A005347 -id:A005347 - cp's OEIS frontend

A000103 Number of n-node triangulations of sphere in which every node has degree >= 4.

0, 0, 1, 1, 2, 5, 12, 34, 130, 525, 2472, 12400, 65619, 357504, 1992985, 11284042, 64719885, 375126827, 2194439398, 12941995397, 76890024027, 459873914230, 2767364341936, 16747182732792

Offset: 4

N. J. A. Sloane

			a(4)=0, a(5)=0 because the tetrahedron and the 5-bipyramid both have vertices of degree 3. a(6)=1 because of the A000109(6)=2 triangulations with 6 nodes (abcdef) the one corresponding to the octahedron (bcde,afec,abfd,acfe,adfb,bedc) has no node of degree 3, whereas the other triangulation (bcdef,afec,abed,ace,adcbf,aeb) has 2 such nodes.

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).

R. Bowen and S. Fisk, Generation of triangulations of the sphere, Math. Comp., 21 (1967), 250-252.
R. Bowen and S. Fisk, Generation of triangulations of the sphere [Annotated scanned copy]
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
D. A. Holton and B. D. McKay, The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices, J. Combinat. Theory B vol 45, iss. 3 (1988) 305-319.
D. A. Holton and B. D. McKay, Erratum, J. Combinat. Theory B vol 47, iss. 2 (1989) 248.
J. Lederberg, Dendral-64, II, Report to NASA, Dec 1965 [Annotated scanned copy]
Thom Sulanke, Generating triangulations of surfaces (surftri), (also subpages).
Eric Weisstein's World of Mathematics, Cubic Polyhedral Graph

Cf. all triangulations: A000109, triangulations with minimum degree 5: A081621.

More terms from Hugo Pfoertner, Mar 24 2003

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm) from the Surftri web site, May 05 2007

A000109 Number of simplicial polyhedra with n vertices; simple planar graphs with n vertices and 3n-6 edges; maximal simple planar graphs with n vertices; planar triangulations with n vertices; triangulations of the sphere with n vertices; 3-connected cubic planar graphs on 2n-4 vertices.

Original entry on oeis.org

1, 1, 1, 2, 5, 14, 50, 233, 1249, 7595, 49566, 339722, 2406841, 17490241, 129664753, 977526957, 7475907149, 57896349553, 453382272049, 3585853662949, 28615703421545

Offset: 3

N. J. A. Sloane

Every planar triangulation on n >= 4 vertices is 3-connected (the connectivity either 3, 4, or 5) and its dual graph is a 3-connected cubic planar graph on 2n-4 vertices. - Manfred Scheucher, Mar 17 2023

G. Brinkmann and Brendan McKay, in preparation. [Looking at http://users.cecs.anu.edu.au/~bdm/publications.html, there are a few papers with Brinkmann that seem relevant, in particular #126 but also #97, 81, 158. Perhaps the right one is 126.]
M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.
C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
B. Grünbaum, Convex Polytopes. Wiley, NY, 1967, p. 424.
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).

