cp's OEIS Frontend

Number of binary bracelets of n beads, 3 of them 0. For n >= 3, a(n-3) is the number of binary bracelets of n beads, 3 of them 0, with 00 prohibited. - Washington Bomfim, Aug 27 2008

Also number of partitions of n-3 into parts 1, 2, and 3. - Joerg Arndt, Sep 05 2013

Number of incongruent triangles with integer sides that have perimeter 2n-3 (see the Jordan et al. link). - Freddy Barrera, Aug 18 2018

Number of ordered triples (x,y,z) of nonnegative integers such that x+y+z=n and xDennis P. Walsh, Apr 19 2019

Number of incongruent triangles formed from any 3 vertices of a regular n-gon. - Frank M Jackson, Sep 11 2022

Also a(n-3) for n > 2, otherwise 0 is the number of incongruent scalene triangles formed from the vertices of a regular n-gon. - Frank M Jackson, Nov 27 2022

In the following guide to related sequences: M=max(x,y,z), m=min(x,y,z), and R=range=M-m. In some cases, it is an offset of the listed sequence which fits the conditions shown for w,x,y. Each sequence satisfies a linear recurrence relation, some of which are identified in the list by the following code (signature):

A: 2, 0, -2, 1, i.e., a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4);

B: 3, -2, -2, 3, -1;

C: 4, -6, 4, -1;

D: 1, 2, -2, -1, 1;

E: 2, 1, -4, 1, 2, -1;

F: 2, -1, 1, -2, 1;

G: 2, -1, 0, 1, -2, 1;

H: 2, -1, 2, -4, 2, -1, 2, -1;

I: 3, -3, 2, -3, 3, -1;

J: 4, -7, 8, -7, 4, -1.

...

