A146535 -id:A146535 - cp's OEIS frontend

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

N. J. A. Sloane, Dec 11 1996

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

			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

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.

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).

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).

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

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

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 *)

A016777(n):=3*n+1$
makelist(A016777(n),n,0,30); /* Martin Ettl, Oct 31 2012 */

a(n)=3*n+1 \\ Charles R Greathouse IV, Jul 28 2015

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

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

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

Partially edited by Joerg Arndt, Mar 11 2010

A007494 Numbers that are congruent to 0 or 2 mod 3.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107

Offset: 0

Christopher Lam Cham Kee (Topher(AT)CyberDude.Com)

The map n -> a(n) (where a(n) = 3n/2 if n even or (3n+1)/2 if n odd) was studied by Mahler, in connection with "Z-numbers" and later by Flatto. One question was whether, iterating from an initial integer, one eventually encountered an iterate = 1 (mod 4). - Jeff Lagarias, Sep 23 2002
Partial sums of 0,2,1,2,1,2,1,2,1,... . - Paul Barry, Aug 18 2007
a(n) = numbers k such that antiharmonic mean of the first k positive integers is not integer. A169609(a(n-1)) = 3. See A146535 and A169609. Complement of A016777. - Jaroslav Krizek, May 28 2010
Range of A173732. - Reinhard Zumkeller, Apr 29 2012
Number of partitions of 6n into two odd parts. - Wesley Ivan Hurt, Nov 15 2014
Numbers m such that 3 divides A000217(m). - Bruno Berselli, Aug 04 2017
Maximal length of a snake like polyomino that fits in a 2 X n rectangle. - Alain Goupil, Feb 12 2020

L. Flatto, Z-numbers and beta-transformations, in Symbolic dynamics and its applications (New Haven, CT, 1991), 181-201, Contemp. Math., 135, Amer. Math. Soc., Providence, RI, 1992.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1002.
Attila Máder, The Use of Experimental Mathematics in the Classroom, in Interesting Mathematical Problems in Sciences and Everyday Life - 2011.
Kurt Mahler, An unsolved problem on the powers of 3/2, J. Austral. Math. Soc., Vol. 8 (1968), pp. 313-321.
P. Sabinin and M. G. Stone, Transforming n-gons by Folding the Plane, Amer. Math. Monthly, Vol. 102, No. 7 (1995), pp. 620-627.
Eric Weisstein's World of Mathematics, Folding.
Robert G. Wilson v, Notes with attachment.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A001651, A032766, A035361, A063574, A132462.
Complement of A016777.
Range of A002517.
Cf. A274406. [Bruno Berselli, Jun 26 2016]

a007494 =  flip div 2 . (+ 1) . (* 3) -- Reinhard Zumkeller, Dec 12 2014

[(6*n+1)/4-(-1)^n/4: n in [0..80]]; // Vincenzo Librandi, Aug 20 2011

a[0]:=0:a[1]:=2:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..71); # Zerinvary Lajos, Mar 16 2008
A007494:=n->floor((3*n+1)/2); seq(A007494(k), k=0..100); # Wesley Ivan Hurt, Sep 27 2013

Flatten[{#,#+2}&/@(3Range[0,40])] (* Harvey P. Dale, May 15 2011 *)
Table[2n - Floor[n/2], {n,0,100}] (* Wesley Ivan Hurt, Sep 27 2013 *)

