A169609 -id:A169609 - cp's OEIS frontend

A168615 Inverse binomial transform of A169609, or of A144437 preceded by 1.

1, 2, -2, 0, 6, -18, 36, -54, 54, 0, -162, 486, -972, 1458, -1458, 0, 4374, -13122, 26244, -39366, 39366, 0, -118098, 354294, -708588, 1062882, -1062882, 0, 3188646, -9565938, 19131876, -28697814, 28697814, 0, -86093442, 258280326, -516560652

Offset: 0

Paul Curtz, Dec 01 2009

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-3).

Cf. A169609, A144437, A000748, A057083, A057682.

[ n le 2 select n else n eq 3 select -2 else -3*Self(n-1)-3*Self(n-2): n in [1..37] ]; // Klaus Brockhaus, Dec 03 2009

Join[{1,2,-2}, LinearRecurrence[{-3, -3}, {0, 6}, 25]] (* G. C. Greubel, Jul 27 2016 *)
LinearRecurrence[{-3,-3},{1,2,-2},40] (* Harvey P. Dale, Jul 21 2024 *)

a(n) = -3*a(n-1) - 3*a(n-2) for n > 2; a(0) = 1, a(1) = 2, a(2) = -2.
a(n) = 2*A123877(n-1), n>0.
G.f.: 1+2*x*(1+2*x)/(1+3*x+3*x^2).
a(6*m + 3) = 0, m>=0. - G. C. Greubel, Jul 27 2016

Edited and extended by Klaus Brockhaus, Dec 03 2009

A168673 Binomial transform of A169609.

Original entry on oeis.org

1, 4, 10, 20, 38, 74, 148, 298, 598, 1196, 2390, 4778, 9556, 19114, 38230, 76460, 152918, 305834, 611668, 1223338, 2446678, 4893356, 9786710, 19573418, 39146836, 78293674, 156587350, 313174700, 626349398, 1252698794, 2505397588, 5010795178, 10021590358

Offset: 0

Paul Curtz, Dec 02 2009

Sequence and successive differences are identical to their third differences. See principal sequence A024495. Main diagonal of the array of successive differences is A083595 (1,6,8,20,36,...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A169609, A144437, A168615, A024495, A083595, A130772, A062092, A130151.

I:=[1,4,10]; [n le 3 select I[n] else 3*Self(n-1)- 3*Self(n-2)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 30 2016

LinearRecurrence[{3,-3,2},{1,4,10},25] (* G. C. Greubel, Jul 29 2016 *)
RecurrenceTable[{a[0] == 1, a[1] == 4, a[2] == 10, a[n] == 3 a[n-1] - 3 a[n-2] + 2 a[n-3]}, a, {n, 40}] (* Vincenzo Librandi, Jul 30 2016 *)

a(n)=([0,1,0; 0,0,1; 2,-3,3]^n*[1;4;10])[1,1] \\ Charles R Greathouse IV, Jul 30 2016

a(n+1) - 2a(n) = A130772(n).
a(n) = A062092(n) - A130151(n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) for n > 2; a(0) = 1, a(1) = 4, a(2) = 10.
G.f.: (1 + x + x^2)/(1 -3*x +3*x^2 -2*x^3). - Philippe Deléham, Dec 03 2009

Edited and extended by Klaus Brockhaus, Dec 03 2009

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

Original entry on oeis.org

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

A144437 Period 3: repeat [3, 3, 1].

Original entry on oeis.org

3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3

Offset: 1

Paul Curtz, Oct 05 2008

The sequence is generated from numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...
Conjecture: a(n) is the separatix. See A045944.
Also the decimal expansion of the constant 3310/999. - R. J. Mathar, May 21 2009
Continued fraction expansion of A171417.
Greatest common divisor of (n+1)^2-1 and (n+1)^2+2. - Bruno Berselli, Mar 08 2017

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A000217, A045944, A109007, A164306, A167192, A169609, A256095.

Numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...