A212959 ... |w-x|=|x-y| ...... recurrence type A

A212960 ... |w-x| != |x-y| ................... B

A212683 ... |w-x| < |x-y| .................... B

A212684 ... |w-x| >= |x-y| ................... B

A212963 ... see entry for definition ......... B

A212964 ... |w-x| < |x-y| < |y-w| ............ B

A006331 ... |w-x| < y ........................ C

A005900 ... |w-x| <= y ....................... C

A212965 ... w = R ............................ D

A212966 ... 2*w = R

A212967 ... w < R ............................ E

A212968 ... w >= R ........................... E

A077043 ... w = x > R ........................ A

A212969 ... w != x and x > R ................. E

A212970 ... w != x and x < R ................. E

A055998 ... w = x + y - 1

A011934 ... w < floor((x+y)/2) ............... B

A182260 ... w > floor((x+y)/2) ............... B

A055232 ... w <= floor((x+y)/2) .............. B

A011934 ... w >= floor((x+y)/2) .............. B

A212971 ... w < floor((x+y)/3) ............... B

A212972 ... w >= floor((x+y)/3) .............. B

A212973 ... w <= floor((x+y)/3) .............. B

A212974 ... w > floor((x+y)/3) ............... B

A212975 ... R is even ........................ E

A212976 ... R is odd ......................... E

A212978 ... R = 2*n - w - x

A212979 ... R = average{w,x,y}

A212980 ... w < x + y and x < y .............. B

A212981 ... w <= x+y and x < y ............... B

A212982 ... w < x + y and x <= y ............. B

A212983 ... w <= x + y and x <= y ............ B

A002623 ... w >= x + y and x <= y ............ B

A087811 ... w = 2*x + y ...................... A

A008805 ... w = 2*x + 2*y .................... D

A000982 ... 2*w = x + y ...................... F

A001318 ... 2*w = 2*x + y .................... F

A001840 ... w = 3*x + y

A212984 ... 3*w = x + y

A212985 ... 3*w = 3*x + y

A001399 ... w = 2*x + 3*y

A212986 ... 2*w = 3*x + y

A008810 ... 3*x = 2*x + y .................... F

A212987 ... 3*w = 2*x + 2*y

A001972 ... w = 4*x + y ...................... G

A212988 ... 4*w = x + y ...................... G

A212989 ... 4*w = 4*x + y

A008812 ... 5*w = 2*x + 3*y

A016061 ... n < w + x + y <= 2*n ............. C

A000292 ... w + x + y <=n .................... C

A000292 ... 2*n < w + x + y <= 3*n ........... C

A212977 ... n/2 < w + x + y <= n

A143785 ... w < R < x ........................ E

A005996 ... w < R <= x ....................... E

A128624 ... w <= R <= x ...................... E

A213041 ... R = 2*|w - x| .................... A

A213045 ... R < 2*|w - x| .................... B

A087035 ... R >= 2*|w - x| ................... B

A213388 ... R <= 2*|w - x| ................... B

A171218 ... M < 2*m .......................... B

A213389 ... R < 2|w - x| ..................... E

A213390 ... M >= 2*m ......................... E

A213391 ... 2*M < 3*m ........................ H

A213392 ... 2*M >= 3*m ....................... H

A213393 ... 2*M > 3*m ........................ H

A213391 ... 2*M <= 3*m ....................... H

A047838 ... w = |x + y - w| .................. A

A213396 ... 2*w < |x + y - w| ................ I

A213397 ... 2*w >= |x + y - w| ............... I

A213400 ... w < R < 2*w

A069894 ... min(|w-x|,|x-y|) = 1

A000384 ... max(|w-x|,|x-y|) = |w-y|

A213395 ... max(|w-x|,|x-y|) = w

A213398 ... min(|w-x|,|x-y|) = x ............. A

A213399 ... max(|w-x|,|x-y|) = x ............. D

A213479 ... max(|w-x|,|x-y|) = w+x+y ......... D

A213480 ... max(|w-x|,|x-y|) != w+x+y ........ E

A006918 ... |w-x| + |x-y| > w+x+y ............ E

A213481 ... |w-x| + |x-y| <= w+x+y ........... E

A213482 ... |w-x| + |x-y| < w+x+y ............ E

A213483 ... |w-x| + |x-y| >= w+x+y ........... E

A213484 ... |w-x|+|x-y|+|y-w| = w+x+y

A213485 ... |w-x|+|x-y|+|y-w| != w+x+y ....... J

A213486 ... |w-x|+|x-y|+|y-w| > w+x+y ........ J

A213487 ... |w-x|+|x-y|+|y-w| >= w+x+y ....... J

A213488 ... |w-x|+|x-y|+|y-w| < w+x+y ........ J

A213489 ... |w-x|+|x-y|+|y-w| <= w+x+y ....... J

A213490 ... w,x,y,|w-x|,|x-y| distinct

A213491 ... w,x,y,|w-x|,|x-y| not distinct

A213493 ... w,x,y,|w-x|,|x-y|,|w-y| distinct

A213495 ... w = min(|w-x|,|x-y|,|w-y|)

A213492 ... w != min(|w-x|,|x-y|,|w-y|)

A213496 ... x != max(|w-x|,|x-y|)

A213498 ... w != max(|w-x|,|x-y|,|w-y|)

A213497 ... w = min(|w-x|,|x-y|)

A213499 ... w != min(|w-x|,|x-y|)

A213501 ... w != max(|w-x|,|x-y|)

A213502 ... x != min(|w-x|,|x-y|)

...

A211795 includes a guide for sequences that count 4-tuples (w,x,y,z) having all terms in {0,...,n} and satisfying selected properties. Some of the sequences indexed at A211795 satisfy recurrences that are represented in the above list.

Partial sums of the numbers congruent to {1,3} mod 6 (see A047241). - Philippe Deléham, Mar 16 2014

Let M_n be the n X n matrix of the following form: [3 2 1 0 0 0 0 0 0 0 / 2 3 2 1 0 0 0 0 0 0 / 1 2 3 2 1 0 0 0 0 0 / 0 1 2 3 2 1 0 0 0 0 / 0 0 1 2 3 2 1 0 0 0 / 0 0 0 1 2 3 2 1 0 0 / 0 0 0 0 1 2 3 2 1 0 / 0 0 0 0 0 1 2 3 2 1 / 0 0 0 0 0 0 1 2 3 2 / 0 0 0 0 0 0 0 1 2 3]. Then for n > 2 a(n) = det M_(n-2). - Benoit Cloitre, Jun 20 2002