a(n)=n+(n+1)>>1 \\ Charles R Greathouse IV, Jul 25 2011

a(n) = 3*n/2 if n even, otherwise (3*n+1)/2.
If u(1)=0, u(n) = n + floor(u(n-1)/3), then a(n-1) = u(n). - Benoit Cloitre, Nov 26 2002
G.f.: x*(x+2)/((1-x)^2*(1+x)). - Ralf Stephan, Apr 13 2002
a(n) = 3*floor(n/2) + 2*(n mod 2) = A032766(n) + A000035(n). - Reinhard Zumkeller, Apr 04 2005
a(n) = (6*n+1)/4 - (-1)^n/4; a(n) = Sum_{k=0..n-1} (1 + (-1)^(k/2)*cos(k*Pi/2)). - Paul Barry, Aug 18 2007
A145389(a(n)) <> 1. - Reinhard Zumkeller, Oct 10 2008
a(n) = A002943(n) - A173511(n). - Reinhard Zumkeller, Feb 20 2010
a(n) = 3*n - a(n-1) - 1 (with a(0)=0). - Vincenzo Librandi, Nov 18 2010
a(n) = Sum_{k>=0} A030308(n,k)*A042950(k). - Philippe Deléham, Oct 17 2011
a(n) = n + ceiling(n/2). - Arkadiusz Wesolowski, Sep 18 2012
a(n) = 2n - floor(n/2) = floor((3n+1)/2) = n + (n + (n mod 2))/2. - Wesley Ivan Hurt, Oct 19 2013
a(n) = A000217(n+1) - A099392(n+1). - Bui Quang Tuan, Mar 27 2015
a(n) = n + floor(n/2) + (n mod 2). - Bruno Berselli, Apr 04 2016
a(n) = Sum_{i=1..n} numerator(2/i). - Wesley Ivan Hurt, Feb 26 2017
a(n) = Sum_{k=0..n-1} Sum_{i=0..k} C(i,k)+(-1)^(k-i). - Wesley Ivan Hurt, Sep 20 2017
E.g.f.: (3*exp(x)*x + sinh(x))/2. - Stefano Spezia, Feb 11 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = log(3)/2 - Pi/(6*sqrt(3)). - Amiram Eldar, Dec 04 2021

A060747 a(n) = 2*n - 1.

Original entry on oeis.org

-1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151

Offset: 0

Henry Bottomley, Apr 26 2001

If you put n red balls and n blue balls in a bag and draw them one by one without replacement, the probability of never having drawn equal numbers of the two colors before the final ball is drawn is 1/a(n) unsigned.
abs(a(n)) = 2n - 1 + 2*0^n. It has A048495 as binomial transform. - Paul Barry, Jun 09 2003
For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is an integer. A040001(a(n)) = 1. See A145051 and A040001. -  Jaroslav Krizek, May 28 2010
From Jaroslav Krizek, May 28 2010: (Start)
For n >= 1, a(n) = corresponding values of antiharmonic means to numbers from A016777 (numbers k such that antiharmonic mean of the first k positive integers is an integer).
a(n) = A000330(A016777(n)) / A000217(A016777(n)) = A146535(A016777(n)+1). (End)

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000217, A000330, A000984, A001477, A002420, A005408, A005843, A016777, A023443, A040001, A048495, A078008, A145051, A146535, A161680.

a060747 = subtract 1 . (* 2)
a060747_list = [-1, 1 ..] -- Reinhard Zumkeller, Jul 05 2015
-- Reinhard Zumkeller, Jul 05 2015

Table[2*n - 1, {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
LinearRecurrence[{2,-1},{-1,1},80] (* Harvey P. Dale, Mar 27 2020 *)

a(n)=2*n-1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A005408(n)-2 = A005843(n)-1 = -A000984(n)/A002420(n) = A001477(n)+A023443(n).
G.f.: (3*x - 1)/(1 - x)^2.
Abs(a(n)) = Sum_{k=0..n} (A078008(k) mod 4). - Paul Barry, Mar 12 2004
E.g.f.: exp(x)*(2*x-1). - Paul Barry, Mar 31 2007
a(n) = 2*a(n-1) - a(n-2); a(0)=-1, a(1)=1. - Philippe Deléham, Nov 03 2008
a(n) = 4*n - a(n-1) - 4 for n>0, with a(0)=-1. - Vincenzo Librandi, Aug 07 2010
a(n) = A161680(A005843(n))/n for n > 0. - Stefano Spezia, Feb 14 2025

A004273 0 together with odd numbers.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 0