David Wasserman, Table of n, a(n) for n = 3..23
J. Bokowski and P. Schuchert, Equifacetted 3-spheres as topes of nonpolytopal matroid polytopes, Discrete Comput. Geom. 13 (1995), no. 3-4, 347-361.
R. Bowen and S. Fisk, Generation of triangulations of the sphere [Annotated scanned copy]
R. Bowen and S. Fisk, Generation of triangulations of the sphere, Math. Comp., 21 (1967), 250-252.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs
Aharon Davidson, From Planck Area to Graph Theory: Topologically Distinct Black Hole Microstates, arXiv:1907.03090 [gr-qc], 2019.
M. Deza, M. Dutour and P. W. Fowler, Zigzags, railroads and knots in fullerenes, J. Chem. Inf. Comput. Sci., 44 (2004), 1282-1293.
C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
Firsching, Moritz Realizability and inscribability for simplicial polytopes via nonlinear optimization. Math. Program. 166, No. 1-2 (A), 273-295 (2017). Table 1
Komei Fukuda, Hiroyuki Miyata, and Sonoko Moriyama, Complete Enumeration of Small Realizable Oriented Matroids. Discrete Comput. Geom. 49 (2013), no. 2, 359--381. MR3017917. Also arXiv:1204.0645 [math.CO], 2012. - From _N. J. A. Sloane_, Feb 16 2013
Jan Goedgebeur and Patric R. J. Ostergard, Switching 3-Edge-Colorings of Cubic Graphs, arXiv:2105:01363 [math.CO], May 2021. See Table 4.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Lee Zheng Han and Chia Vui Leong, The Walk of Maximal Planar Graphs, 2018.
Jorik Jooken, Computer-assisted graph theory: a survey, arXiv:2508.20825 [math.CO], 2025. See p. 5.
Paul Jungeblut, Edge Guarding Plane Graphs, Master Thesis, Karlsruhe Institute of Technology (Germany, 2019).
J. Lederberg, Dendral-64, II, Report to NASA, Dec 1965 [Annotated scanned copy]
J. Lederberg, Hamilton circuits of convex trivalent polyhedra (up to 18 vertices), Am. Math. Monthly, 74 (1967), 522-527.
J. Lederberg, Hamilton circuits of convex trivalent polyhedra (up to 18 vertices), Am. Math. Monthly, 74 (1967), 522-527. (Annotated scanned copy)
F. H. Lutz, Triangulated manifolds with few vertices: Combinatorial Manifolds, arXiv:math/0506372 [math.CO], 2005.
G. P. Michon, Counting Polyhedra
Manfred Scheucher, Hendrik Schrezenmaier, and Raphael Steiner, A Note On Universal Point Sets for Planar Graphs, arXiv:1811.06482 [math.CO], 2018.
I. Sciriha and P. W. Fowler, Nonbonding Orbitals in Fullerenes: Nuts and Cores in Singular Polyhedral Graphs, J. Chem. Inf. Model., 47, 5, 1763 - 1775, 2007.
A. Stoimenow, A theorem on graph embedding with a relation to hyperbolic volume, Combinatorica, October 2016, Volume 36, Issue 5, pp 557-589.
Thom Sulanke, Generating triangulations of surfaces (surftri), (also subpages).
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.
William T. Tutte, On the enumeration of convex polyhedra, J. Combin. Theory Ser. B 28 (1980), 105-126.
Eric Weisstein's World of Mathematics, Cubic Polyhedral Graph
Eric Weisstein's World of Mathematics, Simple Polyhedron
Eric Weisstein's World of Mathematics, Triangulated Graph
Index entries for "core" sequences

Cf. A005964, A007816, A058378, A253882.

Cf. A000944, A007021, A111358.

From William P. Orrick, Apr 07 2021: (Start)
a(n) >= A007816(n-3)/n! = binomial(n,2)*(4*n-11)!/(n!*(3*n-6)!) for all n >= 4.
a(n) ~ A007816(n-3)/n! = binomial(n,2)*(4*n-11)!/(n!*(3*n-6)!) ~ (1/64)*sqrt(1/(6*Pi))*n^(-7/2)*(256/27)^(n-2), using the theorem that the automorphism group of a maximal planar graph is almost certainly trivial as n gets large. (Tutte)
(End)

Extended by Brendan McKay and Gunnar Brinkmann using their program "plantri", Dec 19 2000

Definition clarified by Manfred Scheucher, Mar 17 2023

A000994 Shifts 2 places left under binomial transform.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 13, 36, 109, 359, 1266, 4731, 18657, 77464, 337681, 1540381, 7330418, 36301105, 186688845, 995293580, 5491595645, 31310124067, 184199228226, 1116717966103, 6968515690273, 44710457783760, 294655920067105, 1992750830574681, 13817968813639426

Offset: 0

N. J. A. Sloane

a(n) is the number of permutations of [n-1] that avoid both of the dashed patterns 1-23 and 3-12 and start with a descent (or are a singleton). For example, a(5)=5 counts 2143, 3142, 3214, 3241, 4321. - David Callan, Nov 21 2011

			A(x) = 1 + x^2/(1-x) + x^4/((1-x)^2*(1-2x)) + x^6/((1-x)^2*(1-2x)^2*(1-3x)) +...

Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
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).

Alois P. Heinz, Table of n, a(n) for n = 0..650 (first 101 terms from T. D. Noe)
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
N. J. A. Sloane, Transforms
S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Cf. A000995, A051139, A051140. Cf. A007318.

