A006918 -id:A006918 - cp's OEIS frontend

A211795 Number of (w,x,y,z) with all terms in {1,...,n} and wx < 2y*z.

0, 1, 11, 58, 177, 437, 894, 1659, 2813, 4502, 6836, 10008, 14121, 19449, 26117, 34372, 44422, 56597, 71044, 88160, 108115, 131328, 158074, 188773, 223604, 263172, 307719, 357715, 413493, 475690, 544480, 620632, 704381, 796413

Offset: 0

Clark Kimberling, Apr 27 2012

nonn

Each sequence in the following guide counts 4-tuples
(w,x,y,z) such that the indicated relation holds and the four numbers w,x,y,z are in {1,...,n}.  The notation "m div" means that m divides every term of the sequence.
A211058 ... wx <= yz
A211787 ... wx <= 2yz
A211795 ... wx < 2yz
A211797 ... wx > 2yz
A211809 ... wx >= 2yz
A211812 ... wx <= 3yz
A211917 ... wx < 3yz
A211918 ... wx > 3yz
A211919 ... wx >= 3yz
A211920 ... 2wx < 3yz
A211921 ... 2wx <= 3yz
A211922 ... 2wx > 3yz
A211923 ... 2wx >= 3yz
A212019 ... wx = 2yz ..... 2 div
A212020 ... wx = 3yz ..... 2 div
A212021 ... 2wx = 3yz .... 2 div
A212047 ... wx = 4yz
A212048 ... 3wx = 4yz .... 2 div
A212049 ... wx = 5yz ..... 2 div
A212050 ... 2wx = 5yz .... 2 div
A212051 ... 3wx = 5yz .... 2 div
A212052 ... 4wx = 5yz .... 2 div
A209978 ... wx = yz + 1 .. 2 div
A212053 ... wx <= yz + 1
A212054 ... wx > yz + 1
A212055 ... wx <= yz + 2
A212056 ... wx > yz + 2
A197168 ... wx = yz + 2 .. 2 div
A061201 ... w = xyz
A212057 ... w < xyz
A212058 ... w >= xyz
A212059 ... w = xyz - 1
A212060 ... w = xyz - 2
A212061 ... wx = (yz)^2
A212062 ... w^2 = xyz
A212063 ... w^2 < xyz
A212064 ... w^2 >= xyz
A212065 ... w^2 <= xyz
A212066 ... w^2 > xyz
A212067 ... w^3 = xyz
A002623 ... w = 2x + y + z
A006918 ... w = 2x + 2y + z
A000601 ... w = x + 2y + 3z (except for initial 0's)
A212068 ... 2w = x + y + z
A212069 ... 3w = x + y + z (w = average{x,y,z})
A212088 ... 3w < x + y + z
A212089 ... 3w >= x + y + z
A212090 ... w < x + y + z
A000332 ... w >= x + y + z
A212145 ... w < 2x + y + z
A001752 ... w >= 2x + y + z
A001400 ... w = 2x +3y + 4z
A005900 ... w = -x + y + z
A192023 ... w = -x + y + z + 2
A212091 ... w^2 = x^2 + y^2 + z^2 ... 3 div
A212087 ... w^2 + x^2 = y^2 + z^2
A212092 ... w^2 < x^2 + y^2 + z^2
A212093 ... w^2 <= x^2 + y^2 + z^2
A212094 ... w^2 > x^2 + y^2 + z^2
A212095 ... w^2 >= x^2 + y^2 + z^2
A212096 ... w^3 = x^3 + y^3 + z^3 ... 6 div
A212097 ... w^3 < x^3 + y^3 + z^3
A212098 ... w^3 <= x^3 + y^3 + z^3
A212099 ... w^3 > x^3 + y^3 + z^3
A212100 ... w^3 >= x^3 + y^3 + z^3
A212101 ... wx^2 = yz^2
A212102 ... 1/w = 1/x + 1/y + 1/z
A212103 ... 3/w = 1/x + 1/y + 1/z; w = h.m. of {x,y,z}
A212104 ... 3/w >= 1/x + 1/y + 1/z; w >= h.m.
A212105 ... 3/w < 1/x + 1/y + 1/z; w < h.m.
A212106 ... 3/w > 1/x + 1/y + 1/z; w > h.m.
A212107 ... 3/w <= 1/x + 1/y + 1/z; w <= h.m.
A212133 ... median(w,x,y,z) = mean(w,x,y,z)
A212134 ... median(w,x,y,z) <= mean(w,x,y,z)
A212135 ... median(w,x,y,z) > mean(w,x,y,z)
A212241 ... wx + yz > n
A212243 ... 2wx + yz = n
A212244 ... w = xyz - n
A212245 ... w = xyz - 2n
A212246 ... 2w = x + y + z - n
A212247 ... 3w = x + y + z + n
A212249 ... 3w < x + y + z + n
A212250 ... 3w >= x + y + z + n
A212251 ... 3w = x + y + z + n + 1
A212252 ... 3w = x + y + z + n + 2
A212254 ... w = x + 2y + 3z - n
A212255 ... w^2 = mean(x^2, y^2, z^2)
A212256 ... 4/w = 1/x + 1/y +1/z + 1/n
In the list above, if the relation in the second column is of the form "w rel ax + by + cz" then the sequence is linearly recurrent.  In the list below, the same is true for expressions involving more than one relation.
A000332 ... w < x <= y < z .... C(n,4)
A000914 ... w < x <= y < z .... Stirling 1st kind
A000914 ... w < x <= y >= z ... Stirling 1st kind
A050534 ... w < x < y >= z .... tritriangular
A001296 ... w <= x <= y >= z .. 4-dim pyramidal
A006322 ... x < x > y >= z
A002418 ... w < x >= y < z
A050534 ... w < x >=y >= z
A212415 ... w < x >= y <= z
A001296 ... w < x >= y <= z
A212246 ... w <= x > y <= z
A006322 ... w <= x >= y <= z
A212501 ... w > x < y >= z
A212503 ... w < 2x and y < 2z ..... A (note below)
A212504 ... w < 2x and y > 2z ..... A
A212505 ... w < 2x and y >= 2z .... A
A212506 ... w <= 2x and y <= 2z ... A
A212507 ... w < 2x and y <= 2z .... B
A212508 ... w < 2x and y < 3z ..... C
A212509 ... w < 2x and y <= 3z .... C
A212510 ... w < 2x and y > 3z ..... C
A212511 ... w < 2x and y >= 3z .... C
A212512 ... w <= 2x and y < 3z .... C
A212513 ... w <= 2x and y <= 3z ... C
A212514 ... w <= 2x and y > 3z .... C
A212515 ... w <= 2x and y >= 3z ... C
A212516 ... w > 2x and y < 3z ..... C
A212517 ... w > 2x and y <= 3z .... C
A212518 ... w > 2x and y > 3z ..... C
A212519 ... w > 2x and y >= 3z .... C
A212520 ... w >= 2x and y < 3z .... C
A212521 ... w >= 2x and y <= 3z ... C
A212522 ... w >= 2x and y > 3z .... C