N. J. A. Sloane

Also continued fraction for tanh(1) (A073744 is decimal expansion). - Rick L. Shepherd, Aug 07 2002
From Jaroslav Krizek, May 28 2010: (Start)
For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is an integer. A040001(a(n)) = 1. See A145051 and A040001.
For n >= 1, a(n) = corresponding values of antiharmonic means to numbers from A016777 (numbers k such that antiharmonic mean of the first k positive integers is an integer).
a(n) = A000330(A016777(n)) / A000217(A016777(n)) = A146535(A016777(n)+1). (End)
If the n-th prime is denoted by p(n) then it appears that a(j) = distinct, increasing values of (Sum of the quadratic non-residues of p(n) - Sum of the quadratic residues of p(n)) / p(n) for each j. - Christopher Hunt Gribble, Oct 05 2010
A214546(a(n)) > 0. - Reinhard Zumkeller, Jul 20 2012
Dimension of the space of weight 2n+2 cusp forms for Gamma_0(6).
The size of a maximal 2-degenerate graph of order n-1 (this class includes 2-trees and maximal outerplanar graphs (MOPs)). - Allan Bickle, Nov 14 2021
Numbers not considered even for the purpose of roulette. J. Lowell, Apr 29 2025

			G.f. = x + 3*x^2 + 5*x^3 + 7*x^4 + 9*x^5 + 11*x^6 + 13*x^7 + 15*x^8 + 17*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Allan Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.
Index entries for linear recurrences with constant coefficients, signature (2, -1).

Cf. A110185, continued fraction expansion of 2*tanh(1/2), and A204877, continued fraction expansion of 3*tanh(1/3). [Bruno Berselli, Jan 26 2012]

Cf. A005408.

Concatenation([0],List([1,3..141])); # Muniru A Asiru, Jul 28 2018

[2*n-Floor((n+2) mod (n+1)): n in [0..70]]; // Vincenzo Librandi, Sep 21 2011

Join[{0}, Range[1, 200, 2]] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

a(n)=max(2*n-1,n) \\ Charles R Greathouse IV, May 14 2014

def A004273(n): return (n<<1)-1 if n else 0 # Chai Wah Wu, Jul 13 2024

def a(n) : return( dimension_cusp_forms( Gamma0(6), 2*n+2) ); # Michael Somos, Jul 03 2014

G.f.: x*(1+x)/(-1+x)^2. - R. J. Mathar, Nov 18 2007
a(n) = lodumo_2(A057427(n)). - Philippe Deléham, Apr 26 2009
Euler transform of length 2 sequence [3, -1]. - Michael Somos, Jul 03 2014
a(n) = (4*n - 1 - (-1)^(2^n))/2. - Luce ETIENNE, Jul 11 2015

A060789 a(n) = n / (gcd(n,2) * gcd(n,3)).

Original entry on oeis.org

1, 1, 1, 2, 5, 1, 7, 4, 3, 5, 11, 2, 13, 7, 5, 8, 17, 3, 19, 10, 7, 11, 23, 4, 25, 13, 9, 14, 29, 5, 31, 16, 11, 17, 35, 6, 37, 19, 13, 20, 41, 7, 43, 22, 15, 23, 47, 8, 49, 25, 17, 26, 53, 9, 55, 28, 19, 29, 59, 10, 61, 31, 21, 32, 65, 11, 67, 34, 23, 35, 71, 12, 73, 37, 25, 38, 77

Offset: 1

Len Smiley, Apr 26 2001

a(n+2) is absolute value of numerator of determinant of n X n matrix with M(i,j) = 2/(i(i+1)) if i=j otherwise 1. - Alexander Adamchuk, May 19 2006
Numerator of n/(n+6). - Gerry Martens, Aug 06 2015
In addition to being multiplicative, this sequence is also a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n, m)) for n, m >= 1. In particular, it follows that a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 20 2019
Connected to the Diophantine equation x^2 + (x + 1)^2 + … + (x + k)^2 = C, C an integer, x >= 0, k >= 0. Rewritten it is (k + 1)*x^2 + k*(k+1)*x + k*(k + 1)*(2*k + 1)/6 = C. The existence of its solutions depends on gcd(k + 1, k*(k + 1), k*(k + 1)*(2*k + 1)/6). - Ctibor O. Zizka, Oct 04 2023

Harry J. Smith, Table of n, a(n) for n=1..1000
Wikipedia, Quasi-polynomial
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A011782, A109047, A146535, A220466.