&cat [[3, 3, 1]^^30]; // Wesley Ivan Hurt, Jul 02 2016

seq(op([3, 3, 1]), n=1..50); # Wesley Ivan Hurt, Jul 02 2016

A144437[n_]:=Denominator[n/3]; Array[A144437,100] (* Enrique Pérez Herrero, Oct 05 2011 *)
CoefficientList[Series[(3 + 3 x + x^2)/(1 - x^3), {x, 0, 120}], x] (* Michael De Vlieger, Jul 02 2016 *)
Table[Mod[2*n^2 + 1, 3,1], {n,1,50}] (* G. C. Greubel, Aug 24 2017 *)

a(n)=if(n%3,3,1) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = (7-4*cos(2*Pi*n/3))/3. - Jaume Oliver Lafont, Nov 23 2008
G.f.: x*(3 + 3*x + x^2)/((1 - x)*(1 + x + x^2)). - R. J. Mathar, May 21 2009
a(n) = 3/gcd(n,3). - Reinhard Zumkeller, Oct 30 2009
a(n) = denominator(n^k/3), where k>0 is an integer. - Enrique Pérez Herrero, Oct 05 2011
a(n) = gcd(T(n+1), T(2)) = A256095(n+1, 2), with the triangular numbers T = A000217, for n >= 1. - Wolfdieter Lang, Mar 17 2015
a(n) = a(n-3) for n>3; a(n) = A169609(n) for n>0. - Wesley Ivan Hurt, Jul 02 2016
E.g.f.: (1/3)*(7*exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2) - 3). - G. C. Greubel, Aug 24 2017
From Nicolas Bělohoubek, Nov 11 2021: (Start)
a(n) = 9/(a(n-2)*a(n-1)).
a(n) = 7 - a(n-2) - a(n-1). See also A052901 or A069705. (End)

Edited by R. J. Mathar, May 21 2009

A146535 Numerator of (2*n-1)/3.

Original entry on oeis.org

1, 1, 5, 7, 3, 11, 13, 5, 17, 19, 7, 23, 25, 9, 29, 31, 11, 35, 37, 13, 41, 43, 15, 47, 49, 17, 53, 55, 19, 59, 61, 21, 65, 67, 23, 71, 73, 25, 77, 79, 27, 83, 85, 29, 89, 91, 31, 95, 97, 33, 101, 103, 35, 107, 109, 37, 113, 115, 39, 119, 121, 41, 125, 127, 43, 131, 133, 45

Offset: 1

Artur Jasinski, Oct 31 2008

From Jaroslav Krizek, May 28 2010: (Start)
a(n+1) = numerators of antiharmonic mean of the first n positive integers for n >= 1.
See A169609(n-1) - denominators of antiharmonic mean of the first n positive integers for n >= 1. (End)

			Fractions begin with 1/6, 1/2, 5/6, 7/6, 3/2, 11/6, 13/6, 5/2, 17/6, 19/6, 7/2, 23/6, ...

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A146306, A146307, A146308, A146309, A146310, A146311, A146312, A146313.
Cf. A061347, A102283.
Trisections: A016921, A005408, A016969. [R. J. Mathar, Nov 21 2008]

Table[Numerator[(2 n - 1)/6], {n, 1, 100}]
LinearRecurrence[{0,0,2,0,0,-1},{1,1,5,7,3,11},100] (* Harvey P. Dale, Feb 24 2015 *)

a(n) = numerator((2*n-1)/3); \\ Altug Alkan, Apr 13 2018

From R. J. Mathar, Nov 21 2008: (Start)
a(n) = 2*a(n-3) - a(n-6).
G.f.: x(1+x)(1+5x^2+x^4)/((1-x)^2*(1+x+x^2)^2). (End)
Sum_{k=1..n} a(k) ~ (7/9) * n^2. - Amiram Eldar, Apr 04 2024
a(n) = (2*n - 1)*(7 - A061347(n) +3*A102283(n))/9. - Stefano Spezia, Feb 14 2025

Name edited by Altug Alkan, Apr 13 2018

A201277 T(n,k) = number of n X k 0..2 arrays with every row and column nondecreasing rightwards and downwards, and the number of instances of each value within one of each other.

