A101881 -id:A101881 - cp's OEIS frontend

A000096 a(n) = n*(n+3)/2.

0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377, 1430, 1484, 1539, 1595, 1652, 1710, 1769

Offset: 0

N. J. A. Sloane

For n >= 1, a(n) is the maximal number of pieces that can be obtained by cutting an annulus with n cuts. See illustration. - Robert G. Wilson v
n(n-3)/2 (n >= 3) is the number of diagonals of an n-gon. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
n(n-3)/2 (n >= 4) is the degree of the third-smallest irreducible presentation of the symmetric group S_n (cf. James and Kerber, Appendix 1).
a(n) is also the multiplicity of the eigenvalue (-2) of the triangle graph Delta(n+1). (See p. 19 in Biggs.) - Felix Goldberg (felixg(AT)tx.technion.ac.il), Nov 25 2001
For n > 3, a(n-3) = dimension of the traveling salesman polytope T(n). - Benoit Cloitre, Aug 18 2002
Also counts quasi-dominoes (quasi-2-ominoes) on an n X n board. Cf. A094170-A094172. - Jon Wild, May 07 2004
Coefficient of x^2 in (1 + x + 2*x^2)^n. - Michael Somos, May 26 2004
a(n) is the number of "prime" n-dimensional polyominoes. A "prime" n-polyomino cannot be formed by connecting any other n-polyominoes except for the n-monomino and the n-monomino is not prime. E.g., for n=1, the 1-monomino is the line of length 1 and the only "prime" 1-polyominoes are the lines of length 2 and 3. This refers to "free" n-dimensional polyominoes, i.e., that can be rotated along any axis. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005
Solutions to the quadratic equation q(m, r) = (-3 +- sqrt(9 + 8(m - r))) / 2, where m - r is included in a(n). Let t(m) = the triangular number (A000217) less than some number k and r = k - t(m). If k is neither prime nor a power of two and m - r is included in A000096, then m - q(m, r) will produce a value that shares a divisor with k. - Andrew S. Plewe, Jun 18 2005
Sum_{k=2..n+1} 4/(k*(k+1)*(k-1)) = ((n+3)*n)/((n+2)*(n+1)). Numerator(Sum_{k=2..n+1} 4/(k*(k+1)*(k-1))) = (n+3)*n/2. - Alexander Adamchuk, Apr 11 2006
Number of rooted trees with n+3 nodes of valence 1, no nodes of valence 2 and exactly two other nodes. I.e., number of planted trees with n+2 leaves and exactly two branch points. - Theo Johnson-Freyd (theojf(AT)berkeley.edu), Jun 10 2007
If X is an n-set and Y a fixed 2-subset of X then a(n-2) is equal to the number of (n-2)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
For n >= 1, a(n) is the number of distinct shuffles of the identity permutation on n+1 letters with the identity permutation on 2 letters (12). - Camillia Smith Barnes, Oct 04 2008
If s(n) is a sequence defined as s(1) = x, s(n) = kn + s(n-1) + p for n > 1, then s(n) = a(n-1)*k + (n-1)*p + x. - Gary Detlefs, Mar 04 2010
The only primes are a(1) = 2 and a(2) = 5. - Reinhard Zumkeller, Jul 18 2011
a(n) = m such that the (m+1)-th triangular number minus the m-th triangular number is the (n+1)-th triangular number: (m+1)(m+2)/2 - m(m+1)/2 = (n+1)(n+2)/2. - Zak Seidov, Jan 22 2012
For n >= 1, number of different values that Sum_{k=1..n} c(k)*k can take where the c(k) are 0 or 1. - Joerg Arndt, Jun 24 2012
On an n X n chessboard (n >= 2), the number of possible checkmate positions in the case of king and rook versus a lone king is 0, 16, 40, 72, 112, 160, 216, 280, 352, ..., which is 8*a(n-2). For a 4 X 4 board the number is 40. The number of positions possible was counted including all mirror images and rotations for all four sides of the board. - Jose Abutal, Nov 19 2013
If k = a(i-1) or k = a(i+1) and n = k + a(i), then C(n, k-1), C(n, k), C(n, k+1) are three consecutive binomial coefficients in arithmetic progression and these are all the solutions. There are no four consecutive binomial coefficients in arithmetic progression. - Michael Somos, Nov 11 2015
a(n-1) is also the number of independent components of a symmetric traceless tensor of rank 2 and dimension n >= 1. - Wolfdieter Lang, Dec 10 2015
Numbers k such that 8k + 9 is a square. - Juri-Stepan Gerasimov, Apr 05 2016
Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rho-th derivative of f, expressed as a polynomial in the averaged symmetric polynomials in the roots of f. [See the Wojnar et al. link] The "last" term of phi_(D,rho) is a multiple of the product of all roots of f; the coefficient is expressible as a polynomial h_D(N) in N:=D-rho. These polynomials are of the form h_D(N)= ((-1)^D/(D-1)!)*(D-N)*N^chi*g_D(N) where chi = (1 if D is odd, 0 if D is even) and g_D(N) is a monic polynomial of degree (D-2-chi). Then a(n) are the negated coefficients of the next to the highest order term in the polynomials N^chi*g_D(N), starting at D=3. - Gregory Gerard Wojnar, Jul 19 2017
For n >= 2, a(n) is the number of summations required to solve the linear regression of n variables (n-1 independent variables and 1 dependent variable). - Felipe Pedraza-Oropeza, Dec 07 2017
For n >= 2, a(n) is the number of sums required to solve the linear regression of n variables: 5 for two variables (sums of X, Y, X^2, Y^2, X*Y), 9 for 3 variables (sums of X1, X2, Y1, X1^2, X1*X2, X1*Y, X2^2, X2*Y, Y^2), and so on. - Felipe Pedraza-Oropeza, Jan 11 2018
a(n) is the area of a triangle with vertices at (n, n+1), ((n+1)*(n+2)/2, (n+2)*(n+3)/2), ((n+2)^2, (n+3)^2). - J. M. Bergot, Jan 25 2018
Number of terms less than 10^k: 1, 4, 13, 44, 140, 446, 1413, 4471, 14141, 44720, 141420, 447213, ... - Muniru A Asiru, Jan 25 2018
a(n) is also the number of irredundant sets in the (n+1)-path complement graph for n > 2. - Eric W. Weisstein, Apr 11 2018
a(n) is also the largest number k such that the largest Dyck path of the symmetric representation of sigma(k) has exactly n peaks, n >= 1. (Cf. A237593.) - Omar E. Pol, Sep 04 2018
For n > 0, a(n) is the number of facets of associahedra. Cf. A033282 and A126216 and their refinements A111785 and A133437 for related combinatorial and analytic constructs. See p. 40 of Hanson and Sha for a relation to projective spaces and string theory. - Tom Copeland, Jan 03 2021
For n > 0, a(n) is the number of bipartite graphs with 2n or 2n+1 edges, no isolated vertices, and a stable set of cardinality 2. - Christian Barrientos, Jun 13 2022
For n >= 2, a(n-2) is the number of permutations in S_n which are the product of two different transpositions of adjacent points. - Zbigniew Wojciechowski, Mar 31 2023
a(n) represents the optimal stop-number to achieve the highest running score for the Greedy Pig game with an (n-1)-sided die with a loss on a 1. The total at which one should stop is a(s-1),  e.g. for a 6-sided die, one should pass the die at 20. See Sparks and Haran. - Nicholas Stefan Georgescu, Jun 09 2024