Cf. other sequences given by the formula n/gcd(n,k) = numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

List([1..80],n->n/(Gcd(n,2)*Gcd(n,3))); # Muniru A Asiru, Feb 20 2019

[n/(Gcd(n,2)*Gcd(n,3)) : n in [1..100]]; // Wesley Ivan Hurt, Aug 06 2015

I:=[1,1,1,2,5,1,7,4,3,5,11,2]; [n le 12 select I[n] else 2*Self(n-6)-Self(n-12): n in [1..80]]; // Vincenzo Librandi, Aug 07 2015

a := proc(n): if n = 1 then 1 else denom((4*n-6)/n) fi: end: seq(a(n), n=1..77); # Johannes W. Meijer, Dec 19 2012

Numerator[Table[(-1)^(n+1) Det[ DiagonalMatrix[ Table[ 2/(i(i+1)) - 1, {i, 1, n-2} ] ] + 1 ], {n, 30} ]] (* Alexander Adamchuk, May 19 2006 *)
Table[Numerator[(n+3)/(n+2)/(n+1)/n],{n,60}] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)
Table[n/(GCD[n, 2] GCD[n, 3]), {n, 100}] (* Wesley Ivan Hurt, Aug 06 2015 *)
LinearRecurrence[{0,0,0,0,0,2,0,0,0,0,0,-1}, {1,1,1,2,5,1,7,4,3,5,11,2}, 80] (* Vincenzo Librandi, Aug 07 2015 *)

a(n) = { n / (gcd(n, 2) * gcd(n, 3)) } \\ Harry J. Smith, Jul 11 2009

[lcm(n,6)/6 for n in range(1, 78)] # Zerinvary Lajos, Jun 07 2009

G.f.: x*(1 + x + x^2 + 2*x^3 + 5*x^4 + x^5 + 5*x^6 + 2*x^7 + x^8 + x^9 + x^10)/(1 - x^6)^2.
Multiplicative with a(2^e)=2^(e-1), a(3^e)=3^(e-1), a(p^e)=p^e, p>3. - Vladeta Jovovic, Sep 09 2004
a(n) = Numerator[(-1)^(n+1)*Det[DiagonalMatrix[Table[2/(i(i+1))-1, {i,1,n-2}]]+1]], n>2. - Alexander Adamchuk, May 19 2006
a(n) divides n. a(6k) = k for integer k>0. a(p^k) = p^k for prime p>3 and integer k>0. - Alexander Adamchuk, Sep 20 2006
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109047(n)/6.
Dirichlet g.f. zeta(s-1)*(1-1/2^s-2/3^s+2/6^s). (End)
a(n) = denominator((4*n-6)/n), n >= 2, with a(1) = 1. - Johannes W. Meijer, Dec 19 2012
a((2*n-1)*2^p) = A011782(p)*A146535(n), p >= 0. - Johannes W. Meijer, Feb 06 2013
a(n) = 2*a(n-6) - a(n-12) for n >= 12. - Robert Israel, Aug 06 2015
a(n) = gcd((n-1)*n*(n+1)/6, n). - Lechoslaw Ratajczak, Feb 16 2017
From Peter Bala, Feb 13 2019: (Start)
a(n) = n/gcd(n,n + 6) = n/gcd(n,6).
a(n) = n/b(n), where b(n) is the purely periodic sequence [1,2,3,2,1,6,...] with period 6.
a(n) is a quasi-polynomial in n: a(6*n+1) = 6*n + 1; a(6*n+2) = 3*n + 1; a(6*n+3) = 2*n + 1; a(6*n+4) = 3*n + 2; a(6*n+5) = 6*n + 5; a(6*n) = n.
a(n) = numerator(n/(n + 6)); a(n) = denominator((n + 6)/n).
(End)
Sum_{k=1..n} a(k) ~ 7*n^2/24. - Vaclav Kotesovec, Aug 09 2022
From Ctibor O. Zizka, Oct 04 2023: (Start)
For k >=0, a(k) = gcd(k + 1, k*(k + 1), k*(k + 1)*(2*k + 1)/6).
If (k mod 6) = 0 or 4 then a(k) = (k + 1).
If (k mod 6) = 1 or 3 then a(k) = (k + 1)/2.
If (k mod 6) = 2 then a(k) = (k + 1)/3.
If (k mod 6) = 5 then a(k) = (k + 1)/6. (End)

A096431 Denominator of (9(n^4 - 2n^3 + 2n^2 - n) + 2)/(2(2*n - 1)).