Column k=2 of A143983. Cf. A007476, A088022, A086880.

a000994 n = a000994_list !! n
a000994_list = 1 : 0 : us where
  us = 1 : 1 : f 2 where
    f x = (1 + sum (zipWith (*) (map (a007318' x) [2..x]) us)) : f (x + 1)
-- Reinhard Zumkeller, Jun 02 2013

A000994 := proc(n) local k; option remember; if n <= 1 then 1 else 1 + add(binomial(n, k)*A000994(k - 2), k = 2 .. n); fi; end;

a[n_] := a[n] = 1 + Sum[Binomial[n, k]*a[k-2], {k, 2, n}]; Join[{1, 0}, Table[a[n], {n, 0, 24}]] (* Jean-François Alcover, Oct 11 2011, after Maple *)

a(n)=polcoeff(sum(k=0, n, x^(2*k)*(1-k*x)/prod(j=0, k, 1-j*x+x*O(x^n))^2), n) \\ Paul D. Hanna, Nov 02 2006

Since this satisfies a recurrence similar to that of the Bell numbers (A000110), the asymptotic behavior is presumably just as complicated - see A000110 for details.
However, a(n)/A000995(n) (e.g., 77464/63117) -> 1.228..., the constant in A051148 and A051149.
O.g.f.: A(x) = Sum_{n>=0} x^(2*n)*(1-n*x)/Product_{k=0..n} (1-k*x)^2. - Paul D. Hanna, Nov 02 2006
Let S(n) = Sum_{k >= 1} k^n/k!^2. Then S(n) = a(n)*S(0) + A000995(n)*S(1) is stated in A086880, where S(0) = 2.279585302... (see A070910) and S(1) = 1.590636854... (see A096789). Cf. A088022. - Peter Bala, Jan 27 2015
G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Aug 09 2020

A000679 Number of groups of order 2^n.

Original entry on oeis.org

1, 1, 2, 5, 14, 51, 267, 2328, 56092, 10494213, 49487367289

Offset: 0

N. J. A. Sloane

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 51*x^5 + 267*x^6 + 2328*x^7 + ...

James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
M. F. Newman, Groups of prime-power order (1990). In Groups—Canberra 1989 (pp. 49-62). Springer, Berlin, Heidelberg. See Table 1.
M. F. Newman and E. A. O'Brien, A CAYLEY library for the groups of order dividing 128, Group theory (Singapore, 1987), 437-442, de Gruyter, Berlin-New York, 1989.
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. Applebaum, J. Clikeman, J. A. Davis, J. F. Dillon, J. Jedwab, T. Rabbani, K. Smith, and W. Yolland, Constructions of difference sets in nonabelian 2-groups, Alg. Num. Theor. (2023) Vol. 17, No. 2. 359-396. See p. 396.
Hans Ulrich Besche and Bettina Eick, Construction of finite groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
Hans Ulrich Besche and Bettina Eick, The groups of order at most 1000 except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
Hans Ulrich Besche, Bettina Eick and E. A. O'Brien, The groups of order at most 2000, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.
David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3.
Hans Ulrich Besche, The Small Groups library
Bettina Eick and E. A. O'Brien, Enumerating p-groups. Group theory. J. Austral. Math. Soc. Ser. A 67 (1999), no. 2, 191-205.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Rodney James and John Cannon, Computation of isomorphism classes of p-groups, Mathematics of Computation 23.105 (1969): 135-140.
Rodney James, M. F. Newman, and E. A. O'Brien, The Groups of Order 128, J. Algebra 129, 136-158, 1990.
G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
E. A. O'Brien, The Groups of Order 256 J. Algebra 143, 219-235, 1991.
Eugene Rodemich, The groups of order 128, J. Algebra 67 (1980), no. 1, 129-142.
Eric Weisstein's World of Mathematics, Finite Group.
Marcel Wild, The groups of order 16 made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
Index entries for sequences related to groups.

Cf. A000001, A046058.

A000679 := List([0..8],n -> NumberSmallGroups(2^n)); # Muniru A Asiru, Oct 15 2017

seq(GroupTheory:--NumGroups(2^n),n=0..10); # Robert Israel, Oct 15 2017

Join[{1}, FiniteGroupCount[2^Range[10]]] (* Vincenzo Librandi, Mar 28 2018 *)

a(n) = 2^((2/27)n^3 + O(n^(8/3))).

a(n) = A000001(2^n). - Amiram Eldar, Mar 10 2024

a(9) and a(10) found by Eamonn O'Brien

a(10) corrected by David Burrell, Jun 06 2022

A001680 The partition function G(n,3).