A212523 ... w + x < y + z
A212560 ... w + x <= y + z
A212561 ... w + x = 2y + 2z
A212562 ... w + x < 2y + 2z ....... B
A212563 ... w + x <= 2y + 2z ...... B
A212564 ... w + x > 2y + 2z ....... B
A212565 ... w + x >= 2y + 2z ...... B
A212566 ... w + x = 3y + 3z
A212567 ... 2w + 2x = 3y + 3z
A212570 ... |w - x| = |x - y| + |y - z|
A212571 ... |w - x| < |x - y| + |y - z| ... B ... 4 div
A212572 ... |w - x| <= |x - y| + |y - z| .. B
A212573 ... |w - x| > |x - y| + |y - z| ... B ... 2 div
A212574 ... |w - x| >= |x - y| + |y - z| .. B
A212575 ... 2|w - x| = |x - y| + |y - z|
A212576 ... |w - x| = 2|x - y| + 2|y - z|
A212577 ... |w - x| = 2|x - y| - |y - z|
A212578 ... 2|w - x| = |x - y| - |y - z|
A212579 ... min{|w-x|,|w-y|} = min{|x-y|,|x-z|}
A212692 ... w = |x - y| + |y - z| ............... 2 div
A212568 ... w < |x - y| + |y - z| ............... 2 div
A212573 ... w <= |x - y| + |y - z| .............. 2 div
A212574 ... w > |x - y| + |y - z|
A212575 ... w >= |x - y| + |y - z|
A212676 ... w + x = |x - y| + |y - z| ......... H
A212677 ... w + y = |x - y| + |y - z|
A212678 ... w + x + y = |x - y| + |y - z|
A006918 ... w + x + y + z = |x - y| + |y - z| . H
A212679 ... |x - y| = |y - z| ................. H
A212680 ... |x - y| = |y - z| + 1 ..............H 2 div
A212681 ... |x - y| < |y - z| ................... 2 div
A212682 ... |x - y| >= |y - z|
A212683 ... |x - y| = w + |y - z| ............... 2 div
A212684 ... |x - y| = n - w + |y - z|
A212685 ... |w - x| = w + |y - z|
A186707 ... |w - x| < w + |y - z| ... (Note D)
A212714 ... |w - x| >= w + |y - z| .......... H . 2 div
A212686 ... 2*|w - x| = n + |y - z| ............. 4 div
A212687 ... 2*|w - x| < n + |y - z| ......... B
A212688 ... 2*|w - x| < n + |y - z| ......... B . 2 div
A212689 ... 2*|w - x| > n + |y - z| ......... B . 2 div
A212690 ... 2*|w - x| <= n + |y - z| ........ B
A212691 ... w + |x - y| = |x - z| + |y - z| . E . 2 div
...
In the above lists, all the terms of (w,x,y,z) are in {1,...,n}, but in the next lists they are all in {0,...,n}, and sequences are all linearly recurrent.
  R=range{w,x,y,z}=max{w,x,y,z}-min{w,x,y,z}.
A212740 ... max{w,x,y,z} < 2*min{w,x,y,z} .... A
A212741 ... max{w,x,y,z} >= 2*min{w,x,y,z} ... A
A212742 ... max{w,x,y,z} <= 2*min{w,x,y,z} ... A
A212743 ... max{w,x,y,z} > 2*min{w,x,y,z} .... A . 2 div
A212744 ... w=range (=max-min) ............... E
A212745 ... w=max{w,x,y,z} - 2*min{w,x,y,z}
A212746 ... R is in {w,x,y,z} ................ E
A212569 ... R is not in {w,x,y,z} ............ E
A212749 ... w=R or xA212750 ... w=R or x=R or yA212751 ... w=R or x=R or yA212752 ... wR ......... A
A212753 ... wR or z>R ......... D
A212754 ... wR or y>R or z>R ......... D
A002415 ... w = x + R ........................ D
A212755 ... |w - x| = R ...................... D
A212756 ... 2w = x + R
A212757 ... 2w = R
A212758 ... w = floor(R/2)
A002413 ... w = floor((x+y+z/2))
A212759 ... w, x, y are even
A212760 ... w is even and x = y + z .......... E
A212761 ... w is odd and x and y are even .... F . 2 div
A212762 ... w and x are odd y is even ........ F . 2 div
A212763 ... w, x, y are odd .................. F
A212764 ... w, x, y are even and z is odd .... F
A030179 ... w and x are even and y and z odd
A212765 ... w is even and x,y,z are odd ...... F
A212766 ... w is even and x is odd ........... A . 2 div
A212767 ... w and x are even and w+x=y+z ..... E
A212889 ... R is even ........................ A
A212890 ... R is odd ......................... A . 2 div
A212742 ... w-x, x-y, y-z are all even ....... A
A212892 ... w-x, x-y, y-z are all odd ........ A
A212893 ... w-x, x-y, y-z have same parity ... A
A005915 ... min{|w-x|, |x-y|, |y-z|} = 0
A212894 ... min{|w-x|, |x-y|, |y-z|} = 1
A212895 ... min{|w-x|, |x-y|, |y-z|} = 2
A179824 ... min{|w-x|, |x-y|, |y-z|} > 0
A212896 ... min{|w-x|, |x-y|, |y-z|} <= 1
A212897 ... min{|w-x|, |x-y|, |y-z|} > 1
A212898 ... min{|w-x|, |x-y|, |y-z|} <= 2
A212899 ... min{|w-x|, |x-y|, |y-z|} > 2
A212901 ... |w-x| = |x-y| = |y-z|
A212900 ... |w-x|, |x-y|, |y-z| are distinct . G
A212902 ... |w-x| < |x-y| < |y-z| ............ G
A212903 ... |w-x| <= |x-y| <= |y-z| .......... G
A212904 ... |w-x| + |x-y| + |y-z| = n ........ H
A212905 ... |w-x| + |x-y| + |y-z| = 2n ....... H
...
Note A: A212503-A212506 (and others) have these recurrence coefficients: 2,2,-6,0,6,-2,-2,1.
B: 3,-1,-5,5,1,-3,1
C: 0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1
D: 4,-5,0,5,-4,1
E: 1,3,-3,-3,3,1,-1
F: 1,4,-4,-6,6,4,-4,-1,1
G: 2,1,-3,-1,1,3,-1,-2,1
H: 2,1,-4,1,2,-1

		Examples
					a(2)=11 counts these (w,x,y,z): (1,1,1,1), (1,1,1,2), (1,1,2,1), (2,1,2,1), (2,1,1,2), (1,2,2,1), (1,2,1,2), (1,1,2,2), (1,2,2,2), (2,1,2,2), (2,2,2,2).
		