Original entry on oeis.org

1, 3, 1, 7, 9, 11, 13, 3, 17, 19, 21, 23, 5, 27, 29, 31, 33, 7, 37, 39, 41, 43, 9, 47, 49, 51, 53, 11, 57, 59, 61, 63, 13, 67, 69, 71, 73, 15, 77, 79, 81, 83, 17, 87, 89, 91, 93, 19, 97, 99, 101, 103, 21, 107, 109, 111, 113, 23, 117, 119, 121, 123, 25, 127, 129, 131, 133

Offset: 1

Eric W. Weisstein, Aug 09 2004

From Altug Alkan, Apr 13 2018: (Start)
Also numerator of (2*n-1)/5.
Proof: Since 9*(n^4-2*n^3+2*n^2-n)+2 = 9*n^4-18*n^3+18*n^2-9*n+2 = (3*n^2-3*n+1)*(3*n^2-3*n+2), this is an even number. So we can see that 9*n^4-18*n^3+18*n^2-9*n+2 = (4*n^3-7*n^2+5*n-2)*(2*n-1) + n^2*(n^2+1) and we should focus on the n^2*(n^2+1)/(2*n-1) part for the denominator, n^2+1 = ((2*n-1)/4)*((2*n+1)+5/(2*n-1)) and n^2*(n^2+1)/(2*n-1) = (n^2/4)*(2*n+1+5/(2*n-1)).
Since gcd(n^2, 2*n-1) = 1 and 4 is always killed by the numerator part independent of denominator of 5/(2*n-1), the denominator of (9*(n^4-2*n^3+2*n^2-n)+2)/(2*(2*n-1)) will always be determined by the denominator of 5/(2*n-1).
In other words, this is the numerator of (2*n-1)/5. (End)

			1, 28/3, 38, 703/7, 1891/9, 4186/11, ...

G. C. Greubel, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Magic Hexagon
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

Cf. A096430 (numerators), A097361, A146535.

A096431:= func< n | Numerator((2*n-1)/5) >; [A096431(n): n in [1..60]]; // G. C. Greubel, Oct 14 2024

Table[ (9(n^4-2n^3+2n^2-n)+2)/(2(2n-1)),{n,80}]//Denominator (* or *)
LinearRecurrence[{0,0,0,0,2,0,0,0,0,-1},{1,3,1,7,9,11,13,3,17,19},80] (* Harvey P. Dale, Aug 25 2021 *)

a(n) = numerator((2*n-1)/5); \\ Altug Alkan, Apr 13 2018

first(n) = my(res = vector(n, i, 2*i - 1)); forstep(i = 3, n, 5, res[i]/=5); res \\ David A. Corneth, Apr 15 2018

def A096431(n): return numerator((2*n-1)/5)
[A096431(n) for n in range(1,61)] # G. C. Greubel, Oct 14 2024

Satisfies a linear recurrence with characteristic polynomial (1-x^5)^2.

G.f.: x*(1+x)*(1+2*x-x^2+8*x^3+x^4+8*x^5-x^6+2*x^7+x^8)/((1-x)^2*(1+x+x^2+x^3+x^4)^2). - R. J. Mathar, Mar 11 2011

A194767 Denominator of the fourth increasing diagonal of the autosequence of second kind from (-1)^n / (n+1).