Largest possible size for the directed Cayley graph on two generators having diameter n - 2. - Ralf Stephan, Apr 27 2003

It seems that for n >= 2, a(n) is the maximum number of non-overlapping 1 X 3 rectangles that can be packed into an n X n square. Rectangles can only be placed parallel to the sides of the square. Verified with Lobato's tool, see links. - Dmitry Kamenetsky, Aug 03 2009

Maximum number of edges in a K4-free graph with n vertices. - Yi Yang, May 23 2012

3a(n) + 1 = y^2 if n is not 0 mod 3 and 3a(n) = y^2 otherwise. - Jon Perry, Sep 10 2012

Apart from the initial term this is the elliptic troublemaker sequence R_n(1, 3) (also sequence R_n(2, 3)) in the notation of Stange (see Table 1, p. 16). For other elliptic troublemaker sequences R_n(a, b) see the cross references below. - Peter Bala, Aug 08 2013

The number of partitions of 2n into exactly 3 parts. - Colin Barker, Mar 22 2015

a(n-1) is the maximum number of non-overlapping triples (i,k), (i+1, k+1), (i+2, k+2) in an n X n matrix. Details: The triples are distributed along the main diagonal and 2*(n-1) other diagonals. Their maximum number is floor(n/3) + 2*Sum_{k = 1..n-1} floor(k/3) = floor((n-1)^2/3). - Gerhard Kirchner, Feb 04 2017

Conjecture: a(n) is the number of intersection points of n cevians that cut a triangle into the maximum number of pieces (see A007980). - Anton Zakharov, May 07 2017

From Gus Wiseman, Oct 05 2020: (Start)

Also the number of unimodal triples (meaning the middle part is not strictly less than both of the other two) of positive integers summing to n + 1. The a(2) = 1 through a(6) = 12 triples are:

(1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5)

(1,2,1) (1,2,2) (1,2,3) (1,2,4)

(2,1,1) (1,3,1) (1,3,2) (1,3,3)

(2,2,1) (1,4,1) (1,4,2)

(3,1,1) (2,2,2) (1,5,1)

(2,3,1) (2,2,3)

(3,2,1) (2,3,2)

(4,1,1) (2,4,1)

(3,2,2)

(3,3,1)

(4,2,1)

(5,1,1)

(End)

a(n-3) is the number of aperiodic necklaces (Lyndon words) with 3 black beads and n-3 white beads.

Number of triangular partitions (see Almkvist).

Consists of arithmetic progression quadruples of common difference n+1 starting at A045943(n). Refers to the least number of coins needed to be rearranged in order to invert the pattern of a (n+1)-rowed triangular array. For instance, a 5-rowed triangular array requires a minimum of a(4)=5 rearrangements (shown bracketed here) for it to be turned upside down.

.....{*}..................{*}*.*{*}{*}

.....*.*....................*.*.*.{*}

....*.*.*....---------\......*.*.*

..{*}*.*.*...---------/.......*.*

{*}{*}*.*{*}..................{*}

- Lekraj Beedassy, Oct 13 2003

Partial sums of 1,1,1,2,2,2,3,3,3,4,4,4,... - Jon Perry, Mar 01 2004

Sum of three successive terms is a triangular number in natural order starting with 3: a(n)+a(n+1)+a(n+2) = T(n+2) = (n+2)*(n+3)/2. - Amarnath Murthy, Apr 25 2004

Apply Riordan array (1/(1-x^3),x) to n. - Paul Barry, Apr 16 2005

Absolute values of numbers that appear in A145919. - Matthew Vandermast, Oct 28 2008

In the Moree definition, (-1)^n*a(n) is the 3rd Witt transform of A033999 and (-1)^n*A004524(n) with 2 leading zeros dropped is the 2nd Witt transform of A033999. - R. J. Mathar, Nov 08 2008

Column sums of:

1 2 3 4 5 6 7 8 9.....

1 2 3 4 5 6.....

1 2 3.....

........................