Original entry on oeis.org

1, 1, 2, 5, 14, 46, 166, 652, 2780, 12644, 61136, 312676, 1680592, 9467680, 55704104, 341185496, 2170853456, 14314313872, 97620050080, 687418278544, 4989946902176, 37286121988256, 286432845428192, 2259405263572480, 18280749571449664, 151561941235370176

Offset: 0

N. J. A. Sloane

nonn

Number of '12-3 and 21-3'-avoiding permutations.
Set partitions into sets of size at most 3. The e.g.f. for partitions into sets of size at most s is exp( sum(j=1..s, x^j/j!) ). [Joerg Arndt, Dec 07 2012]
Also called restricted Stirling numbers of the second kind (see Mezo). - N. J. A. Sloane, Nov 27 2013

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).

Seiichi Manyama, Table of n, a(n) for n = 0..653 (terms 0..250 from Alois P. Heinz)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Filippo Disanto and Thomas Wiehe, Some instances of a sub-permutation problem on pattern avoiding permutations, arXiv preprint arXiv:1210.6908 [math.CO], 2012.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 18
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1.
F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.

Cf. A001681, A189886.

Column k=3 of A229223.

G:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0,
       add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
a:= n-> G(n, 3):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 20 2012
# Recurrence:
rec := {(-n^2-3*n-2)*f(n)+(-2*n-4)*f(n+1)-2*f(n+2)+2*f(n+3)=0,f(0)=1,f(1)=1,f(2)=2}:
aList := gfun:-rectoproc(rec,f(n),list): aList(25); # Peter Luschny, Feb 26 2018

Table[Sum[n!/(m!2^(n+j-2m)3^(m-j))Binomial[m,j]Binomial[j,n+2j-3m],{m,0,n},{j,0,3m-n}],{n,0,15}]

E.g.f.: exp ( x + x^2 / 2 + x^3 / 6 ).
a(n) = n! * sum(k=1..n, 1/k! * sum(j=0..k, C(k,j) * C(j,n-3*k+2*j) * 2^(-n+2*k-j) * 3^(j-k))). [Vladimir Kruchinin, Jan 25 2011]
a(n) = G(n,3) with G(0,i) = 1, G(n,i) = 0 for n>0 and i<1, otherwise G(n,i) = Sum_{j=0..floor(n/i)} G(n-i*j,i-1) * n!/(i!^j*(n-i*j)!*j!). - Alois P. Heinz, Apr 20 2012
D-finite with recurrence 2*a(n) -2*a(n-1) +2*(-n+1)*a(n-2) -(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jan 25 2013
Proof of foregoing recurrence: The partition containing n can be a singleton (a(n-1) partitions of the remaining terms), a doubleton ((n-1) choices for its companion times a(n-2) partitions of the remaining terms) or a tripleton ((n-1) choose 2 choices for its companions times a(n-3) partitions for the remaining terms), so a(n) = a(n-1) + (n-1)a(n-2) + (n-1)*(n-2)/2 * a(n-3). - Micah E. Fogel, Feb 14 2013
a(n) ~ n^(2*n/3)*exp(1/2*(2*n)^(2/3)+2/3*(2*n)^(1/3)-2*n/3-4/9)/(sqrt(3)*2^(n/3)). - Vaclav Kotesovec, May 29 2013

More terms added May 13 2009

A001149 A self-generating sequence: a(1)=1, a(2)=2, a(n+1) chosen so that a(n+1)-a(n-1) is the first number not obtainable as a(j)-a(i) for 1<=i

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 17, 26, 34, 45, 54, 67, 81, 97, 115, 132, 153, 171, 198, 228, 256, 288, 323, 357, 400, 439, 488, 530, 581, 627, 681, 732, 790, 843, 908, 963, 1029, 1085, 1152, 1213, 1284, 1346, 1418, 1484, 1561, 1630, 1710, 1785, 1867, 1945, 2034, 2116

Offset: 1

N. J. A. Sloane

nonn

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).

Manfred Scheucher, Table of n, a(n) for n = 1..2000
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
Manfred Scheucher, Python Script

Cf. A005282, A054540.