Original entry on oeis.org

2, 2, 12, 20, 10, 42, 56, 24, 90, 110, 44, 156, 182, 70, 240, 272, 102, 342, 380, 140, 462, 506, 184, 600, 650, 234, 756, 812, 290, 930, 992, 352, 1122, 1190, 420, 1332, 1406, 494, 1560, 1640, 574, 1806, 1892, 660, 2070, 2162, 752, 2352, 2450, 850, 2652, 2756, 954, 2970, 3080, 1064, 3306, 3422, 1180, 3660

Offset: 0

Paul Curtz, Sep 02 2011

The autosequence of first kind from (-1)^n/(n+1) is A189733.
For the second kind (the second increasing diagonal is (-1)^n/(n+1), half of the main one):
      2,     1,     0,  -1/2,  -1/3,   1/6,   1/2,  5/12,
     -1,    -1,  -1/2,   1/6,   1/2,   1/3, -1/12, -7/20,
      0,   1/2,   2/3,   1/3,  -1/6, -5/12, -4/15,  1/12,
    1/2,   1/6,  -1/3,  -1/2,  -1/4,  3/20,  7/20, 13/60,
   -1/3,  -1/2,  -1/6,   1/4,   2/5,   1/5, -2/15, -3/10,
   -1/6,   1/3,  5/12,  3/20,  -1/5,  -1/3,  -1/6,  5/42,
    1/2,  1/12, -4/15, -7/20, -2/15,   1/6,   2/7,   1/7,
  -5/12, -7/20, -1/12, 13/60,  3/10,  5/42,  -1/7,  -1/4.
Main diagonal: (period 2:repeat 2, -1)/A026741(n+1).
Second (increasing) diagonal: (-1)^n / (n+1).
Third (increasing) diagonal: (-1)^(n+1)*A026741(n) / A045896(n).
Fourth (increasing) diagonal: (-1)^(n+1)*A146535(n)/ a(n).

OEIS Wiki, Autosequence.
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A001504 = 2*A060544, A049450 = 2*A000326, A045945 = 6*A005449.

Cf. A026741, A045896, A051176, A146535, A189733, A306368.

c = Table[1/9 (7 n + 7 n^2 + 2 n Cos[2 n *Pi/3] + 2 n^2 Cos[2 n *Pi/3] + 2 Sqrt[3] n Sin[2 n *Pi/3] + 2 Sqrt[3] n^2 Sin[2 n *Pi/3]), {n, 1, 50}] (* Roger Bagula, Mar 25 2012 *)
a[n_] := (n+1) * Numerator[(n+2)/3]; Array[a, 60, 0] (* Amiram Eldar, Sep 17 2023 *)
LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{2,2,12,20,10,42,56,24,90},60] (* Harvey P. Dale, May 15 2025 *)

a(3*n) = (3*n+1)*(3*n+2), a(3*n+1) = (n+1)*(3*n+2), a(3*n+2) = 3*(n+1)*(3*n+4).
G.f.: 2*(1+x+6*x^2+7*x^3+2*x^4+3*x^5+x^6)/(1-x^3)^3. - Jean-François Alcover, Nov 11 2016
a(n+2) = 2 * A306368(n) for n >= 0. - Joerg Arndt, Aug 25 2023
a(n) = (n+1) * A051176(n+2) for n >= 0. - Paul Curtz, Sep 13 2023
Sum_{n>=0} 1/a(n) = 1 + log(3) - Pi/(3*sqrt(3)). - Amiram Eldar, Sep 17 2023

A206771 0 followed by the numerators of the reduced (A001803(n) + A001790(n)) / (2*A046161(n)).

Original entry on oeis.org

0, 1, 1, 9, 5, 175, 189, 1617, 429, 57915, 60775, 508079, 264537, 8788507, 9100525, 75218625, 9694845, 5109183315, 5250613995, 43106892675, 22090789875, 723694276305, 740104577355, 6049284520695, 1543768261425, 201547523019375

