

A016777


a(n) = 3*n + 1.


270



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
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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
Hilbert series for twisted cubic curve.  Paul Barry, Aug 11 2006
If Y is a 3subset of an nset X then, for n >= 3, a(n3) is the number of 3subsets 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 nth power of polynomial x^3+x^2+x+1.  Artur Jasinski, Oct 06 2008
a(n) = numbers k such that the antiharmonic mean of the first k positive integers is an integer.
See A005408 (odd positive integers) for corresponding values A146535(a(n)). (End)
Also the maximum number of triangles that n + 2 noncoplanar points can determine in 3D space.  Carmine Suriano, Oct 08 2010
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 ncocktail party graph.  Eric W. Weisstein, Sep 21 2017
Also the number of (not necessarily maximal) cliques in the nladder rung graph.  Eric W. Weisstein, Nov 29 2017
Also the number of maximal and maximum cliques in the nbook graph.  Eric W. Weisstein, Dec 01 2017
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(n2) 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 3permutations 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(n1) = 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 {2n1, 2n, ..., 3n2}. 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(n1) = 3*n2 is the dimension of the vector space of all n X n tridiagonal matrices, equals the number of nonzero coefficients: n + 2*(n1) (see Wikipedia link).  Bernard Schott, Mar 03 2023


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

Eric Weisstein's World of Mathematics, Clique


FORMULA

G.f.: (1+2*x)/(1x)^2.
a(n) = 3 + a(n1).
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.: 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) = denominator(Sum_{k=0..n1} 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)*(1hypergeom([1, 1], [4/3], x/(1x)))/(1x). (End)


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

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)sergeihaller.de), Dec 21 2006
(Haskell)
a016777 = (+ 1) . (* 3)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS

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


STATUS

approved