----------------------

1 2 3 5 7 9 12 15 18 - Jon Perry, Nov 16 2010

a(n) is the sum of the positive integers <= n that have the same residue modulo 3 as n. They are the additive counterpart of the triple factorial numbers. - Peter Luschny, Jul 06 2011

a(n+1) is the number of 3-tuples (w,x,y) with all terms in {0,...,n} and w=3*x+y. - Clark Kimberling, Jun 04 2012

a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x-y = (1 mod 3), and x+y < n. - Clark Kimberling, Jul 02 2012

a(n+1) is the number of partitions of n into two sorts of part(s) 1 and one sort of (part) 3. - Joerg Arndt, Jun 10 2013

Arrange A004523 in rows successively shifted to the right two spaces and sum the columns:

1 2 2 3 4 4 5 6 6...

1 2 2 3 4 4 5...

1 2 2 3 4...

1 2 2...

1...

------------------------------

1 2 3 5 7 9 12 15 18... - L. Edson Jeffery, Jul 30 2014

a(n) = A258708(n+1,1) for n > 0. - Reinhard Zumkeller, Jun 23 2015

Also the number of triples of positive integers summing to n + 4, the first less than each of the other two. Also the number of triples of positive integers summing to n + 2, the first less than or equal to each of the other two. - Gus Wiseman, Oct 11 2020

Also the lower matching number of the (n+1)-triangular honeycomb king graph = n-triangular grid graph (West convention). - Eric W. Weisstein, Dec 14 2024

Equivalently, sums of three distinct powers of 2.

Appears to give all n such that 64 is the highest power of 2 dividing A005148(n). - Benoit Cloitre, Jun 22 2002

From Gus Wiseman, Oct 05 2020: (Start)

These are numbers k such that the k-th composition in standard order has length 3. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. The sequence together with the corresponding standard compositions begins:

7: (1,1,1) 44: (2,1,3) 97: (1,5,1)

11: (2,1,1) 49: (1,4,1) 98: (1,4,2)

13: (1,2,1) 50: (1,3,2) 100: (1,3,3)

14: (1,1,2) 52: (1,2,3) 104: (1,2,4)

19: (3,1,1) 56: (1,1,4) 112: (1,1,5)

21: (2,2,1) 67: (5,1,1) 131: (6,1,1)

22: (2,1,2) 69: (4,2,1) 133: (5,2,1)

25: (1,3,1) 70: (4,1,2) 134: (5,1,2)

26: (1,2,2) 73: (3,3,1) 137: (4,3,1)

28: (1,1,3) 74: (3,2,2) 138: (4,2,2)

35: (4,1,1) 76: (3,1,3) 140: (4,1,3)

37: (3,2,1) 81: (2,4,1) 145: (3,4,1)

38: (3,1,2) 82: (2,3,2) 146: (3,3,2)

41: (2,3,1) 84: (2,2,3) 148: (3,2,3)

42: (2,2,2) 88: (2,1,4) 152: (3,1,4)

(End)

If the two leading 0's are dropped, this becomes the essentially identical sequence A103221, with g.f. 1/((1-x^2)*(1-x^3)), which arises in many contexts. For example, 1/((1-x^4)*(1-x^6)) is the Poincaré series [or Poincare series] for modular forms of weight w for the full modular group. As generators one may take the Eisenstein series E_4 (A004009) and E_6 (A013973).

Dimension of the space of weight 2n+8 cusp forms for Gamma_0( 1 ).

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).

a(n) is the number of ways n can be written as the sum of a positive even number and a nonnegative multiple of 3 and so the number of ways (n-2) can be written as the sum of a nonnegative even number and a nonnegative multiple of 3 and also the number of ways (n+3) can be written as the sum of a positive even number and a positive multiple of 3.