			G.f. = 2*x + 5*x^2 + 9*x^3 + 14*x^4 + 20*x^5 + 27*x^6 + 35*x^7 + 44*x^8 + 54*x^9 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993.
G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Maths. and its Appls., Vol. 16, Addison-Wesley, 1981, Reading, MA, U.S.A.
D. G. Kendall et al., Shape and Shape Theory, Wiley, 1999; see p. 4.
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).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
D. Applegate, R. Bixby, V. Chvatal and W. Cook, On the solution of traveling salesman problem, In : Int. Congress of mathematics (Berlin 1998), Documenta Math., Extra Volume ICM 1998, Vol. III, pp. 645-656.
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
L. Euler, Sur une contradiction apparente dans la doctrine des lignes courbes, Mémoires de l'Académie des Sciences de Berlin, 4, 219-233, 1750 Reprinted in Opera Omnia, Series I, Vol. 26. pp. 33-45.
Mareike Fischer, Extremal values of the Sackin balance index for rooted binary trees, arXiv:1801.10418 [q-bio.PE], 2018.
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Theorem 1.
A. Hanson and J. Sha, A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6, arXiv preprint arXiv:0510064 [math-ph], 2005.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
S. P. Humphries, Home page
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1018 [dead link]
Milan Janjic, Two Enumerative Functions
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 13.
A. McLeod and W. O. J. Moser, Counting cyclic binary strings, Math. Mag., 80 (No. 1, 2007), 29-37.
Yashar Memarian, On the Maximum Number of Vertices of Minimal Embedded Graphs, arXiv:0910.2469 [math.CO], 2009-15. [_Jonathan Vos Post_, Oct 14 2009]
Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
E. Pérez Herrero, Binomial Matrix (I), Psychedelic Geometry Blogspot 09/22/09. [_Enrique Pérez Herrero_, Sep 22 2009]
T. Manneville and V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015.
P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
P. Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
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
Maria J. Rodriguez, Black holes in all, The KIAS Newsletter, Vol.3, pp.29-34, Korea Institute for Advanced Study, Dec. 2010. [Wayback archive]
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Wayback archive]
E. Sandifer, How Euler Did It: Cramer's Paradox, 2004.
N. J. A. Sloane, How to cut an annulus into 9 pieces with three cuts.
Ben Sparks and Brady Haran, The Math of Being a Greedy Pig, Numberphile video, 2021.
Eric Weisstein's World of Mathematics, Cramer-Euler Paradox
Eric Weisstein's World of Mathematics, Irredundant Set
Eric Weisstein's World of Mathematics, Path Complement Graph
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for sequences related to Chebyshev polynomials.