	
References
			A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

		
Links
			Bo Gyu Jeong, Table of n, a(n) for n = 0..200

		
CrossrefsCf. A000583 (n^4), A210000, A211809, A212959.
Programs
			
				Mathematica
				t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w*x < 2 y*z, s = s + 1], {w, 1, #},
      {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211795 *)
(* Peter J. C. Moses, Apr 13 2012 *)
				

		
Formulaa(n) = n^4 - A211809(n).

A002623 Expansion of 1/((1-x)^4*(1+x)).

Original entry on oeis.org

1, 3, 7, 13, 22, 34, 50, 70, 95, 125, 161, 203, 252, 308, 372, 444, 525, 615, 715, 825, 946, 1078, 1222, 1378, 1547, 1729, 1925, 2135, 2360, 2600, 2856, 3128, 3417, 3723, 4047, 4389, 4750, 5130, 5530, 5950, 6391, 6853, 7337, 7843, 8372, 8924, 9500

Offset: 0

N. J. A. Sloane

Also a(n) is the number of nondegenerate triangles that can be made from rods of lengths 1 to n+1. - Alfred Bruckstein; corrected by Hans Rudolf Widmer, Nov 02 2023
Also number of circumscribable (or escrible) quadrilaterals that can be made from rods of length 1,2,3,4,...,n. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
Also number of 2 X n binary matrices up to row and column permutation (see the link: Binary matrices up to row and column permutations). - Vladeta Jovovic
Also partial sum of alternate triangular numbers (1, 3, 1+6, 3+10, 1+6+15, 3+10+21, etc.); and also number of triangles pointing in opposite direction to largest triangle in triangular matchstick arrangement of side n+2 (cf. A002717, also the Larsen article). - Henry Bottomley, Aug 08 2000
Ordered union of A002412(n+1) and A016061(n+1). - Lekraj Beedassy, Oct 13 2003
Also Molien series for certain 4-D representation of cyclic group of order 2. - N. J. A. Sloane, Jun 12 2004
From Radu Grigore (radugrigore(AT)gmail.com), Jun 19 2004: (Start)
a(n) = floor( (n+2)*(n+4)*(2n+3) / 24 ). E.g., a(2) = floor(4*6*7/24) = 7 because there are 7 upside down triangles (6 of size one and 1 of size two) in the matchstick figure:
     /\
    /\/\
   /\/\/\
  /\/\/\/\
(End)
Number of non-congruent non-parallelogram trapezoids with positive integer sides (trapezints) and perimeter 2n+5. Also with perimeter 2n+8. - Michael Somos, May 12 2005
a(n) = A108561(n+4,n) for n > 0. - Reinhard Zumkeller, Jun 10 2005
Also number of nonisomorphic planes with n points and 2 lines. E.g., a(0)=1 because with no points, we just have two empty lines. a(1)=3 because the one point may belong to 0, 1 or 2 lines. a(2)=7 because there are 7 ways to determine which of 2 points belong to which of 2 lines, up to isomorphism, i.e., up to a bijection f on the sets of points and a bijection g on the sets of lines, such that A belongs to a iff f(A) belongs to g(a). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
a(n-2) is the number of ways to pick two non-overlapping subwords of equal nonzero length from a word of length n. E.g., a(5-2)=a(3)=13 since the word 12345 of length 5 has the following subword pairs: 1,2; 1,3; 1,4; 1,5; 2,3; 2,4; 2,5; 3,4; 3,5; 4,5; 12,34; 12,45; 23,45. - Michael Somos, Oct 22 2006
Partial sums of A002620. - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007
From Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007: (Start)
Also number of squares of any size in a staircase of n steps built with unit squares:
   
  ||__
  ||__|
  ||__||
For a staircase of 3 steps 6 squares of size 1 and 1 square of size 2, hence c(3)=7.
Columns sums of:
  1 3 6 10 15 21 28 ...
      1  3  6 10 15 ...
            1  3  6 ...
                  1 ...
  ---------------------
  1 3 7 13 22 34 50 ...
(End)
a(n) = sum of row n+1 of triangle A134446. Also, binomial transform of [1, 2, 2, 0, 1, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Oct 25 2007
Let b(n) be the number of 4-tuples (w,x,y,z) having all terms in {1,...,n} and 2w=x+y+z+n; then b(n+3) = a(n) for n >= 0. - Clark Kimberling, May 08 2012
a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and w >= x+y and x <= y. - Clark Kimberling, Jun 04 2012
Also, number of unlabeled bipartite graphs with two left vertices and n right vertices. - Yavuz Oruc, Jan 14 2018
Also number of triples (x,y,z) with 0 < x <= y <= z <= n + 1, x + y > z. - Ralf Steiner, Feb 06 2020
Bisections A002412 and A016061: a(2*k) = k*(k+1)*(4*k-1)/3! and a(2*k+1) = (k+1)*(k+2)*(4*k+9)/3!, for k >= 0. See the Woolhouse link, II. Solution by Stephen Watson, p. 65, with index shifts. - Mo Li, Apr 02 2020
Also, Wiener index of the square of the path graph P_(n+2). - Allan Bickle, Aug 01 2020
Maximum Wiener index of all maximal 2-degenerate graphs with n+2 vertices.  (A maximal 2-degenerate graph can be constructed from a 2-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to two existing vertices.)  The extremal graphs are squares of paths, so the bound also applies to 2-trees and maximal outerplanar graphs. - Allan Bickle, Sep 15 2022