It appears that this is also the number of partitions of 2n+6 that are 4-term arithmetic progressions. - John W. Layman, May 01 2009 [verified by Wesley Ivan Hurt, Jan 17 2021]

a(n) is the number of (n+3)-digit fixed points under the base-3 Kaprekar map A164993 (see A164997 for the list of fixed points). - Joseph Myers, Sep 04 2009

Starting from n=10 also the number of balls in new consecutive hexagonal edges, if an (infinite) chain of balls is winded spirally around the first ball at the center, such that each six steps make an entire winding. - K. G. Stier, Dec 21 2012

In any three consecutive terms at least two of them are equal to each other. - Michael Somos, Mar 01 2014

Number of partitions of (n-2) into parts 2 and 3. - David Neil McGrath, Sep 05 2014

a(n), n >= 0, is also the dimension of S_{2*(n+4)}, the complex vector space of modular cusp forms of weight 2*(n+4) and level 1 (full modular group). The dimension of S_0, S_2, S_4 and S_6 is 0. See, e.g., Ash and Gross, p. 178. Table 13.1. - Wolfdieter Lang, Sep 16 2016

From Wolfdieter Lang, May 08 2017: (Start)

a(n-2) = floor((n-2)/2) - floor((n-2)/3) = floor(n/2) - floor((n+1)/3) is for n >=0 the number of integers k in the interval (n+1)/3 < k <= floor(n/2). This problem appears in the computation of the number of zeros of Chebyshev S(n, x) polynomials (coefficients in A049310) in the open interval (-1, +1). See a comment there. This computation was motivated by a conjecture given in A008611 by Michel Lagneau, Mar 31 2017.

a(n) is also the number of integers k in the closed interval (n+1)/3 <= k <= floor(n/2), which is floor(n/2) - (ceiling((n+1)/3) - 1) for n >= 0 (proof trivial for n+1 == 0 (mod 3) and otherwise). From the preceding statement this a(n) is also a(n-2) + [n == 2 (mod 3)] for n >= 0 (with [statement] = 1 if the statement is true and zero otherwise). This proves the recurrence given by Michael Somos in the formula section. (End)

Assuming the Collatz conjecture to be true, for n > 1, a(n+7) is the row length of the n-th row of A340985. That is, the number of weakly connected components of the Collatz digraph of order n. - Sebastian Karlsson, Feb 23 2021

Molien series for 4-dimensional representation of S_4 [Nebe, Rains, Sloane, Chap. 7].

Also number of pure 2-complexes on 4 nodes with n multiple 2-simplexes. - Vladeta Jovovic, Dec 27 1999

Also number of different integer triangles with perimeter <= n+3. Also number of different scalene integer triangles with perimeter <= n+9. - Reinhard Zumkeller, May 12 2002

a(n) is the coefficient of q^n in the expansion of (m choose 4)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Also number of partitions of n into parts <= 4. a(n) = A026820(n,4), for n > 3. - Reinhard Zumkeller, Jan 21 2010

Number of different distributions of n+10 identical balls in 4 boxes as x,y,z,p where 0 < x < y < z < p. - Ece Uslu and Esin Becenen, Jan 11 2016

Number of partitions of 5n+8 or 5n+12 into 4 parts (+-) 3 mod 5. a(4) = 5 partitions of 28: [7,7,7,7], [12,7,7,2], [12,12,2,2], [17,7,2,2], [22,2,2,2]. a(3) = 3 partitions of 27: [8,8,8,3], [13,8,3,3], [18,3,3,3]. - Richard Turk, Feb 24 2016

a(n) is the total number of non-isomorphic geodetic graphs of diameter n homeomorphic to a complete graph K4. - Carlos Enrique Frasser, May 24 2018

Also, a(n) is the number of n-rowed binary matrices with all row sums equal to 2, up to row and column permutation (see Jovovic's formula). Also, a(n) is the limit of A192517(m,n) as m grows. - Max Alekseyev, Oct 18 2017

Row sums of the triangle defined by the Multiset Transformation of A076864,

1 ;

0 1;

0 2 1;

0 5 2 1;

0 12 8 2 1;

0 33 22 8 2 1;

0 103 72 26 8 2 1;

0 333 229 87 26 8 2 1;

0 1183 782 295 92 26 8 2 1;

0 4442 2760 1036 315 92 26 8 2 1;

0 17576 10270 3735 1129 321 92 26 8 2 1;

0 72810 39770 13976 4117 1154 321 92 26 8 2 1;

0 314595 160713 54132 15547 4237 1161 321 92 26 8 2 1;

- R. J. Mathar, Jul 18 2017

Also the number of non-isomorphic set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018