Complement of A007401. Column 2 of A145324. Column of triangle A014473, first skew subdiagonal of A033282, a diagonal of A079508.
Occurs as a diagonal in A074079/A074080, i.e., A074079(n+3, n) = A000096(n-1) for all n >= 2. Also A074092(n) = 2^n * A000096(n-1) after n >= 2.
Cf. A000124, A000217, A005581-A005584, A023531, A034856, A067550, A127672 (Chebyshev C).
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488.
Similar sequences are listed in A316466.
Cf. A033282, A111785, A126216, A133437.

a := List([0..1000], n -> n*(n+3)/2); # Muniru A Asiru, Jan 25 2018

a000096 n = n * (n + 3) `div` 2
a000096_list = [x | x <- [0..], a023531 x == 1]
-- Reinhard Zumkeller, Feb 14 2015, Dec 04 2012

[n*(n+3)/2: n in [0..60]]; // Juri-Stepan Gerasimov, Apr 05 2016

[n: n in [0..2300] | IsSquare(8*n+9)]; // Juri-Stepan Gerasimov, Apr 05 2016

A000096 := n->n*(n+3)/2; seq(A000096(n), n=0..50);
A000096 :=z*(-2+z)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

Table[n*(n+3)/2, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)
LinearRecurrence[{3,-3,1}, {0,2,5}, 60] (* Harvey P. Dale, Apr 30 2013 *)

{a(n) = n * (n+3)/2}; \\ Michael Somos, May 26 2004

first(n) = Vec(x*(2-x)/(1-x)^3 + O(x^n), -n) \\ Iain Fox, Dec 12 2017

G.f.: A(x) = x*(2-x)/(1-x)^3. a(n) = binomial(n+1, n-1) + binomial(n, n-1).
Connection with triangular numbers: a(n) = A000217(n+1) - 1.
a(n) = a(n-1) + n + 1. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005
a(n) = 2*t(n) - t(n-1) where t() are the triangular numbers, e.g., a(5) = 2*t(5) - t(4) = 2*15 - 10 = 20. - Jon Perry, Jul 23 2003
a(-3-n) = a(n). - Michael Somos, May 26 2004
2*a(n) = A008778(n) - A105163(n). - Creighton Dement, Apr 15 2005
a(n) = C(3+n, 2) - C(3+n, 1). - Zerinvary Lajos, Dec 09 2005
a(n) = A067550(n+1) / A067550(n). - Alexander Adamchuk, May 20 2006
a(n) = A126890(n,1) for n > 0. - Reinhard Zumkeller, Dec 30 2006
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Paul Curtz, Jan 02 2008
Starting (2, 5, 9, 14, ...) = binomial transform of (2, 3, 1, 0, 0, 0, ...). - Gary W. Adamson, Jul 03 2008
For n >= 0, a(n+2) = b(n+1) - b(n), where b(n) is the sequence A005586. - K.V.Iyer, Apr 27 2009
A002262(a(n)) = n. - Reinhard Zumkeller, May 20 2009
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=1, a(n-1)=coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jul 08 2010
a(n) = Sum_{k=1..n} (k+1)!/k!. - Gary Detlefs, Aug 03 2010
a(n) = n(n+1)/2 + n = A000217(n) + n. - Zak Seidov, Jan 22 2012
E.g.f.: F(x) = 1/2*x*exp(x)*(x+4) satisfies the differential equation F''(x) - 2*F'(x) + F(x) = exp(x). - Peter Bala, Mar 14 2012
a(n) = binomial(n+3, 2) - (n+3). - Robert G. Wilson v, Mar 15 2012
a(n) = A181971(n+1, 2) for n > 0. - Reinhard Zumkeller, Jul 09 2012
a(n) = A214292(n+2, 1). - Reinhard Zumkeller, Jul 12 2012
G.f.: -U(0) where U(k) = 1 - 1/((1-x)^2 - x*(1-x)^4/(x*(1-x)^2 - 1/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 27 2012
A023532(a(n)) = 0. - Reinhard Zumkeller, Dec 04 2012
a(n) = A014132(n,n) for n > 0. - Reinhard Zumkeller, Dec 12 2012
a(n-1) = (1/n!)*Sum_{j=0..n} binomial(n,j)*(-1)^(n-j)*j^n*(j-1). - Vladimir Kruchinin, Jun 06 2013
a(n) = 2n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = Sum_{i=2..n+1} i. - Wesley Ivan Hurt, Jun 28 2013
Sum_{n>0} 1/a(n) = 11/9. - Enrique Pérez Herrero, Nov 26 2013
a(n) = Sum_{i=1..n} (n - i + 2). - Wesley Ivan Hurt, Mar 31 2014
A023531(a(n)) = 1. - Reinhard Zumkeller, Feb 14 2015
For n > 0: a(n) = A101881(2*n-1). - Reinhard Zumkeller, Feb 20 2015
a(n) + a(n-1) = A008865(n+1) for all n in Z. - Michael Somos, Nov 11 2015
a(n+1) = A127672(4+n, n), n >= 0, where A127672 gives the coefficients of the Chebyshev C polynomials. See the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 10 2015
a(n) = (n+1)^2 - A000124(n). - Anton Zakharov, Jun 29 2016
Dirichlet g.f.: (zeta(s-2) + 3*zeta(s-1))/2. - Ilya Gutkovskiy, Jun 30 2016
a(n) = 2*A000290(n+3) - 3*A000217(n+3). - J. M. Bergot, Apr 04 2018
a(n) = Stirling2(n+2, n+1) - 1. - Peter Luschny, Jan 05 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/3 - 5/9. - Amiram Eldar, Jan 10 2021
From Amiram Eldar, Jan 20 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = 3.
Product_{n>=1} (1 - 1/a(n)) = 3*cos(sqrt(17)*Pi/2)/(4*Pi). (End)
Product_{n>=0} a(4*n+1)*a(4*n+4)/(a(4*n+2)*a(4*n+3)) = Pi/6. - Michael Jodl, Apr 05 2025

A034856 a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1.