			G.f. = 1 + 3*x + 7*x^2 + 13*x^3 + 22*x^4 + 34*x^5 + 50*x^6 + 70*x^7 + 95*x^8 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.
P. Diaconis, R. L. Graham and B. Sturmfels, Primitive partition identities, in Combinatorics: Paul Erdős is Eighty, Vol. 2, Bolyai Soc. Math. Stud., 2, 1996, pp. 173-192.
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
I. Siap, Linear codes over F_2 + u*F_2 and their complete weight enumerators, in Codes and Designs (Ohio State, May 18, 2000), pp. 259-271. De Gruyter, 2002.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Atmaca and A. Yavuz Oruç, On the size of two families of unlabeled bipartite graphs, AKCE International Journal of Graphs and Combinatorics, 2017.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, Lemma 6 5th line.
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Shalosh B. Ekhad and Doron Zeilberger, Computerizing the Andrews-Fraenkel-Sellers Proofs on the Number of m-ary partitions mod m (and doing MUCH more!), arXiv preprint arXiv:1511.06791 [math.CO], 2015.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
Enrique González-Jiménez and Xavier Xarles, On a conjecture of Rudin on squares in Arithmetic Progressions, arXiv preprint arXiv:1301.5122 [math.NT], 2013.
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051. doi:10.1109/T-C.1973.223649.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 203
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 413
Vladeta Jovovic, Binary matrices up to row and column permutations
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]. See page 79.
W. Lanssens, B. Demoen, and P.-L. Nguyen, The Diagonal Latin Tableau and the Redundancy of its Disequalities, Report CW 666, July 2014, Department of Computer Science, KU Leuven
M. E. Larsen, The eternal triangle - a history of a counting problem, College Math. J., 20 (1989), 370-392.
P. Lisonek, Quasi-polynomials: A case study in experimental combinatorics, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)
Math StackExchange, cycle index for S_2 X S_4, Apr. 2021
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218. See page 217.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=2]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Giovanni Resta, Illustration for a(8)=70.
Marko Riedel, The O.g.f. of inequivalent colorings of a 2xN board with C colors in N is (1/2) 1/(1-w)^(C^2) + (1/2) 1/(1+w)^(C(C-1)/2)*1/(1-w)^(C(C+1)/2) by PET
Sebastian Sassi, Aula Al-Adulrazzaq, Matti Heikinheimo, and Kimmo Tuominen, Fast numerical evaluation of dark matter direct detection event rates, arXiv:2504.19714 [hep-ph], 2025. See p. 7.
Atsuto Seko, Tutorial: Systematic development of polynomial machine learning potentials for elemental and alloy systems, J. Appl. Phys. (2023) Vol. 133, 011101.
J. Silverman, V. E. Vickers and J. M. Mooney, On the number of Costas arrays as a function of array size, Proc. IEEE, 76 (1988), 851-853.
Eric Weisstein's World of Mathematics, Triangle Counting.
W. S. B. Woolhouse, Problem 2420. On the probability of the number of triangles, Mathematical questions with their solutions, v. 9 (June 1868), pp. 63-65.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A002620 (first differences), A000292, A001752 (partial sums), A062109 (binomial transf.).
Bisections A002412, A016061.
Cf. also A002717 (a companion sequence), A002727, A006148, A057524, A134446, A014125, A122046, A122047.
Cf. A000217, A006918, A058187, A108561.
The maximum Wiener index of all maximal k-degenerate graphs for k=1..6 are given in A000292, A002623 (this sequence), A014125, A122046, A122047, A175724, respectively.

A002623 := n->(1/16)*(1+(-1)^n)+(n+1)/8+binomial(n+2,2)/4+binomial(n+3,3)/2;
seq( ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4,n=1..47); # Lewis
a := n -> ((-1)^n*3 + 45 + 68*n + 30*n^2 + 4*n^3) / 48:
seq(a(n), n=0..46); # Peter Luschny, Jan 22 2018

CoefficientList[Series[1/((1-x)^3(1-x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{1,3,7,13,22},50] (* Harvey P. Dale, Jul 19 2011 *)
Table[((2 n^3 + 15 n^2 + 34 n + 45 / 2 + (3/2) (-1)^n) / 24), {n, 0, 100}] (* Vincenzo Librandi, Jan 15 2018 *)
a[ n_] := Floor[(n + 2)*(n + 4)*(2*n + 3)/24]; (* Michael Somos, Feb 19 2024 *)

{a(n) = (8 + 34/3*n + 5*n^2 + 2/3*n^3) \ 8}; /* Michael Somos, Sep 04 1999 */

x='x+O('x^50); Vec(1/((1 - x)^3 * (1 - x^2))) \\ Indranil Ghosh, Apr 04 2017

def A002623(n): return ((n+2)*(n+4)*((n<<1)+3)>>3)//3 # Chai Wah Wu, Mar 25 2024

a(n+1) = a(n) + {(k-1)*k if n=2*k} or {k*k if n=2*k+1}.
a(n)+a(n+1) = A000292(n+1).
a(n) = a(n-2) + A000217(n+1) = A002717(n+2) - A000292(n+1).
Also: a(n) = C(n+3, 3) - a(n-1) with a(0)=1. - Labos Elemer, Apr 26 2003
From Paul Barry, Jul 01 2003: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+3,3).
The signed version 1, -3, 7, ... has the formula:
a(n) = (4*n^3 + 30*n^2 + 68*n + 45)*(-1)^n/48 + 1/16.
This is the partial sums of the signed version of A000292. (End)
From Paul Barry, Jul 21 2003: (Start)
a(n) = Sum_{k=0..n} floor((k+2)^2/4).
a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} (1+(-1)^i)/2. (End)
a(n) = a(n - 2) + (n*(n - 1))/2, with n>2, a(1)=0, a(2)=1; a(n) = (4*n^3+6*n^2-4*n+3*(-1)^n-3)/48, with offset 2. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004 (formula simplified by Bruno Berselli, Aug 29 2013)
a(n) = ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4, with offset 1. - Jerry W. Lewis (JLewis(AT)wyeth.com), Mar 23 2005
a(n) = 2*a(n-1) - a(n-2) + 1 + floor(n/2). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
A002620(n+3) = a(n+1) - a(n). - Michael Somos, Sep 04 1999
Euler transform of length 2 sequence [ 3, 1]. - Michael Somos, Sep 04 2006
a(n) = -a(-5-n) for all n in Z. - Michael Somos, Sep 04 2006
Let P(i,k) be the number of integer partitions of n into k parts, then with k=2 we have a(n) = sum_{m=1}^{n} sum_{i=k}^{m} P(i,k). For k=1 we get A000217 = triangular numbers. - Thomas Wieder, Feb 18 2007
a(n) = (n+(3+(-1)^n)/2)*(n+(7+(-1)^n)/2)*(2*n+5-2*(-1)^n)/24. - Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007 (corrected by Bruno Berselli, Aug 30 2013)
From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A006918(n+1) + A006918(n).
a(n) = A058187(n-2) + 2*A058187(n-1) + A058187(n). (End)
a(0)=1, a(1)=3, a(2)=7, a(3)=13, a(4)=22; for n > 4, a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). - Harvey P. Dale, Jul 19 2011
a(n) = Sum_{i=0..n+2} floor(i/2)*ceiling(i/2). - Bruno Berselli, Aug 30 2013
a(n) = 15/16 + (1/16)*(-1)^n + (17/12)*n + (5/8)*n^2 + (1/12)*n^3. - Robert Israel, Jul 07 2014
a(n) = Sum_{i=0..n+2} (n+1-i)*floor(i/2+1). - Bruno Berselli, Apr 04 2017
a(n) = 1 + floor((2*n^3 + 15*n^2 + 34*n) / 24). - Allan Bickle, Aug 01 2020
E.g.f.: ((24 + 51*x + 21*x^2 + 2*x^3)*cosh(x) + (21 + 51*x + 21*x^2 + 2*x^3)*sinh(x))/24. - Stefano Spezia, Jun 02 2021