Original entry on oeis.org

3, 3, 3, 1, 5, 1, 3, 4, 4, 3, 3, 12, 7, 12, 3, 1, 16, 14, 14, 16, 1, 3, 9, 21, 85, 21, 9, 3, 3, 27, 41, 199, 199, 41, 27, 3, 1, 33, 54, 143, 556, 143, 54, 33, 1, 3, 16, 86, 740, 442, 442, 740, 86, 16, 3, 3, 48, 120, 1274, 2827, 1260, 2827, 1274, 120, 48, 3, 1, 56, 168, 759, 5680

Offset: 1

R. H. Hardin, Nov 29 2011

Table starts
.3..3...1....3.....3.....1......3.......3.......1........3........3........1
.3..5...4...12....16.....9.....27......33......16.......48.......56.......25
.1..4...7...14....21....41.....54......86.....120......168......218......307
.3.12..14...85...199...143....740....1274.....759.....3416.....5312.....2746
.3.16..21..199...556...442...2827....5680....3651....19960....35039....19820
.1..9..41..143...442..1260...3113....7331...15969....32737....63942...120318
.3.27..54..740..2827..3113..27008...71704...59070...408600...894812...621910
.3.33..86.1274..5680..7331..71704..215577..200867..1537010..3717215..2850024
.1.16.120..759..3651.15969..59070..200867..613916..1755850..4655000.11705266
.3.48.168.3416.19960.32737.408600.1537010.1755850.16442049.47956581.43851889

			Some solutions for n=7 k=5
..0..0..0..0..0....0..0..0..0..0....0..0..0..0..0....0..0..0..0..1
..0..0..0..1..1....0..0..0..0..1....0..0..0..0..2....0..0..0..1..1
..0..1..1..1..1....0..0..1..1..1....0..0..0..1..2....0..0..0..1..2
..0..1..2..2..2....1..1..2..2..2....1..1..1..1..2....0..0..1..2..2
..0..1..2..2..2....1..1..2..2..2....1..1..1..2..2....1..1..1..2..2
..0..1..2..2..2....1..1..2..2..2....1..1..2..2..2....1..1..2..2..2
..1..1..2..2..2....1..1..2..2..2....1..2..2..2..2....1..2..2..2..2

R. H. Hardin, Table of n, a(n) for n = 1..684

Columns k=1-7 give: A169609, A201271, A201272, A201273, A201274, A201275, A201276.

Main diagonal gives A201270.

T(n,1) = binomial(3,n modulo 3). For a 0..z array, T(n,1) = binomial(z+1, n modulo (z+1)).

A194880 The numerators of the inverse Akiyama-Tanigawa algorithm from A001045(n).

Original entry on oeis.org

0, -1, -1, -4, -5, -2, -7, -8, -3, -10, -11, -4, -13, -14, -5, -16, -17, -6, -19, -20, -7, -22, -23, -8, -25, -26, -9, -28, -29, -10, -31, -32, -11, -34, -35, -12, -37, -38, -13, -40, -41, -14, -43, -44, -15, -46, -47, -16, -49, -50, -17, -52, -53, -18, -55, -56, -19, -58, -59, -20

Offset: 0

Paul Curtz, Sep 07 2011

0,  -1, -1, -4/3, -5/3, -2, -7/3, -8/3, -3, -10/3, -11/3, -4, -13/4, -14/3, -5,    = a(n)/b(n),
1,    0,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
1,   -2, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
3,   -2,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
5,   -6, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
11, -10,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
21, -22, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
Vertical: A001045(n), -A078008(n), (-1)^(n+1)*A000012(n), (-1)^(n+1)*A010709(n)/A010701(n), (-1)^(n+1)*A010716(n+1)/A010701(n), A007395(n), .. .
a(n)=0, 1 before (-A145064(n+1)=-A051176(n+3).
b(n)=1, 1 before A169609(n). b(n)=1, 1, 1 before A144437(n+1).
a(n+5)-a(n+2)=b(n+5)  (=-1,-3,-3,=-A169609(n)).