Original entry on oeis.org

1, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, 89, 103, 118, 134, 151, 169, 188, 208, 229, 251, 274, 298, 323, 349, 376, 404, 433, 463, 494, 526, 559, 593, 628, 664, 701, 739, 778, 818, 859, 901, 944, 988, 1033, 1079, 1126, 1174, 1223, 1273, 1324, 1376, 1429, 1483

Offset: 1

N. J. A. Sloane

Number of 1's in the n X n lower Hessenberg (0,1)-matrix (i.e., the matrix having 1's on or below the superdiagonal and 0's above the superdiagonal).
If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-2) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007
Number of binary operations which have to be added to Moisil's algebras to obtain algebraic counterparts of n-valued Łukasiewicz logics. See the Wójcicki and Malinowski book, page 31. - Artur Jasinski, Feb 25 2010
Also (n + 1)!(-1)^(n + 1) times the determinant of the n X n matrix given by m(i,j) = i/(i+1) if i=j and otherwise 1. For example, (5+1)! * ((-1)^(5+1)) * Det[{{1/2, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1}, {1, 1, 3/4, 1, 1}, {1, 1, 1, 4/5, 1}, {1, 1, 1, 1, 5/6}}] = 19 = a(5), and (6+1)! * ((-1)^(6+1)) * Det[{{1/2, 1, 1, 1, 1, 1}, {1, 2/3, 1, 1, 1, 1}, {1, 1, 3/4, 1, 1, 1}, {1, 1, 1, 4/5, 1, 1}, {1, 1, 1, 1, 5/6, 1}, {1, 1, 1, 1, 1, 6/7}}] = 26 = a(6). - John M. Campbell, May 20 2011
2*a(n-1) = n*(n+1) - 4, n>=0, with a(-1) = -2 and a(0) = -1, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 17 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013
a(n) is not divisible by 3, 5, 7, or 11. - Vladimir Shevelev, Feb 03 2014
With a(0) = 1 and a(1) = 2, a(n-1) is the number of distinct values of 1 +- 2 +- 3 +- ... +- n, for n > 0. - Derek Orr, Mar 11 2015
Also, numbers m such that 8*m+17 is a square. - Bruno Berselli, Sep 16 2015
Omar E. Pol's formula from Apr 23 2008 can be interpreted as the position of an element located on the third diagonal of an triangular array (read by rows) provided n > 1. - Enrique Pérez Herrero, Aug 29 2016
a(n) is the sum of the numerator and denominator of the fraction that is the sum of 2/(n-1) + 2/n; all fractions are reduced and n > 2. - J. M. Bergot, Jun 14 2017
a(n) is also the number of maximal irredundant sets in the (n+2)-path complement graph for n > 1. - Eric W. Weisstein, Apr 12 2018
From Klaus Purath, Dec 07 2020: (Start)
a(n) is not divisible by primes listed in A038890. The prime factors are given in A038889 and the prime terms of the sequence are listed in A124199.
Each odd prime factor p divides exactly 2 out of any p consecutive terms with the exception of 17, which appears only once in such an interval of terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -3 (mod p), see examples.
If A is a sequence satisfying the recurrence t(n) = 5*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 4 or A(0) = -1, A(1) = n-1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i for i>0. (End)
Mark each point on a 4^n grid with the number of points that are visible from the point; for n > 1, a(n) is the number of distinct values in the grid. - Torlach Rush, Mar 23 2021
The sequence gives the number of "ON" cells in the cellular automaton on a quadrant of a square grid after the n-th stage, where the "ON" cells lie only on the external perimeter and the perimeter of inscribed squares having the cell (1,1) as a unique common vertex. See Spezia link. - Stefano Spezia, May 28 2025