A115994 Triangle read by rows: T(n,k) is number of partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 2, 6, 5, 7, 8, 8, 14, 9, 20, 1, 10, 30, 2, 11, 40, 5, 12, 55, 10, 13, 70, 18, 14, 91, 30, 15, 112, 49, 16, 140, 74, 1, 17, 168, 110, 2, 18, 204, 158, 5, 19, 240, 221, 10, 20, 285, 302, 20, 21, 330, 407, 34, 22, 385, 536, 59, 23, 440, 698, 94, 24, 506, 896, 149, 25

Offset: 1

Emeric Deutsch, Feb 11 2006

Row n has floor(sqrt(n)) terms. Row sums yield A000041. Column 2 yields A006918. sum(k*T(n,k),k=1..floor(sqrt(n)))=A115995.
T(n,k) is number of partitions of n-k^2 into parts of 2 kinds with at most k of each kind.
The limit of the diagonals is A000712 (partitions into parts of two kinds). In particular, if 0<=m<=n, T(n(n+1)/2 + m, n) = A000712(m). These partitions in this range can be viewed as an equilateral right triangle of side n, with one partition appended on the top (at the left) and another appended on the right. - Franklin T. Adams-Watters, Jan 11 2006
Successive columns approach closer and closer to A000712. - N. J. A. Sloane, Mar 10 2007

			T(5,2) = 2 because the only partitions of 5 having Durfee square of size 2 are [3,2] and [2,2,1]; the other five partitions ([5], [4,1], [3,1,1], [2,1,1,1] and [1,1,1,1,1]) have Durfee square of size 1.
Triangle starts:
  1;
  2;
  3;
  4,  1;
  5,  2;
  6,  5;
  7,  8;
  8, 14;
  9, 20,  1;
  ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Alois P. Heinz, Rows n = 1..1000, flattened
E. R. Canfield, From recursions to asymptotics: Durfee and dilogarithmic deductions, Advances in Applied Mathematics, Volume 34, Issue 4, May 2005, Pages 768-797
E. R. Canfield, S. Corteel, C. D. Savage, Durfee Polynomials, Electronic Journal of Combinatorics 5 (1998), #R32.
S. B. Ekhad, D. Zeilberger, A Quick Empirical Reproof of the Asymptotic Normality of the Hirsch Citation Index (First proved by Canfield, Corteel, and Savage), arXiv preprint arXiv:1411.0002, 2014.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 45
Eric Weisstein's World of Mathematics, Durfee Square.

For another version see A115720. Row lengths A000196.

Cf. A115995, A115721, A115722, A008284, A006918.

g:=sum(t^k*q^(k^2)/product((1-q^j)^2,j=1..k),k=1..40): gser:=series(g,q=0,32): for n from 1 to 27 do P[n]:=coeff(gser,q^n) od: for n from 1 to 27 do seq(coeff(P[n],t^j),j=1..floor(sqrt(n))) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
T:= (n, k)-> add(b(m, k)*b(n-k^2-m, k), m=0..n-k^2):
seq(seq(T(n, k), k=1..floor(sqrt(n))), n=1..30); # Alois P. Heinz, Apr 09 2012

Map[Select[#,#>0&]&,Drop[Transpose[Table[CoefficientList[ Series[x^(n^2)/Product[1-x^i,{i,1,n}]^2,{x,0,nn}],x],{n,1,10}]],1]] //Grid (* Geoffrey Critzer, Sep 27 2013 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; T[n_, k_] := Sum[b[m, k]*b[n-k^2-m, k], {m, 0, n-k^2}]; Table[T[n, k], {n, 1, 30}, {k, 1, Sqrt[n]}] // Flatten (* Jean-François Alcover, Dec 25 2015, after Alois P. Heinz *)

G.f.: sum(k>=1, t^k*q^(k^2)/product(j=1..k, (1-q^j)^2 ) ).

T(n,k) = Sum_{i=0}^{n-k^2} P*(i,k)*P*(n-k^2-i), where P*(n,k) = P(n+k,k) is the number of partitions of n objects into at most k parts.

Edited and verified by Franklin T. Adams-Watters, Mar 11 2006

A212959 Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

Original entry on oeis.org

1, 4, 11, 20, 33, 48, 67, 88, 113, 140, 171, 204, 241, 280, 323, 368, 417, 468, 523, 580, 641, 704, 771, 840, 913, 988, 1067, 1148, 1233, 1320, 1411, 1504, 1601, 1700, 1803, 1908, 2017, 2128, 2243, 2360, 2481, 2604, 2731, 2860, 2993, 3128, 3267

Offset: 0

Clark Kimberling, Jun 01 2012

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

			a(1)=4 counts these (x,y,z): (0,0,0), (1,1,1), (0,1,0), (1,0,1).
Numbers congruent to {1, 3} mod 6: 1, 3, 7, 9, 13, 15, 19, ...
a(0) = 1;
a(1) = 1 + 3 = 4;
a(2) = 1 + 3 + 7 = 11;
a(3) = 1 + 3 + 7 + 9 = 20;
a(4) = 1 + 3 + 7 + 9 + 13 = 33;
a(5) = 1 + 3 + 7 + 9 + 13 + 15 = 48; etc. - _Philippe Deléham_, Mar 16 2014

A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

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

Cf. A047241, A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] == Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 50]]   (* A212959 *)

a(n)=(6*n^2+8*n+3)\/4 \\ Charles R Greathouse IV, Jul 28 2015

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1+2*x+3*x^2)/((1+x)*(1-x)^3).
a(n) + A212960(n) = (n+1)^3.
a(n) = (6*n^2 + 8*n + 3 + (-1)^n)/4. - Luce ETIENNE, Apr 05 2014
a(n) = 2*A069905(3*(n+1)+2) - 3*(n+1). - Ayoub Saber Rguez, Aug 31 2021

A008805 Triangular numbers repeated.

