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A016777 a(n) = 3n + 1. 202
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 n such that the concatenation of the first n 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

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

REFERENCES

W. Decker, C. Lossen, Computing in Algebraic Geometry, Springer, 2006, p. 22

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 16.

Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269.

LINKS

Table of n, a(n) for n=0..62.

Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017.

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")

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.

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

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] - Benoit Cloitre, Apr 05 2002

(1 + 4x + 7x^2 + 10x^3 + ...) = (1 + 2x + 3x^2 ...)/(1 - 2x + 4x^2 - 8x^3 ...). - Gary W. Adamson, Jul 03 2003

E.g.f.: exp(x)*(1+3x). - 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

MAPLE

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-1]+3 od: seq(a[n], n=1..63); # Zerinvary Lajos, Mar 16 2008

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).

Sequence in context: A190084 A112335 A145289 * A143460 A143459 A143458

Adjacent sequences:  A016774 A016775 A016776 * A016778 A016779 A016780

KEYWORD

nonn,easy,changed

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 December 15 18:26 EST 2017. Contains 296047 sequences.