			From _Bruno Berselli_, Mar 09 2015: (Start)
By the definition (first formula):
----------------------------------------------------------------------
  1       4         8           13            19              26
----------------------------------------------------------------------
                                                              X
                                              X              X X
                                X            X X            X X X
                    X          X X          X X X          X X X X
          X        X X        X X X        X X X X        X X X X X
  X      X X      X X X      X X X X      X X X X X      X X X X X X
          X        X X        X X X        X X X X        X X X X X
----------------------------------------------------------------------
(End)
From _Klaus Purath_, Dec 07 2020: (Start)
Assuming a(i) is divisible by p with 0 < i < p and a(k) is the next term divisible by p, then from i + k == -3 (mod p) follows that k = min(p*m - i - 3) != i for any integer m.
(1) 17|a(7) => k = min(17*m - 10) != 7 => m = 2, k = 24 == 7 (mod 17). Thus every a(17*m + 7) is divisible by 17.
(2) a(9) = 53 => k = min(53*m - 12) != 9 => m = 1, k = 41. Thus every a(53*m + 9) and a(53*m + 41) is divisible by 53.
(3) 101|a(273) => 229 == 71 (mod 101) => k = min(101*m - 74) != 71 => m = 1, k = 27. Thus every a(101*m + 27) and a(101*m + 71) is divisible by 101. (End)
From _Omar E. Pol_, Aug 08 2021: (Start)
Illustration of initial terms:                             _ _
.                                           _ _           |_|_|_
.                              _ _         |_|_|_         |_|_|_|_
.                   _ _       |_|_|_       |_|_|_|_       |_|_|_|_|_
.          _ _     |_|_|_     |_|_|_|_     |_|_|_|_|_     |_|_|_|_|_|_
.   _     |_|_|    |_|_|_|    |_|_|_|_|    |_|_|_|_|_|    |_|_|_|_|_|_|
.  |_|    |_|_|    |_|_|_|    |_|_|_|_|    |_|_|_|_|_|    |_|_|_|_|_|_|
.
.   1       4         8          13            19              26
------------------------------------------------------------------------ (End)

A. S. Karpenko, Łukasiewicz's Logics and Prime Numbers, 2006 (English translation).
G. C. Moisil, Essais sur les logiques non-chrysippiennes, Ed. Academiei, Bucharest, 1972.
Wójcicki and Malinowski, eds., Łukasiewicz Sentential Calculi, Wrocław: Ossolineum, 1977.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 471.
Milan Janjic, Two Enumerative Functions.
W. F. Klostermeyer, M. E. Mays, L. Soltes, and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, Vol. 35, No. 4 (1997), pp. 318-328.
Stavros Konstantinidis, António Machiavelo, Nelma Moreira, and Rogério Reis, On the size of partial derivatives and the word membership problem, Acta Informatica Vol. 58, 357-375.
G. C. Moisil, Recherches sur les logiques non-chrysippiennes, Ann. Sci. Univ. Jassy, 26 (1940), 431-466.
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.
D. D. Olesky, B. L. Shader, and P. van den Driessche, Permanents of Hessenberg (0,1)-matrices, Electronic Journal of Combinatorics, 12 (2005), #R70.
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev., Vol. 4, No. 5 (1960), pp. 473-478.
Stefano Spezia, Illustrations for n = 1..9.
Renzo Sprugnoli, Alternating Weighted Sums of Inverses of Binomial Coefficients, J. Integer Sequences, 15 (2012), #12.6.3. - From _N. J. A. Sloane_, Nov 29 2012
Eric Weisstein's World of Mathematics, Maximal Irredundant Set.
Eric Weisstein's World of Mathematics, Path Complement Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for sequences related to Łukasiewicz.

Subsequence of A165157.
Triangular numbers (A000217) minus two.
Third diagonal of triangle in A059317.
Cf. A000096, A000124, A000217, A000290, A005408, A005843, A008277, A024206, A027379.
Cf. A028347, A038889, A038890, A049600, A091823, A101881, A113452, A113453, A113454.
Cf. A113455, A124199, A130296, A131818, A134225, A161680, A168244, A253909, A267633.

a034856 = subtract 1 . a000096 -- Reinhard Zumkeller, Feb 20 2015

[Binomial(n + 1, 2) + n - 1: n in [1..60]]; // Vincenzo Librandi, May 21 2011

a := n -> hypergeom([-2, n-1], [1], -1);
seq(simplify(a(n)), n=1..53); # Peter Luschny, Aug 02 2014

f[n_] := n (n + 3)/2 - 1; Array[f, 55] (* or *) k = 2; NestList[(k++; # + k) &, 1, 55] (* Robert G. Wilson v, Jun 11 2010 *)
Table[Binomial[n + 1, 2] + n - 1, {n, 53}] (* or *)
Rest@ CoefficientList[Series[x (1 + x - x^2)/(1 - x)^3, {x, 0, 53}], x] (* Michael De Vlieger, Aug 29 2016 *)