Description corrected and moved to name line by Franklin T. Adams-Watters, Nov 01 2009

More terms from Manfred Scheucher, Jul 01 2015

A001402 Number of partitions of n into at most 6 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760, 8442, 9192

Offset: 0

N. J. A. Sloane

Also number of partitions of n into parts <= 6: a(n) = A026820(n,6). - Reinhard Zumkeller, Jan 21 2010
Counts unordered closed walks of weight n on a single vertex graph containing 6 loops of weights 1, 2, 3, 4, 5 and 6. - David Neil McGrath, Apr 11 2015
Number of different distributions of n+21 identical balls in 6 boxes as x,y,z,p,q,m where 0Ece Uslu and Esin Becenen, Jan 11 2016
a(n) could be the total number of non-isomorphic geodetic graphs of diameter n>=2 homeomorphic to the Petersen graph. - Carlos Enrique Frasser, May 24 2018

			The number of partitions of 6 into parts less than or equal to 6 is a(6)=11. These are (6)(51)(42)(33)(411)(321)(222)(3111)(2211)(21111)(111111). - _David Neil McGrath_, Apr 11 2015
a(4) = 5, i.e., {1,2,3,4,5,10},{1,2,3,4,6,9},{1,2,3,4,7,8},{1,2,3,5,6,8},{1,2,4,5,6,7} Number of different distributions of 25 identical balls in 6 boxes as x,y,z,p,q,m where 0 < x < y < z < p < q < m. - _Ece Uslu_, Esin Becenen, Jan 11 2016

A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
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).

Washington Bomfim, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
G. E. Andrews, Partitions: At the Interface of q-Series and Modular Forms, The Ramanujan Journal 7, 385-400 (2003), Eq.(3.10).
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, American Journal of Mathematics, Vol. 2, No. 1 (Mar., 1879), pp. 71-84 (14 pages).
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419. [Annotated scanned copy]
C. E. Frasser and G. N. Vostrov, Geodetic Graphs Homeomorphic to a Given Geodetic Graph, arXiv:1611.01873 [cs.DM], 2016. [p. 27]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 355
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz), arXiv:1108.4391 [math.CO], 2011.
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).

Essentially same as A026812. Cf. A037145 (first differences), A288341 (partial sums).
a(n) = A008284(n+6, 6), n >= 0.
A194197(n) = a(60*n). - Alois P. Heinz, Aug 23 2011

with(combstruct):ZL7:=[S,{S=Set(Cycle(Z,card<7))}, unlabeled]: seq(count(ZL7,size=n),n=0..50);  # Zerinvary Lajos, Sep 24 2007
a:= n-> (Matrix(21, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 1, 0, 0, -1, 0, -2, 0, 1, 1, 1, 1, 0, -2, 0, -1, 0, 0, 1, 1, -1][i] else 0 fi)^n)[1,1]; seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2008
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=6)},unlabelled]: seq(combstruct[count](B, size=n), n=0..50); # Zerinvary Lajos, Mar 21 2009
## more efficient for large arguments (try with 10^100 or 100^1000):
a:= proc(n) local m, r; m := iquo (n, 60, 'r');
(167 +(2325 +(15400 +(47250 +54000*m +4500*r)*m +3150*r +150*r^2)*m
+[0, 795, 1875, 3030, 4500, 6075, 7995, 10050, 12480, 15075, 18075, 21270, 24900, 28755, 33075, 37650, 42720, 48075, 53955, 60150, 66900, 73995, 81675, 89730, 98400, 107475, 117195, 127350, 138180, 149475, 161475, 173970, 187200, 200955, 215475, 230550, 246420, 262875, 280155, 298050, 316800, 336195, 356475, 377430, 399300, 421875, 445395, 469650, 494880, 520875, 547875, 575670, 604500, 634155, 664875, 696450, 729120, 762675, 797355, 832950][r+1])*m
+[0, 63, 207, 348, 570, 795, 1143, 1482, 1968, 2475, 3135, 3828, 4722, 5643, 6795, 8010, 9468, 11007, 12843, 14760, 17010, 19383, 22107, 24978, 28260, 31695, 35583, 39672, 44238, 49035, 54375, 59958, 66132, 72603, 79695, 87120, 95238, 103707, 112923, 122550, 132960, 143823, 155547, 167748, 180870, 194535, 209163, 224382, 240648, 257535, 275535, 294228, 314082, 334683, 356535, 379170, 403128, 427947, 454143, 481260][r+1])*m/6
+[1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760, 8442, 9192, 9975, 10829, 11720, 12692, 13702, 14800, 15944, 17180, 18467][r+1] end:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 22 2011
A := [1,1,2,3,5,7,11,14,20,26,35,44,58,71,90,110,136,163,199,235,282];
a := proc(n) option remember; if n < 21 then A[n+1] else 1+(a(n-2)+a(n-3)+a(n-4))-(2*a(n-7)+2*a(n-8)+a(n-9))+(a(n-11)+2*a(n-12)+2*a(n-13))-(a(n-16)+a(n-17)+a(n-18))+(a(n-20)) fi end:
seq(a(i),i=0..50); # Peter Luschny, Aug 23 2011
## program using quasi-polynomials; see article by Sills and Zeilberger:
a:= m-> subs (n=m, add ([[n^5/86400 +7*n^4/11520 +77*n^3/6480 +245*n^2/2304 +43981*n/103680 +199577/345600], [-n^2/768 -7*n/256 -581/4608, n^2/768 +7*n/256 +581/4608], [-n/162 -19/324, -n/162 -23/324, n/81 +7/54], [1/32, -1/32, -1/32, 1/32], [1/25, 0, -1/25, -2/25, 2/25], [1/36, -1/36, -1/18, -1/36, 1/36, 1/18]][r][1 +irem (m-1+r, r)], r=1..6)):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 24 2011
## using Andrews-style expressions; see article by Sills and Zeilberger:
a:= n-> 1 +31*n^2/288 +floor(n/4)/16 -floor(n/4 +1/2)/16 +7*n^4/11520 +floor(n/5)/5 +n^5/86400 -(n^2/384 +7*n/128 +581/2304)*n +(n^2/192 +7*n/64 +581/1152) *floor(n/2) -(n/54 +61/324)*n +(n/54 +19/108) *floor((n+1)/3) +(n/27 +7/18) *floor(n/3) +floor(n/6)/18 -floor(n/6 +2/3)/36 +floor(n/6 +1/3)/18 +floor((n+1)/6)/12 +713*n/1800 +77*n^3/6480:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 24 2011

CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)*(1 - x^5)*(1 - x^6)), {x, 0, 60} ], x ]
(* Second program: *)
T[n_, k_] := T[n, k] = If[n==0 || k==1, 1, T[n, k-1] + If[k>n, 0, T[n-k, k]]]; a[n_] := T[n, 6]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 12 2017, after Alois P. Heinz's code for A026820 *)
Table[Length[IntegerPartitions[n,6]],{n,0,50}] (* Harvey P. Dale, Jul 30 2025 *)

a(n)=floor((6*n^5+315*n^4+6160*n^3+55125*n^2+(216705+9600*(n%3<1))*n+527500)/518400+(n+1)*(n+20)*(-1)^n/768) \\ Tani Akinari, May 27 2014

a(n)={round((n+11)*((6*n^4+249*n^3+2071*n^2-4931*n+40621)/518400+n\2*(n+10) /192+( (n+1)\3+ n\3*2 )/54))};
vector(60,n,n--; a(n)) \\ Washington Bomfim, Jan 16 2021

a(n) = 1 + (a(n-2) + a(n-3) + a(n-4)) - (2*a(n-7) + 2*a(n-8) + a(n-9)) + (a(n-11) + 2*a(n-12) + 2*a(n-13)) - (a(n-16) + a(n-17) + a(n-18)) + (a(n-20)). - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000
G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)). - Alois P. Heinz, Aug 22 2011
a(n) ~ n^5 / 86400. - Charles R Greathouse IV, Aug 23 2011
a(n) = (167 + (2325 + (15400 + (47250 + 54000*m + 4500*r)*m + 3150*r + 150*r^2)*m + X(r))*m + Y(r))*m/6 + Z(r) where m = floor(n/60), r = n mod 60 and X, Y, Z are functions of r (see Maple program below). - Alois P. Heinz, Aug 23 2011
a(n) = floor((2 + 3*(floor(n/3) + floor(-n/3))) * (floor(n/3)+1)/54 + (6*n^5 + 315*n^4 + 6160*n^3 + 55125*n^2 + 219905*n + 485700)/518400 + (n+1)*(n+20)*(-1)^n/768). - Tani Akinari, Aug 05 2013
a(n) = a(n-1) + a(n-2) - a(n-5) - 2*a(n-7) + a(n-9) + a(n-10) + a(n-11) + a(n-12) - 2*a(n-14) - a(n-16) + a(n-19) + a(n-20) - a(n-21). - David Neil McGrath, Apr 11 2015
a(n+6) = a(n) + A001401(n). - Ece Uslu, Esin Becenen, Jan 11 2016
a(n) = round((n+11)*((6*n^4 + 249*n^3 + 2071*n^2 - 4931*n + 40621)/518400 + floor(n/2)*(n+10)/192 + (floor((n+1)/3) + 2*floor(n/3))/54)). - Washington Bomfim, Jan 15 2021

A000107 Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates.