G. C. Greubel, Table of n, a(n) for n = 0..5000
D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

a[0]=0; a[1]=-1; a[n_] := (-n-1)/Max[1, 2*Mod[n, 3]-1]; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Sep 18 2012 *)

a(3*n)=-3*n-1 except a(0)=0; a(3*n+1)=-3*n-2 except a(1)=-1; a(3*n+2)=-n-1.
From Chai Wah Wu, May 07 2024: (Start)
a(n) = 2*a(n-3) - a(n-6) for n > 7.
G.f.: x*(x^6 + x^5 - 3*x^3 - 4*x^2 - x - 1)/(x^6 - 2*x^3 + 1). (End)

A171419 Decimal expansion of (5+sqrt(65))/10.

Original entry on oeis.org

1, 3, 0, 6, 2, 2, 5, 7, 7, 4, 8, 2, 9, 8, 5, 4, 9, 6, 5, 2, 3, 6, 6, 6, 1, 3, 2, 3, 0, 3, 0, 3, 7, 7, 1, 1, 3, 1, 1, 3, 4, 3, 9, 6, 3, 0, 5, 6, 0, 8, 5, 7, 3, 3, 8, 7, 9, 3, 6, 5, 9, 2, 3, 8, 9, 2, 6, 3, 8, 7, 4, 9, 5, 1, 0, 2, 5, 6, 8, 8, 2, 0, 2, 9, 6, 0, 1, 5, 4, 2, 6, 6, 7, 1, 7, 2, 3, 0, 6, 4, 3, 6, 3, 7, 4

Offset: 1

Klaus Brockhaus, Dec 08 2009

Continued fraction expansion of (5+sqrt(65))/10 is A169609.

			(5+sqrt(65))/10 = 1.30622577482985496523....

Cf. A010517 (decimal expansion of sqrt(65)), A169609 (repeat 1, 3, 3).

RealDigits[(5+Sqrt[65])/10,10,120][[1]]  (* Harvey P. Dale, Apr 02 2011 *)

A222591 Numerators of (n*(n - 3)/6) + 1, arising as the maximum possible number of triple lines for an n-element set.

Original entry on oeis.org

1, 5, 8, 4, 17, 23, 10, 38, 47, 19, 68, 80, 31, 107, 122, 46, 155, 173, 64, 212, 233, 85, 278, 302, 109, 353, 380, 136, 437, 467, 166, 530, 563, 199, 632, 668, 235, 743, 782, 274, 863, 905, 316, 992, 1037, 361, 1130, 1178, 409, 1277, 1328

Offset: 3

Jonathan Vos Post, Feb 25 2013

Numerators of (n*(n - 3)/6) + 1, which arises as the maximum possible number of triple lines for an n-element set, according to Green and Tao, cited in Elekes. The fractions for n = 3, 4, 5, 6, ... are 1/1, 5/3, 8/3, 4/1, 17/3, 23/3, 10/1, 38/3, 47/3, 19/1, 68/3, 80/3, 31/1, 107/3, 122/3, 46/1, 155/3, 173/3, 64/1, 212/3, 233/3, 85/1, 278/3, 302/3, 109/1, 353/3, 380/3, 136/1, 437/3, 467/3, 166/1, 530/3, 563/3, 199/1, 632/3, 668/3, 235/1, 743/3, 782/3, 274/1, 863/3, 905/3, 316/1, 992/3, 1037/3, 361/1, 1130/3, 1178/3, 409/1, 1277/3, 1328/3. The corresponding denominators are A169609.

			a(10) = 38 because (10*(10 - 3)/6) + 1 = 38/3.

György Elekes, Endre Szabó, On Triple Lines and Cubic Curves --- the Orchard Problem revisited, arXiv:1302.5777 [math.CO], Feb 23, 2013.
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A169609.

Numerator[Table[(n(n-3))/6+1,{n,3,60}]] (* or *) LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{1,5,8,4,17,23,10,38,47},60] (* Harvey P. Dale, Feb 11 2015 *)

cp's OEIS Frontend

A168615 Inverse binomial transform of A169609, or of A144437 preceded by 1.

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A168673 Binomial transform of A169609.

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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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A144437 Period 3: repeat [3, 3, 1].

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A146535 Numerator of (2*n-1)/3.

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