A034856(n) := block(
        n-1+(n+1)*n/2
)$ /* R. J. Mathar, Mar 19 2012 */

A034856(n)=(n+3)*n\2-1 \\ M. F. Hasler, Jan 21 2015

def A034856(n): return n*(n+3)//2 -1 # G. C. Greubel, Jun 15 2025

G.f.: A(x) = x*(1 + x - x^2)/(1 - x)^3.
a(n) = A049600(3, n-2).
a(n) = binomial(n+2, 2) - 2. - Paul Barry, Feb 27 2003
With offset 5, this is binomial(n, 0) - 2*binomial(n, 1) + binomial(n, 2), the binomial transform of (1, -2, 1, 0, 0, 0, ...). - Paul Barry, Jul 01 2003
Row sums of triangle A131818. - Gary W. Adamson, Jul 27 2007
Binomial transform of (1, 3, 1, 0, 0, 0, ...). Also equals A130296 * [1,2,3,...]. - Gary W. Adamson, Jul 27 2007
Row sums of triangle A134225. - Gary W. Adamson, Oct 14 2007
a(n) = A000217(n+1) - 2. - Omar E. Pol, Apr 23 2008
From Jaroslav Krizek, Sep 05 2009: (Start)
a(n) = a(n-1) + n + 1 for n >= 1.
a(n) = n*(n-1)/2 + 2*n - 1.
a(n) = A000217(n-1) + A005408(n-1) = A005843(n-1) + A000124(n-1). (End)
a(n) = Hyper2F1([-2, n-1], [1], -1). - Peter Luschny, Aug 02 2014
a(n) = floor[1/(-1 + Sum_{m >= n+1} 1/S2(m,n+1))], where S2 is A008277. - Richard R. Forberg, Jan 17 2015
a(n) = A101881(2*(n-1)). - Reinhard Zumkeller, Feb 20 2015
a(n) = A253909(n+3) - A000217(n+3). - David Neil McGrath, May 23 2015
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - David Neil McGrath, May 23 2015
For n > 1, a(n) = 4*binomial(n-1,1) + binomial(n-2,2), comprising the third column of A267633. - Tom Copeland, Jan 25 2016
From Klaus Purath, Dec 07 2020: (Start)
a(n) = A024206(n) + A024206(n+1).
a(2*n-1) = -A168244(n+1).
a(2*n) = A091823(n). (End)
Sum_{n>=1} 1/a(n) = 3/2 + 2*Pi*tan(sqrt(17)*Pi/2)/sqrt(17). - Amiram Eldar, Jan 06 2021
a(n) + a(n+1) = A028347(n+2). - R. J. Mathar, Mar 13 2021
a(n) = A000290(n) - A161680(n-1). - Omar E. Pol, Mar 26 2021
E.g.f.: 1 + exp(x)*(x^2 + 4*x - 2)/2. - Stefano Spezia, Jun 05 2021
a(n) = A024916(n) - A244049(n). - Omar E. Pol, Aug 01 2021
a(n) = A000290(n) - A000217(n-2). - Omar E. Pol, Aug 05 2021

More terms from Zerinvary Lajos, May 12 2006

A177994 Triangle read by rows, A177990 * A070909.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1

Offset: 0

Gary W. Adamson, May 16 2010

Row sums = A101881: (1, 2, 4, 5, 8, 9, 13, 14,...).

Double Riordan array ( 1/((1 - x)*(1 - x^2)); x*(1 - x^2), x/(1 - x^2) ) as defined in Davenport et al. The set of double Riordan arrays of the form ( g(x); x*f_1(x), x*f_2(x) ), where f_1(x)*f_2(x) = 1, forms a group under matrix multiplication. Here g, f_1 and f_2 denote power series with constant term equal to 1. This is the array (( 1/((1 - x)*(1 - x^2)), 1/(1 - x) )) in the notation of the Bala link. - Peter Bala, Aug 26 2021

			First few rows of the triangle =
1;
1, 1;
2, 1, 1;
2, 1, 1, 1;
3, 1, 2, 1, 1;
3, 1, 2, 1, 1, 1;
4, 1, 3, 1, 2, 1, 1;
4, 1, 3, 1, 2, 1, 1, 1;
5, 1, 4, 1, 3, 1, 2, 1, 1;
5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
...

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Peter Bala, Matrices with repeated columns - the generalised Appell groups
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Cf. A177991 = A070909 * A177990.

Cf. A101881.

a177994 n k = a177994_tabl !! n !! k
a177994_row n = a177994_tabl !! n
a177994_tabl = [1] : [1,1] : map f a177994_tabl
               where f xs@(x:_) = (x + 1) : 1 : xs
-- Reinhard Zumkeller, Feb 20 2015

As infinite lower triangular matrices, A177990 * A070909

A165157 Zero followed by partial sums of A133622.

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 12, 13, 18, 19, 25, 26, 33, 34, 42, 43, 52, 53, 63, 64, 75, 76, 88, 89, 102, 103, 117, 118, 133, 134, 150, 151, 168, 169, 187, 188, 207, 208, 228, 229, 250, 251, 273, 274, 297, 298, 322, 323, 348, 349, 375, 376, 403, 404, 432, 433, 462, 463, 493, 494, 525

Offset: 0

Jaroslav Krizek, Sep 05 2009

A133622 is a toothed sequence.

Interleaving of A055998 and A034856.

			From _Stefano Spezia_, Jul 10 2020: (Start)
Illustration of the initial terms for n > 0:
o    o      o      o         o        o
     o o    o o    o o       o o      o o
            o      o         o        o
                   o o o     o o o    o o o
                             o        o
                                      o o o o
(1)  (3)   (4)    (7)       (8)      (12)
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Equals -1+A101881.
a(n) = A117142(n+2)-2 = A055802(n+6)-3 = A114220(n+5)-3 = A134519(n+3)-3.
Cf. A133622, A055998, A034856.

a165157 n = a165157_list !! n
a165157_list = scanl (+) 0 a133622_list
-- Reinhard Zumkeller, Feb 20 2015

m:=60; T:=[ 1+(1+(-1)^n)*n/4: n in [1..m] ]; [0] cat [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..m] ]; // Klaus Brockhaus, Sep 06 2009