Original entry on oeis.org

0, 1, 2, 5, 13, 35, 95, 262, 727, 2033, 5714, 16136, 45733, 130046, 370803, 1059838, 3035591, 8710736, 25036934, 72069134, 207727501, 599461094, 1731818878, 5008149658, 14496034714, 41993925955, 121747732406, 353221737526, 1025471857282, 2978995353959, 8658997820084

Offset: 0

N. J. A. Sloane

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 61, 62 (2.1.8-2.1.10).
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
Bernhard Gittenberger, Emma Yu Jin, Michael Wallner, On the shape of random Pólya structures, arXiv|1707.02144 [math.CO], 2017-2018; Discrete Math., 341 (2018), 896-911.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. Harary, R. W. Robinson, Isomorphic factorizations VIII: bisectable trees, Combinatorica 4 (2) (1984) 169-179, eq. (4.12).
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 123
R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Table 6.
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Row sums of A339067.
INVERT transform of A000081.
Column k=1 of A008295.

with(numtheory): b:= proc(n) option remember; `if`(n<2, n, add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1) /(n-1)) end: a:= proc(n) option remember; b(n) +add(a(n-i) *b(i), i=1..n-1) end: seq(a(n), n=0..26); # Alois P. Heinz, Jun 02 2009

b[0] = 0; b[1] = 1; b[n_] := b[n] = Sum[ Sum[ d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1); a[n_] := a[n] = b[n] + Sum[ a[n-i]*b[i], {i, 1, n-1}]; Table[ a[n], {n, 0, 26}](* Jean-François Alcover, Mar 07 2012, after Alois P. Heinz *)

G.f.: A000081(x)/(1-A000081(x)), where A000081(x) is the g.f. of A000081 [Harary-Robinson]. - R. J. Mathar, Sep 16 2015

a(n) ~ A340310 * A051491^n / sqrt(n). - Vaclav Kotesovec, Jan 04 2021

Better description from Christian G. Bower, Apr 15 1998

A001524 Number of stacks, or arrangements of n pennies in contiguous rows, each touching 2 in row below.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 12, 18, 26, 38, 53, 75, 103, 142, 192, 260, 346, 461, 607, 797, 1038, 1348, 1738, 2234, 2856, 3638, 4614, 5832, 7342, 9214, 11525, 14369, 17863, 22142, 27371, 33744, 41498, 50903, 62299, 76066, 92676, 112666, 136696, 165507, 200018

Offset: 0

N. J. A. Sloane

Also n-stacks with strictly receding left wall.
Weakly unimodal compositions such that each up-step is by at most 1 (and first part 1). By dropping the requirement for weak unimodality one obtains A005169. - Joerg Arndt, Dec 09 2012
The values of a(19) and a(20) in Auluck's table on page 686 are wrong (they have been corrected here). - David W. Wilson, Mar 07 2015
Also the number of overpartitions of n having more overlined parts than non-overlined parts.   For example, a(5) = 5 counts the overpartitions [5'], [4',1'], [3',2'], [3',1',1] and [2',2,1']. - Jeremy Lovejoy, Jan 15 2021

			For a(6)=8 we have the following stacks:
..x
.xx .xx. ..xx .x... ..x.. ...x. ....x
xxx xxxx xxxx xxxxx xxxxx xxxxx xxxxx xxxxxx
From _Franklin T. Adams-Watters_, Jan 18 2007: (Start)
For a(7) = 12 we have the following stacks:
..x. ...x
.xx. ..xx .xxx .xx.. ..xx. ...xx
xxxx xxxx xxxx xxxxx xxxxx xxxxx
and
.x.... ..x... ...x.. ....x. .....x
xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxxx
(End)

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).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)
J. S. Birman, Letter to N. J. A. Sloane, Apr 09 1994
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Erich Friedman, Illustration of initial terms
H. W. Gould, R. K. Guy, and N. J. A. Sloane, Correspondence, 1987.
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.
R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

Cf. A001522, A001523, A171604, A007293, A015128.