Original entry on oeis.org

1, 1, 3, 3, 6, 6, 10, 10, 15, 15, 21, 21, 28, 28, 36, 36, 45, 45, 55, 55, 66, 66, 78, 78, 91, 91, 105, 105, 120, 120, 136, 136, 153, 153, 171, 171, 190, 190, 210, 210, 231, 231, 253, 253, 276, 276, 300, 300, 325, 325, 351, 351, 378, 378, 406, 406, 435, 435

Offset: 0

N. J. A. Sloane

Number of choices for nonnegative integers x,y,z such that x and y are even and x + y + z = n.
Diagonal sums of A002260, when arranged as a number triangle. - Paul Barry, Feb 28 2003
a(n) = number of partitions of n+4 such that the differences between greatest and smallest parts are 2: a(n-4) = A097364(n,2) for n>3. - Reinhard Zumkeller, Aug 09 2004
For n >= i, i=4,5, a(n-i) is the number of incongruent two-color bracelets of n beads, i from them are black (cf. A005232, A032279), having a diameter of symmetry. - Vladimir Shevelev, May 03 2011
Prefixing A008805 by 0,0,0,0 gives the sequence c(0), c(1), ... defined by c(n)=number of (w,x,y) such that w = 2x+2y, where w,x,y are all in {1,...,n}; see A211422. - Clark Kimberling, Apr 15 2012
Partial sums of positive terms of A142150. - Reinhard Zumkeller, Jul 07 2012
The sum of the first parts of the nondecreasing partitions of n+2 into exactly two parts, n >= 0. - Wesley Ivan Hurt, Jun 08 2013
Number of the distinct symmetric pentagons in a regular n-gon, see illustration for some small n in links. - Kival Ngaokrajang, Jun 25 2013
a(n) is the number of nonnegative integer solutions to the equation x + y + z = n such that x + y <= z. For example, a(4) = 6 because we have 0+0+4 = 0+1+3 = 0+2+2 = 1+0+3 = 1+1+2 = 2+0+2. - Geoffrey Critzer, Jul 09 2013
a(n) is the number of distinct opening moves in n X n tic-tac-toe. - I. J. Kennedy, Sep 04 2013
a(n) is the number of symmetry-allowed, linearly-independent terms at n-th order in the series expansion of the T2 X t2 vibronic perturbation matrix, H(Q) (cf. Opalka & Domcke). - Bradley Klee, Jul 20 2015
a(n-1) also gives the number of D_4 (dihedral group of order 4) orbits of an n X n square grid with squares coming in either of two colors and only one square has one of the colors. - Wolfdieter Lang, Oct 03 2016
Also, this sequence is the third column in the triangle of the coefficients of the sum of two consecutive Fibonacci polynomials F(n+1, x) and F(n, x) (n>=0) in ascending powers of x. - Mohammad K. Azarian, Jul 18 2018
In an n-person symmetric matching pennies game (a zero-sum normal-form game) with n > 2 symmetric and indistinguishable players, each with two strategies (viz. heads or tails), a(n-3) is the number of distinct subsets of players that must play the same strategy to avoid incurring losses (single pure Nash equilibrium in the reduced game). The total number of distinct partitions is A000217(n-1). - Ambrosio Valencia-Romero, Apr 17 2022
a(n) is the number of connected bipartite graphs with n+1 edges and a stable set of cardinality 2. - Christian Barrientos, Jun 15 2022
a(n) is the number of 132-avoiding odd Grassmannian permutations of size n+2. - Juan B. Gil, Mar 10 2023
Consider a regular n-gon with all diagonals drawn. Define a "layer" to be the set of all regions sharing an edge with the exterior. Removing a layer creates another layer. Count the layers, removing them until none remain. The number of layers is a(n-2). See illustration. - Christopher Scussel, Nov 07 2023

			a(5) = 6, since (5) + 2 = 7 has three nondecreasing partitions with exactly 2 parts: (1,6),(2,5),(3,4). The sum of the first parts of these partitions = 1 + 2 + 3 = 6. - _Wesley Ivan Hurt_, Jun 08 2013

H. D. Brunk, An Introduction to Mathematical Statistics, Ginn, Boston, 1960; p. 360.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
G. E. Andrews, M. Beck, and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv preprint arXiv:1406.3374 [math.NT], 2014.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 46.
Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 19.
Kival Ngaokrajang, The distinct symmetric 5-gons in a regular n-gon for n = 6..13
D. Opalka and W. Domcke, High-order expansion of T2xt2 Jahn-Teller potential energy surfaces in tetrahedral molecules, J. Chem. Phys., 132, 154108 (2010).
Christopher Scussel, Illustration of layers in regular n-gons with all diagonals drawn
Vladimir Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
Index entries for Molien series

Cf. A000217, A002260, A002620, A006918 (partial sums), A054252, A135276, A142150, A158920 (binomial trans.).

List([0..60], n-> (2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32); # G. C. Greubel, Sep 12 2019

import Data.List (transpose)
a008805 = a000217 . (`div` 2) . (+ 1)
a008805_list = drop 2 $ concat $ transpose [a000217_list, a000217_list]
-- Reinhard Zumkeller, Feb 01 2013

[(2*n+3+(-1)^n)*(2*n+7+(-1)^n)/32 : n in [0..50]]; // Wesley Ivan Hurt, Apr 22 2015

A008805:=n->(2*n+3+(-1)^n)*(2*n+7+(-1)^n)/32: seq(A008805(n), n=0..50); # Wesley Ivan Hurt, Apr 22 2015

CoefficientList[Series[1/(1-x^2)^2/(1-x), {x, 0, 50}], x]
Table[Binomial[Floor[n/2] + 2, 2], {n, 0, 57}] (* Michael De Vlieger, Oct 03 2016 *)

PARI
```
a(n)=(n\2+2)*(n\2+1)/2
```

def A008805(n): return (m:=(n>>1)+1)*(m+1)>>1 # Chai Wah Wu, Oct 20 2023

[(2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32 for n in (0..60)] # G. C. Greubel, Sep 12 2019