[ n le 2 select n-1 else n le 4 select n else 2*Self(n-2)-Self(n-4)+1: n in [1..61] ]; // Klaus Brockhaus, Sep 06 2009

a(0) = 0, a(2*n) = a(2*n-1) + n + 1, a(2*n+1) = a(2*n) + 1.
a(n) = (n^2+10*n)/8 if n is even, a(n) = (n^2+8*n-1)/8 if n is odd.
a(2*k) = A055998(k) = k*(k+5)/2; a(2*k+1) = A034856(k+1) = k*(k+5)/2+1.
a(n) = 2*a(n-2)-a(n-4)+1 for n > 3; a(0)=0, a(1)=1, a(2)=3, a(3)=4. - Klaus Brockhaus, Sep 06 2009
a(n) = (2*n*(n+9)-1+(2*n+1)*(-1)^n)/16. - Klaus Brockhaus, Sep 06 2009
a(n) = n+binomial(1+floor(n/2),2). - Mircea Merca, Feb 18 2012
G.f.: x*(1+2*x-x^2-x^3)/((1-x)^3*(1+x)^2). - Klaus Brockhaus, Sep 06 2009
From Stefano Spezia, Jul 10 2020: (Start)
E.g.f.: (x*(9 + x)*cosh(x) + (-1 + 11*x + x^2)*sinh(x))/8.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4. (End)

Edited and extended by Klaus Brockhaus, Sep 06 2009

A283394 a(n) = 3n(3*n + 7)/2 + 4.

Original entry on oeis.org

4, 19, 43, 76, 118, 169, 229, 298, 376, 463, 559, 664, 778, 901, 1033, 1174, 1324, 1483, 1651, 1828, 2014, 2209, 2413, 2626, 2848, 3079, 3319, 3568, 3826, 4093, 4369, 4654, 4948, 5251, 5563, 5884, 6214, 6553, 6901, 7258, 7624, 7999, 8383, 8776, 9178, 9589, 10009

Offset: 0

Bruno Berselli, Mar 23 2017

Sum_{k = 0..n} (3*k + r)^3 is divisible by 3*n*(3*n + 2*r + 3)/2 + r^2: the sequence corresponds to the case r = 2 of this formula (other cases are listed in Crossrefs section).
Also, Sum_{k = 0..n} (3*k + 2)^3 / a(n) gives 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, ... (A005449).
a(n) is even if n belongs to A014601. No term is divisible by 3, 5, 7 and 11.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Nickolas Arustamyan, Christopher Cox, Erik Lundberg, Sean Perry, and Zvi Rosen, On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force, arXiv:2106.11416 [math-ph], 2021.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005449, A034856, A095794, A101881, A165157, A186349.

Sequences with formula 3*n*(3*n + 2*r + 3)/2 + r^2: A038764 (r=-1), A027468 (r=0), A081271 (r=1), this sequence (r=2), A027468 (r=3; offset: -1), A080855 (r=4; offset: -2).

Magma
```
[3*n*(3*n+7)/2+4: n in [0..50]];
```

Table[3 n (3 n + 7)/2 + 4, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{4,19,43},50] (* Harvey P. Dale, Mar 02 2019 *)

Maxima
```
makelist(3*n*(3*n+7)/2+4, n, 0, 50);
```

a(n) = 3*n*(3*n + 7)/2 + 4; \\ Indranil Ghosh, Mar 24 2017

Python
```
[3*n*(3*n+7)/2+4 for n in range(50)]
```
Sage
```
[3*n*(3*n+7)/2+4 for n in range(50)]
```

O.g.f.: (4 + 7*x - 2*x^2)/(1 - x)^3.
E.g.f.: (8 + 30*x + 9*x^2)*exp(x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A081271(-n-2).
a(n) = 3*A095794(n+1) + 1.
a(n) = A034856(3*n+2) = A101881(6*n+2) = A165157(6*n+3) = A186349(6*n+3).
The inverse binomial transform yields 4, 15, 9, 0 (0 continued), therefore:
a(n) = 4*binomial(n,0) + 15*binomial(n,1) + 9*binomial(n,2).

A101882 Write three numbers, skip one, write three, skip two, write three, skip three... and so on.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 16, 17, 18, 23, 24, 25, 31, 32, 33, 40, 41, 42, 50, 51, 52, 61, 62, 63, 73, 74, 75, 86, 87, 88, 100, 101, 102, 115, 116, 117, 131, 132, 133, 148, 149, 150, 166, 167, 168, 185, 186, 187, 205, 206, 207, 226, 227, 228, 248, 249, 250, 271

Offset: 1

Candace Mills (scorpiocand(AT)yahoo.com), Dec 19 2004

Union of A052905, A052905+1, and A052905+2. - Ivan Neretin, Aug 03 2016

First terms of the 3 repeated terms belong to A052905. - Michael De Vlieger, Aug 03 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A000217, A052905, A101881, A101883.