Row sums of triangle A259095.

s := 1+sum(z^(n*(n+1)/2)/((1-z^(n))*product((1-z^i), i=1..n-1)^2), n=1..50): s2 := series(s, z, 300): for j from 1 to 100 do printf(`%d,`,coeff(s2, z, j)) od: # James Sellers, Feb 27 2001
# second Maple program:
b:= proc(n, i) option remember; `if`(i>n, 0, `if`(
      irem(n, i)=0, 1, 0)+add(j*b(n-i*j, i+1), j=1..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 03 2018

m = 45; CoefficientList[ Series[Sum[ z^(n*(n+1)/2)/((1-z^(n))*Product[(1-z^i), {i, 1, n-1}]^2), {n, 1, m}], {z, 0, m}], z] // Prepend[Rest[#], 1] &
(* Jean-François Alcover, May 19 2011, after Maple prog. *)

{a(n) = if( n<0, 0, polcoeff( sum( k=0,(sqrt(8*n + 1) - 1) / 2, x^((k^2 + k) / 2) / prod( i=1, k, (1 - x^i + x * O(x^n))^((iMichael Somos, Apr 27 2003 */

G.f.: sum(n>=1, q^(n*(n+1)/2) / prod(k=1..n-1, 1-q^k)^2 / (1-q^n) ). [Joerg Arndt, Jun 28 2013]
a(n) = sum_{m>0,k>0,2*k^2+k+2*m<=n-1} A008289(m,k)*A000041(n-k*(1+2k)-2*m-1). - [Auluck eq 29]
From Vaclav Kotesovec, Mar 03 2020: (Start)
Pi * sqrt(2/3) <= n^(-1/2)*log(a(n)) <= Pi * sqrt(5/6). [Auluck, 1951]
log(a(n)) ~ 2*Pi*sqrt(n/5). [Wright, 1971]
a(n) ~ exp(2*Pi*sqrt(n/5)) / (sqrt(2) * 5^(3/4) * (1 + sqrt(5)) * n). (End)
a(n) = A143184(n) - A340659(n). - Vaclav Kotesovec, Jun 06 2021

Corrected by R. K. Guy, Apr 08 1988

More terms from James Sellers, Feb 27 2001

A000719 Number of disconnected graphs with n nodes.

Original entry on oeis.org

0, 0, 1, 2, 5, 13, 44, 191, 1229, 13588, 288597, 12297299, 1031342116, 166123498733, 50668194387427, 29104827043066808, 31455591302381718651, 64032471448906164191208, 245999896712611657677614268, 1787823725136869060356731751124

Offset: 0

N. J. A. Sloane

a(n) is also the number of simple unlabeled graphs on n+1 nodes with diameter 2 and connectivity 1. - Geoffrey Critzer, Oct 23 2016

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..87 (first 31 terms from David Wasserman)
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Amer. Math. Soc. 78 (1955), 445-463.
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967
Eric Weisstein's World of Mathematics, Disconnected Graph.
Eric Weisstein's World of Mathematics, k-Connected Graph.
Eric Weisstein's World of Mathematics, k-Edge-Connected Graph.

Equals (A000088) minus (A001349).

<< "Combinatorica`"; max = 18; A000088 = Table[ NumberOfGraphs[n], {n, 0, max}]; f[x_] = 1 - Product[ 1/(1 - x^k)^b[k], {k, 1, max}]; b[0] = b[1] = b[2] = 1; coes = CoefficientList[ Series[ f[x], {x, 0, max}], x]; sol = First[ Solve[ Thread[ Rest[ coes + A000088 ] == 0]]]; a[n_] := a[n] = A000088[[n+1]] - b[n] /. sol; a[0] = a[1] = 0; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Nov 24 2011 *)

from functools import lru_cache
from itertools import combinations
from fractions import Fraction
from math import prod, gcd, factorial
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A000719(n):
    if n == 0: return 0
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<>1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n)))
    @lru_cache(maxsize=None)
    def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
    return b(n)-sum(mobius(n//d)*c(d) for d in divisors(n,generator=True))//n # Chai Wah Wu, Jul 03 2024

More terms from Christian G. Bower

Further terms from Vladeta Jovovic, Apr 14 2000

cp's OEIS Frontend

A000103 Number of n-node triangulations of sphere in which every node has degree >= 4.

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A000109 Number of simplicial polyhedra with n vertices; simple planar graphs with n vertices and 3n-6 edges; maximal simple planar graphs with n vertices; planar triangulations with n vertices; triangulations of the sphere with n vertices; 3-connected cubic planar graphs on 2n-4 vertices.

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A000994 Shifts 2 places left under binomial transform.

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A000679 Number of groups of order 2^n.

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GAP

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A001680 The partition function G(n,3).

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A001149 A self-generating sequence: a(1)=1, a(2)=2, a(n+1) chosen so that a(n+1)-a(n-1) is the first number not obtainable as a(j)-a(i) for 1<=i

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A001402 Number of partitions of n into at most 6 parts.

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