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 A016777 a(n) = 3*n + 1. 246
 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178, 181, 184, 187 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). 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 n such that Sum_{k=0..n} (binomial(n+k,n-k) mod 2) is even (cf. A007306). - Benoit Cloitre, May 09 2004 Hilbert series for twisted cubic curve. - Paul Barry, Aug 11 2006 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 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 Number of monomials in the n-th power of polynomial x^3+x^2+x+1. - Artur Jasinski, Oct 06 2008 A145389(a(n)) = 1. - Reinhard Zumkeller, Oct 10 2008 Union of A035504, A165333 and A165336. - Reinhard Zumkeller, Sep 17 2009 Hankel transform of A076025. - Paul Barry, Sep 23 2009 From Jaroslav Krizek, May 28 2010: (Start) a(n) = numbers k such that the antiharmonic mean of the first k positive integers is an integer. A169609(a(n-1)) = 1. See A146535 and A169609. Complement of A007494. See A005408 (odd positive integers) for corresponding values A146535(a(n)). (End) Apart from the initial term, A180080 is a subsequence; cf. A180076. - Reinhard Zumkeller, Aug 14 2010 Also the maximum number of triangles that n + 2 noncoplanar points can determine in 3D space. - Carmine Suriano, Oct 08 2010 A089911(4*a(n)) = 3. - Reinhard Zumkeller, Jul 05 2013 The number of partitions of 6*n into at most 2 parts. - Colin Barker, Mar 31 2015 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 Exponents n > 0 for which 1 + x^2 + x^n is reducible. - Ron Knott, Oct 13 2016 Also the number of independent vertex sets in the n-cocktail party graph. - Eric W. Weisstein, Sep 21 2017 Also the number of (not necessarily maximum) cliques in the n-ladder rung graph. - Eric W. Weisstein, Nov 29 2017 Also the number of maximal and maximum cliques in the n-book graph. - Eric W. Weisstein, Dec 01 2017 For n>=1, a(n) is the size of any snake-polyomino with n cells. - Christian Barrientos and Sarah Minion, Feb 27 2018 The sum of two distinct terms of this sequence is never a square. See Lagarias et al. p. 167. - Michel Marcus, May 20 2018 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 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 Number of 3-permutations of n elements avoiding the patterns 132, 231, 312. See Bonichon and Sun. - Michel Marcus, Aug 20 2022 REFERENCES W. Decker, C. Lossen, Computing in Algebraic Geometry, Springer, 2006, p. 22 L. B. W. Jolley, Summation of Series, Dover Publications, 2nd rev. ed., 1961, pp. 16, 38. Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269. LINKS N. J. A. Sloane, Table of n, a(n) for n = 0..10000 Hacène Belbachir, Toufik Djellal, and Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017. Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022. L. Euler, Observatio de summis divisorum p. 9. L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, see p. 9. Tanya Khovanova, Recursive Sequences Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series") J. C. Lagarias, A. M. Odlyzko, and J. B. Shearer, On the density of sequences of integers the sum of no two of which is a square. I. Arithmetic progressions, Journal of Combinatorial Theory. Series A, 33 (1982), pp. 167-185. T. Mansour, Permutations avoiding a set of patterns from S_3 and a pattern from S_4, arXiv:math/9909019 [math.CO], 1999. Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014. Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022. Terrel Trotter, Normal Magic Triangles of Order n, Journal of Recreational Mathematics Vol. 5, No. 1, 1972, pp. 28-32. Eric Weisstein's World of Mathematics, Book Graph Eric Weisstein's World of Mathematics, Clique Eric Weisstein's World of Mathematics, Cocktail Party Graph Eric Weisstein's World of Mathematics, Independent Vertex Set Eric Weisstein's World of Mathematics, Ladder Rung Graph Eric Weisstein's World of Mathematics, Maximal Clique Eric Weisstein's World of Mathematics, Maximum Clique Chengcheng Yang, A Problem of Erdös Concerning Lattice Cubes, arXiv:2011.15010 [math.CO], 2020. See Table p. 27. Index entries for linear recurrences with constant coefficients, signature (2,-1). FORMULA G.f.: (1+2*x)/(1-x)^2. a(n) = 3 + a(n-1). Sum_{n>=1} (-1)^n/a(n) = (1/3)*(Pi/sqrt(3) + log(2)). [Jolley, p. 16, (79)] - Benoit Cloitre, Apr 05 2002 (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 E.g.f.: exp(x)*(1+3*x). - Paul Barry, Jul 23 2003 a(n) = 2*a(n-1) - a(n-2); a(0)=1, a(1)=4. - Philippe Deléham, Nov 03 2008 a(n) = 6*n - a(n-1) - 1 (with a(0) = 1). - Vincenzo Librandi, Nov 20 2010 Sum_{n>=0} 1/a(n)^2 = A214550. - R. J. Mathar, Jul 21 2012 E.g.f.: E(0), where E(k) = 1 + 3*x/(1 - 2*x/(2*x + 6*x*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 05 2013 G.f.: 1 + 4*x/(G(0) - 4*x), where G(k) = 1 + 4*x + 3*k*(x+1) -  x*(3*k+1)*(3*k+7)/G(k+1); (cont. fraction). - Sergei N. Gladkovskii, Jul 05 2013 a(n) = A238731(n+1,n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-5)^k. - Philippe Deléham, Mar 05 2014 Sum_{i=0..n} (a(i)-i) = A000290(n+1). - Ivan N. Ianakiev, Sep 24 2014 From Wolfdieter Lang, Mar 09 2018: (Start) 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)] G.f. for {n/(1 + 3*n)}_{n >= 0} is (1/3)*(1-hypergeom([1, 1], [4/3], -x/(1-x)))/(1-x). (End) a(n) = -A016789(-1-n) for all n in Z. - Michael Somos, May 27 2019 EXAMPLE 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 MATHEMATICA Range[1, 199, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *) (* Start from Eric W. Weisstein, Sep 21 2017 *) 3 Range[0, 20] + 1 Table[3 n + 1, {n, 0, 20}] LinearRecurrence[{2, -1}, {1, 4}, 20] CoefficientList[Series[(1 + 2 x)/(-1 + x)^2, {x, 0, 20}], x] (* End *) PROG (Magma) [3*n+1 : n in [1..30]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006 (Haskell) a016777 = (+ 1) . (* 3) a016777_list = [1, 4 ..]  -- Reinhard Zumkeller, Feb 28 2013, Feb 10 2012 (Maxima) A016777(n):=3*n+1\$ makelist(A016777(n), n, 0, 30); /* Martin Ettl, Oct 31 2012 */ (PARI) a(n)=3*n+1 \\ Charles R Greathouse IV, Jul 28 2015 CROSSREFS A016789(n) = 1 + a(n). First differences of A000326. Cf. A000290, A016933, A017569, A058183. Row sums of A131033. Complement of A007494. - Reinhard Zumkeller, Oct 10 2008 Cf. A051536 (lcm). Cf. A007559 (partial products). Some subsequences: A002476 (primes), A291745 (nonprimes), A135556 (squares), A016779 (cubes). Sequence in context: A190084 A145289 A112335 * A308014 A143460 A338701 Adjacent sequences:  A016774 A016775 A016776 * A016778 A016779 A016780 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Dec 11 1996 EXTENSIONS Better description from T. D. Noe, Aug 15 2002 Partially edited by Joerg Arndt, Mar 11 2010 STATUS approved

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Last modified September 26 08:43 EDT 2022. Contains 356993 sequences. (Running on oeis4.)