Flatten@Table[(n^2 + 5 n - 4)/2 + {0, 1, 2}, {n, 20}] (* Ivan Neretin, Aug 03 2016 *)
Table[Range[#, # + 2] &[(n^2 + 7 n + 2)/2], {n, 0, 20}] // Flatten (* or *)
Rest@ CoefficientList[Series[x (1 + x + x^2 - x^4 - x^5)/((1 + x + x^2)^2 (1 - x)^3), {x, 0, 61}], x] (* Michael De Vlieger, Aug 03 2016 *)
LinearRecurrence[{1,0,2,-2,0,-1,1},{1,2,3,5,6,7,10},70] (* Harvey P. Dale, Dec 26 2019 *)

a(n)=my(k=n%3); if(k==2, n^2+17*n-2, k==1, n^2+19*n-2, n^2+15*n)/18 \\ Charles R Greathouse IV, Aug 03 2016

G.f.: x*(1+x+x^2-x^4-x^5)/ ((1+x+x^2)^2 * (1-x)^3). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; checked and corrected by R. J. Mathar, Sep 16 2009]

a(n) = n + k * (k+1) / 2 where k = floor((n-1) / 3). - Ziad Ahmed, Jun 17 2025

A101883 Write four numbers, skip one, write four, skip two, write four, skip three... and so on.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 19, 20, 21, 22, 27, 28, 29, 30, 36, 37, 38, 39, 46, 47, 48, 49, 57, 58, 59, 60, 69, 70, 71, 72, 82, 83, 84, 85, 96, 97, 98, 99, 111, 112, 113, 114, 127, 128, 129, 130, 144, 145, 146, 147, 162, 163, 164, 165, 181, 182, 183, 184, 201

Offset: 1

Candace Mills (scorpiocand(AT)yahoo.com), Dec 19 2004

Union of A051936, A051936+1, A051936+2, and A051936+3. - Ivan Neretin, Aug 03 2016

First terms of the four repeated terms belong to A051936. - Michael De Vlieger, Aug 03 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A000217, A051936, A101881, A101882.

LinearRecurrence[{1,0,0,2,-2,0,0,-1,1},{1,2,3,4,6,7,8,9,12},80] (* or *) Rest[CoefficientList[Series[x*(1+x+x^2+x^3-x^5-x^6-x^7)/((1+x)^2(x^2+1)^2(1-x)^3),{x,0,80}],x]](* Harvey P. Dale, Jan 19 2016 *)
Table[Range[#, # + 3] &[n (n + 1)/2 - 9], {n, 4, 20}] // Flatten (* Michael De Vlieger, Aug 03 2016 *)

G.f.: x*(1+x+x^2+x^3-x^5-x^6-x^7)/ ((1+x)^2 * (x^2+1)^2 * (1-x)^3). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; checked and corrected by R. J. Mathar, Sep 16 2009]

A368096 Triangle read by rows where T(n,k) is the number of non-isomorphic set-systems of length k and weight n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 5, 8, 3, 1, 0, 1, 8, 18, 13, 3, 1, 0, 1, 9, 32, 37, 15, 3, 1, 0, 1, 13, 55, 96, 59, 16, 3, 1, 0, 1, 14, 91, 209, 196, 74, 16, 3, 1, 0, 1, 19, 138, 449, 573, 313, 82, 16, 3, 1, 0, 1, 20, 206, 863, 1529, 1147, 403, 84, 16, 3, 1

Offset: 0

Gus Wiseman, Dec 28 2023

A set-system is a finite set of finite nonempty sets.

Conjecture: Column k = 2 is A101881.

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   2   1
   0   1   4   3   1
   0   1   5   8   3   1
   0   1   8  18  13   3   1
   0   1   9  32  37  15   3   1
   0   1  13  55  96  59  16   3   1
   0   1  14  91 209 196  74  16   3   1
   0   1  19 138 449 573 313  82  16   3   1
   ...
Non-isomorphic representatives of the set-systems counted in row n = 5:
  .  {12345}  {1}{1234}  {1}{2}{123}  {1}{2}{3}{12}  {1}{2}{3}{4}{5}
              {1}{2345}  {1}{2}{134}  {1}{2}{3}{14}
              {12}{123}  {1}{2}{345}  {1}{2}{3}{45}
              {12}{134}  {1}{12}{13}
              {12}{345}  {1}{12}{23}
                         {1}{12}{34}
                         {1}{23}{24}
                         {1}{23}{45}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A283877, connected case A300913.
For multiset partitions we have A317533.
Counting connected components instead of edges gives A321194.
For set multipartitions we have A334550.
For strict multiset partitions we have A368099.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A049311 counts non-isomorphic set multipartitions, connected A056156.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A316980 counts non-isomorphic strict multiset partitions, connected A319557.
A319559 counts non-isomorphic T_0 set-systems, connected A319566.
Cf. A101881, A255903, A302545, A306005, A317532, A317794, A317795, A319560, A321405, A368094, A368095.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&And@@UnsameQ@@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

Terms a(66) and beyond from Andrew Howroyd, Jan 11 2024

cp's OEIS Frontend

A000096 a(n) = n*(n+3)/2.

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A034856 a(n) = binomial(n+1, 2) + n - 1 = n*(n+3)/2 - 1.

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A177994 Triangle read by rows, A177990 * A070909.

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A165157 Zero followed by partial sums of A133622.

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A283394 a(n) = 3*n*(3*n + 7)/2 + 4.

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A101882 Write three numbers, skip one, write three, skip two, write three, skip three... and so on.

A283394 a(n) = 3n(3*n + 7)/2 + 4.