Offset: 0

Paul Curtz, Jan 10 2013

We write the fractions a(n)/b(n) and higher order differences as a matrix:
0,         1,      1,    9/8,    5/4,...
1,         0,    1/8,    1/8, 15/128,... = (A001790(n)/A046161(n) +Lorbeta(n)) /2
-1,      1/8,      0, -1/128, -1/128,... = (Lorbeta(n+1) +A161200(n+1)/A046161(n+1)) / 2
9/8,    -1/8, -1/128,      0, 1/1024,...
-5/4, 15/128,  1/128, 1/1024,      0,...
Here, Lorbeta(0)=1 and Lorbeta(n) = -A098597(n-1)/A046161(n) for n>0 is the inverse of the Lorentz factor.
The first line with numerators a(n) and denominators b(n) is 0, 1, 1, 9/8, 5/4, 175/128, 189/128, 1617/1024, 429/256, 57915/32768, 60775/32768,... It is an autosequence: Its inverse binomial transform is the signed sequence.
a(n+1)/(2*n-1)= 1, 3, 1, 25, 21, 147, 33, 3861, 3575, 26741,... .
a(n+1)/A146535(n) = 9, 5, 35, 27, 539, 39, 4455,... .
A001790(n)/A046161(n) yields the coefficients of the Lorentz factor (or Lorentz gamma factor). With b for beta and g for gamma:
g = (1-b^2)^-1 = 1 + (b^2)/2 + 3*(b^4)/8 + 5*(b^6)/16 + ... .
b = (1-g^-2)^-1 = 1 - (g^-2)/2 - (g^-4)/8 - (g^-6)/16  - ... .
Are the denominators of the first subdiagonal 1, 1/8, -1/128, 1/1024,...  A061549(n) ?
a(n+1)/(A000108(n)*b(n)) = 1, 1, 9/16, 1/4, 25/256, 9/256, 49/4096, 1/256, 81/65536, 25/65536, 121/1048576,... = A191871(n+1)/ A084623(n+1)^2 ?

			From the first formula: a(1)=1*1, a(2)=1*1, a(3)=3*3, a(4)=1*5, a(5)=5*35, a(6)=3*63.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
OEIS Wiki, Autosequence
Wikipedia, Lorentz Factor.

Cf. A187791, A000108.

/* By definition: */ m:=25; R:=PowerSeriesRing(Rationals(), m); p:=Coefficients(R!(1/(1-x)^(1/2))); q:=Coefficients(R!((1-x)^(-3/2))); A001790:=[Numerator(p[i]): i in [1..m]]; A001803:=[Numerator(q[i]): i in [1..m]]; A046161:=[Denominator(Binomial(2*n,n)/4^n): n in [0..m-1]]; [0] cat [Numerator((A001803[n]+A001790[n])/(2*A046161[n])): n in [1..m]]; // Bruno Berselli, Mar 11 2013

A206771 := proc(n)
        A001790(n)+A001803(n) ;
        %/2/A046161(n) ;
        numer(%) ;
end proc: # R. J. Mathar, Jan 18 2013

max = 25; A001803 = CoefficientList[Series[(1 - x)^(-3/2), {x, 0, max}], x] // Numerator; A001790 = CoefficientList[Series[1/Sqrt[(1 - x)], {x, 0, max}], x] // Numerator; A046161 = Table[Binomial[2n, n]/4^n, {n, 0, max}] // Denominator; a[n_] := (A001803[[n]] + A001790[[n]])/(2*A046161[[n]]) // Numerator; a[0] = 0; Table[a[n], {n, 0, max}]
(* or (from 1st formula) : *) Table[ n*Numerator[4^(1-n)*Binomial[2n-2, n-1]]/2^IntegerExponent[n, 2], {n, 0, max}]
(* or (from 2nd formula) : *) Table[ Numerator[ CatalanNumber[n-1]/2^(2n-1)]*Numerator[n^2/2^n], {n, 0, max}] (* Jean-François Alcover, Jan 31 2013 *)

a(n) = A000265(n) * A001790(n-1).
a(n) = A098597(n-1) * A191871(n). See also A181318
a(n) = numerator of n*binomial(2n-2,n-1)/4^(n-1). - Nathaniel Johnston, Dec 16 2022

a(11)-a(25) from Jean-François Alcover, Jan 13 2013

cp's OEIS Frontend

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

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A007494 Numbers that are congruent to 0 or 2 mod 3.

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A060747 a(n) = 2*n - 1.

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A004273 0 together with odd numbers.

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A060789 a(n) = n / (gcd(n,2) * gcd(n,3)).

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A096431 Denominator of (9*(n^4 - 2*n^3 + 2*n^2 - n) + 2)/(2*(2*n - 1)).

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A096431 Denominator of (9(n^4 - 2n^3 + 2n^2 - n) + 2)/(2(2*n - 1)).