G.f.: 1/((1-x)*(1-x^2)^2) = 1/((1+x)^2*(1-x)^3).
E.g.f.: (exp(x)*(2*x^2 +12*x+ 11) - exp(-x)*(2*x -5))/16.
a(-n) = a(-5+n).
a(n) = binomial(floor(n/2)+2, 2). - Vladimir Shevelev, May 03 2011
From Paul Barry, May 31 2003: (Start)
a(n) = ((2*n +5)*(-1)^n + (2*n^2 +10*n +11))/16.
a(n) = Sum_{k=0..n} ((k+2)*(1+(-1)^k))/4. (End)
From Paul Barry, Apr 16 2005: (Start)
a(n) = Sum_{k=0..n} floor((k+2)/2)*(1-(-1)^(n+k-1))/2.
a(n) = Sum_{k=0..floor(n/2)} floor((n-2k+2)/2). (End)
A signed version is given by Sum_{k=0..n} (-1)^k*floor(k^2/4). - Paul Barry, Aug 19 2003
a(n) = A108299(n-2,n)*(-1)^floor((n+1)/2) for n>1. - Reinhard Zumkeller, Jun 01 2005
a(n) = A004125(n+3) - A049798(n+2). - Carl Najafi, Jan 31 2013
a(n) = Sum_{i=1..floor((n+2)/2)} i. - Wesley Ivan Hurt, Jun 08 2013
a(n) = (1/2)*floor((n+2)/2)*(floor((n+2)/2)+1). - Wesley Ivan Hurt, Jun 08 2013
From Wesley Ivan Hurt, Apr 22 2015: (Start)
a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
a(n) = (2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32. (End)
a(n-1) = A054252(n,1) = A054252(n^2-1), n >= 1. See a Oct 03 2016 comment above. - Wolfdieter Lang, Oct 03 2016
a(n) = A000217(A008619(n)). - Guenther Schrack, Sep 12 2018
From Ambrosio Valencia-Romero, Apr 17 2022: (Start)
a(n) = a(n-1) if n odd, a(n) = a(n-1) + (n+2)/2 if n is even, for n > 0, a(0) = 1.
a(n) = (n+1)*(n+3)/8 if n odd, a(n) = (n+2)*(n+4)/8 if n is even, for n >= 0.
a(n) = A002620(n+2) - a(n-1), for n > 0, a(0) = 1.
a(n) = A142150(n+2) + a(n-1), for n > 0, a(0) = 1.
a(n) = A000217(n+3)/2 - A135276(n+3)/2. (End)

A257990 The side-length of the Durfee square of the partition having Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1

Offset: 1

Emeric Deutsch, May 18 2015

nonn

The Durfee square of a partition is the largest square that fits inside the Ferrers board of the partition.
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n.
First appearance of k is a(prime(k)^k) = k. - Gus Wiseman, Apr 12 2019

			a(9)=2; indeed, 9 = 3*3 is the Heinz number of the partition [2,2] and, clearly its Durfee square has side-length =2.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass. 1976.
G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
M. Bona, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Findstat, St000183: The side length of the Durfee square of an integer partition

Positions of 1's are A093641. Positions of 2's are A325164. Positions of 3's are A307386.

Cf. A006918, A056239, A062457, A065770, A112798, A115720, A117485, A215366, A252464, A325163, A325169.

with(numtheory): a := proc (p) local B, S, i: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: S := {}: for i to nops(B(p)) do if i <= B(p)[nops(B(p))+1-i] then S := `union`(S, {i}) else  end if end do: max(S) end proc: seq(a(n), n = 2 .. 146);
# second Maple program:
a:= proc(n) local l, t;
      l:= sort(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`);
      for t from nops(l) to 1 by -1 do if l[t]>=t then break fi od; t
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, May 10 2016

a[n_] := a[n] = Module[{l, t}, l = Reverse[Sort[Flatten[Table[PrimePi[ f[[1]] ], {f, FactorInteger[n]}, {f[[2]]}]]]]; For[t = Length[l], t >= 1, t--, If[l[[t]] >= t, Break[]]]; t]; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Feb 17 2017, after Alois P. Heinz *)

For a partition (p_1 >= p_2 >= ... > = p_r) the side-length of its Durfee square is the largest i such that p_i >=i.

a(1)=0 prepended by Alois P. Heinz, May 10 2016

A051168 Triangular array T(h,k) for 0 <= k <= h read by rows: T(h,k) = number of binary Lyndon words with k ones and h-k zeros.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 5, 5, 3, 1, 0, 0, 1, 3, 7, 8, 7, 3, 1, 0, 0, 1, 4, 9, 14, 14, 9, 4, 1, 0, 0, 1, 4, 12, 20, 25, 20, 12, 4, 1, 0, 0, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 0, 0, 1, 5, 18, 40, 66, 75, 66, 40, 18, 5, 1, 0, 0, 1, 6

Offset: 0

Clark Kimberling

T(h,k) is the number of classes of aperiodic binary words of k ones and h-k zeros; words u,v are in the same class if v is a cyclic permutation of u (e.g., u=111000, v=110001) and a word is aperiodic if not a juxtaposition of 2 or more identical subwords.
T(2n, n), T(2n+1, n), T(n, 3) match A022553, A000108, A001840, respectively. Row sums match A001037.
From R. J. Mathar, Jul 31 2008: (Start)
This triangle may also be regarded as the square array A(r,n), the n-th term of the r-th Witt transform of the all-1 sequence, r >= 1, n >= 0, read by antidiagonals:
This array begins as follows:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
0 1 2 3 5 7 9 12 15 18 22 26 30 35 40 45 51 57 63
0 1 2 5 8 14 20 30 40 55 70 91 112 140 168 204 240 285 330
0 1 3 7 14 25 42 66 99 143 200 273 364 476 612 775 969 1197 1463
0 1 3 9 20 42 75 132 212 333 497 728 1026 1428 1932 2583 3384 4389 5598
0 1 4 12 30 66 132 245 429 715 1144 1768 2652 3876 5537 7752 10659 14421 19228
0 1 4 15 40 99 212 429 800 1430 2424 3978 6288 9690 14520 21318 30624 43263 60060
0 1 5 18 55 143 333 715 1430 2700 4862 8398 13995 22610 35530 54477 81719 120175
0 1 5 22 70 200 497 1144 2424 4862 9225 16796 29372 49742 81686 130750 204248
0 1 6 26 91 273 728 1768 3978 8398 16796 32065 58786 104006 178296 297160 482885
0 1 6 30 112 364 1026 2652 6288 13995 29372 58786 112632 208012 371384 643842
0 1 7 35 140 476 1428 3876 9690 22610 49742 104006 208012 400023 742900 1337220
0 1 7 40 168 612 1932 5537 14520 35530 81686 178296 371384 742900 1432613 2674440
...
It is essentially symmetric: A(r,r+i) = A(r,r-i+1).
Some of the diagonals are:
A(r,r+1): A000108
A(r,r): A022553
A(r,r-1): A000108
A(r,r+2): A000150
A(r,r+3): A050181
A(r,r+4): A050182
A(r,r+5): A050183
A(r,r-2): A000150 (End)
Fredman (1975) proved that the number S(n, k, v) of vectors (a_0, ..., a_{n-1}) of nonnegative integer components that satisfy a_0 + ... + a_{n-1} = k and Sum_{i=0..n-1} i*a_i = v (mod n) is given by S(n, k, v) = (1/(n + k)) * Sum_{d | gcd(n, k)} A054533(d, v) * binomial((n + k)/d, k/d) = S(k, n, v). This was also proved by Elashvili et al. (1999), who also proved that S(n, k, v) = Sum_{d | gcd(n, k, v)} S(n/d, k/d, 1). Here, S(n, k, 1) = T(n + k, k). - Petros Hadjicostas, Jul 09 2019

