + ...). - _Gary W. Adamson_, Jul 03 2003
%F E.g.f.: exp(x)*(1+3*x). - _Paul Barry_, Jul 23 2003
%F a(n) = 2*a(n-1) - a(n-2); a(0)=1, a(1)=4. - _Philippe Deléham_, Nov 03 2008
%F a(n) = 6*n - a(n-1) - 1 (with a(0) = 1). - _Vincenzo Librandi_, Nov 20 2010
%F Sum_{n>=0} 1/a(n)^2 = A214550. - _R. J. Mathar_, Jul 21 2012
%F a(n) = A238731(n+1,n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-5)^k. - _Philippe Deléham_, Mar 05 2014
%F Sum_{i=0..n} (a(i)-i) = A000290(n+1). - _Ivan N. Ianakiev_, Sep 24 2014
%F From _Wolfdieter Lang_, Mar 09 2018: (Start)
%F a(n) = denominator(Sum_{k=0..n-1} 1/(a(k)*a(k+1)), with the numerator n = A001477(n), where the sum is set to 0 for n = 0. [Jolley, p. 38, (208)]
%F G.f. for {n/(1 + 3*n)}_{n >= 0} is (1/3)*(1-hypergeom([1, 1], [4/3], -x/(1-x)))/(1-x). (End)
%F a(n) = -A016789(-1-n) for all n in Z. - _Michael Somos_, May 27 2019
%e G.f. = 1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 16*x^5 + 19*x^6 + 22*x^7 + ... - _Michael Somos_, May 27 2019
%t Range[1, 199, 3] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)
%t (* Start from _Eric W. Weisstein_, Sep 21 2017 *)
%t 3 Range[0, 70] + 1
%t Table[3 n + 1, {n, 0, 70}]
%t LinearRecurrence[{2, -1}, {1, 4}, 70]
%t CoefficientList[Series[(1 + 2 x)/(-1 + x)^2, {x, 0, 70}], x]
%t (* End *)
%o (Magma) [3*n+1 : n in [1..70]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
%o (Haskell)
%o a016777 = (+ 1) . (* 3)
%o a016777_list = [1, 4 ..] -- _Reinhard Zumkeller_, Feb 28 2013, Feb 10 2012
%o (Maxima) A016777(n):=3*n+1$
%o makelist(A016777(n),n,0,30); /* _Martin Ettl_, Oct 31 2012 */
%o (PARI) a(n)=3*n+1 \\ _Charles R Greathouse IV_, Jul 28 2015
%o (SageMath) [3*n+1 for n in range(1,71)] # _G. C. Greubel_, Mar 15 2024
%Y Cf. A000290, A001477, A005408, A007306, A007494, A016789, A016933.
%Y Cf. A017569, A035504, A045944, A058183, A076025, A089911, A144390.
%Y Cf. A145389, A146535, A165333, A165336, A169609, A180076, A180080.
%Y Cf. A214550, A238731.
%Y Cf. A007559 (partial products), A051536 (lcm).
%Y First differences of A000326.
%Y Row sums of A131033.
%Y Complement of A007494. - _Reinhard Zumkeller_, Oct 10 2008
%Y Some subsequences: A002476 (primes), A291745 (nonprimes), A135556 (squares), A016779 (cubes).
%K nonn,easy,changed
%O 0,2
%A _N. J. A. Sloane_, Dec 11 1996
%E Better description from _T. D. Noe_, Aug 15 2002
%E Partially edited by _Joerg Arndt_, Mar 11 2010
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