A016777 a(n) = 3*n + 1.

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

Offset: 0

Author

N. J. A. Sloane, Dec 11 1996

Keywords

• nonn
• easy

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 maximal) 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

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

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

Examples

			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
		

References

• W. Decker, C. Lossen, Computing in Algebraic Geometry, Springer, 2006, p. 22.
• Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.1 Terminology, p. 264.
• 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.
• Benjamin Bedert, On a problem of Erdős and Sárközy about sequences with no term dividing the sum of two larger terms, arXiv preprint, arXiv:2301.07065 [math.NT], 2023.
• Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
• Paul Erdős and András Sárközy, On the divisibility properties of sequences of integers, Proc. London Math. Soc. (3), 21 (1970), pp. 97-101.
• Leonhard Euler, Observatio de summis divisorum p. 9.
• Leonhard Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, see p. 9.
• L. B. W. Jolley, Summation of Series, Dover, 1961, pp. 16, 38.
• 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
• Wikipedia, Tridiagonal matrix.
• 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).

Crossrefs

Cf. A000290, A001477, A005408, A007306, A007494, A016789, A016933.

Cf. A017569, A035504, A045944, A058183, A076025, A089911, A144390.

Cf. A145389, A146535, A165333, A165336, A169609, A180076, A180080.

Cf. A214550, A238731.

Cf. A007559 (partial products), A051536 (lcm).

First differences of A000326.

Row sums of A131033.

Complement of A007494. - Reinhard Zumkeller, Oct 10 2008

Some subsequences: A002476 (primes), A291745 (nonprimes), A135556 (squares), A016779 (cubes).

Programs

Haskell

a016777 = (+ 1) . (* 3)
a016777_list = [1, 4 ..]  -- Reinhard Zumkeller, Feb 28 2013, Feb 10 2012

Magma

[3*n+1 : n in [1..70]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

Mathematica

Range[1, 199, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
(* Start from Eric W. Weisstein, Sep 21 2017 *)
3 Range[0, 70] + 1
Table[3 n + 1, {n, 0, 70}]
LinearRecurrence[{2, -1}, {1, 4}, 70]
CoefficientList[Series[(1 + 2 x)/(-1 + x)^2, {x, 0, 70}], x]
(* End *)

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

SageMath

[3*n+1 for n in range(1,71)] # G. C. Greubel, Mar 15 2024

Formula

G.f.: (1+2*x)/(1-x)^2.

a(n) = A016789(n) - 1.

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

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

Extensions

Better description from T. D. Noe, Aug 15 2002

Partially edited by Joerg Arndt, Mar 11 2010