			Triangle begins with:
h=0: 1
h=1: 1, 1
h=2: 0, 1, 0
h=3: 0, 1, 1, 0
h=4: 0, 1, 1, 1,  0
h=5: 0, 1, 2, 2,  1,  0
h=6: 0, 1, 2, 3,  2,  1, 0
h=7: 0, 1, 3, 5,  5,  3, 1, 0
h=8: 0, 1, 3, 7,  8,  7, 3, 1, 0
h=9: 0, 1, 4, 9, 14, 14, 9, 4, 1, 0
...
T(6,3) counts classes {111000},{110100},{110010}, each of 6 aperiodic. The class {100100} contains 3 periodic words, counted by T(3,1) as {100}, consisting of 3 aperiodic words 100,010,001.

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Freddy Cachazo and Nick Early, Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes, arXiv:2010.09708 [math.CO], 2020.
A. Elashvili and M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9 (1998), no. 2, 233-238; MR1691428 (2000c:13006).
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173-188; MR1719140 (2000j:05009). See p. 174. - _N. J. A. Sloane_, Aug 06 2014
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
Romeo Meštrović, Different classes of binary necklaces and a combinatorial method for their enumerations, arXiv:1804.00992 [math.CO], 2018.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
P. Stanica and S. Maitra, Rotation symmetric boolean functions - count and cryptographic properties, Disc. Appl. Math. 156 (2008) 1567-1580; see h_{n,w} in eq. (3).
Index entries for sequences related to Lyndon words

Columns 1-11: A000012, A004526(n-1), A001840(n-4), A006918(n-4), A011795(n-1), A011796(n-6), A011797(n-1), A031164(n-9), A011845, A032168, A032169. See also A000150.

Cf. A047996, A052307, A052314, A092964, A185158, A123223, A124814.

A := proc(r,n) local gf,d,genf; genf := 1/(1-x) ; gf := 0 ; for d in numtheory[divisors](r) do gf := gf + numtheory[mobius](d)*(subs(x= x^d,genf))^(r/d) ; od: gf := expand(gf/r) ; coeftayl(gf,x=0,n) ; end proc:
A051168 := proc(n,k) if n<=1 then 1; elif n=0 or n=k then 0; else A(n-k,k) ; end if;
end proc:
seq(seq(A051168(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Mar 29 2011

Table[If[n===0,1,1/n Plus@@(MoebiusMu[ # ]Binomial[n/#,k/# ]&/@ Divisors[GCD[n,k]])],{n,0,12},{k,0,n}] (* Wouter Meeussen, Jul 20 2008 *)

{T(n, k) = local(A, ps, c); if( k<0 || k>n, 0, if( n==0 && k==0, 1, A = x * O(x^n) + y * O(y^n); ps = 1 - x - y + A; for( m=1, n, for( i=0, m, c = polcoeff( polcoeff(ps, i, x), m-i, y); if( m==n && i==k, break(2), ps *= (1 -y^(m-i) * x^i + A)^c))); -c))} /* Michael Somos, Jul 03 2004 */

T(n,k) = if (n==0, 1, (1/n) * sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d,k/d)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 16 2018

T(h, k) = 1 for (h, k) in {(0, 0), (1, 0), (1, 1)}; T(h, k) = 0 if h >= 2 and k = 0 or k = h. Otherwise, T(h, k) = (1/h)*(C(h, k)-S(h, k)), where S(h, k) = Sum_{d <= 2, d|h, d|k} (h/d)*T(h/d, k/d).
1 - x - y = Product_{i,j} (1 - x^i * y^j)^T(i+j, j) where i >= 0, j >= 0 are not both zero. - Michael Somos, Jul 03 2004
The prime rows are given by (1+x)^p/p with noninteger coefficients rounded to zero. E.g., for h = 2 below, (1 + x)^2/2 = (1 + 2*x + x^2)/2 = 0.5 + x + 0.5*x^2 gives (0,1,0). - Tom Copeland, Oct 21 2014
T(n,k) = (1/n) * Sum_{d | gcd(n,k)} mu(d) * binomial(n/d, k/d), for n > 0. - Andrew Howroyd, Mar 26 2017
From Petros Hadjicostas, Jun 16 2019: (Start)
O.g.f. for column k >= 1: (x^k/k) * Sum_{d|k} mu(d)/(1 - x^d)^(k/d).
Bivariate o.g.f.: Sum_{n,k >= 0} T(n, k)*x^n*y^k = 1 - Sum_{d >= 1} (mu(d)/d) *log(1 - x^d * (1 + y^d)).
(End)

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A106255 Triangle composed of triangular numbers, row sums = A006918.

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A084835 a(n) = A000094(n+4) - A006918(n).

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A038376 a(n) = (n-3)*A006918(n-2)/2 for n >= 2, with a(0) = a(1) = 0.

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A211795 Number of (w,x,y,z) with all terms in {1,...,n} and w*x < 2*y*z.

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A002623 Expansion of 1/((1-x)^4*(1+x)).

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A115994 Triangle read by rows: T(n,k) is number of partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

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A212959 Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

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A211795 Number of (w,x,y,z) with all terms in {1,...,n} and wx < 2y*z.