A036918 -id:A036918 - cp's OEIS frontend

A001339 a(n) = Sum_{k=0..n} (k+1)! binomial(n,k).

1, 3, 11, 49, 261, 1631, 11743, 95901, 876809, 8877691, 98641011, 1193556233, 15624736141, 220048367319, 3317652307271, 53319412081141, 909984632851473, 16436597430879731, 313262209859119579, 6282647653285676001, 132266266384961600021, 2916471173788403280463

Offset: 0

N. J. A. Sloane

nonn

Number of arrangements of {1, 2, ..., n, n + 1} containing the element 1. - Emeric Deutsch, Oct 11 2001
From Thomas Wieder, Oct 21 2004: (Start)
"Also the number of hierarchies with unlabeled elements and labeled levels where the levels are permuted.
"Let l_x denote level x, e.g. l_2 is level 2. Let * denote an element. Then l_1*l_2***l_3** denotes a hierarchy of n = 6 unlabeled elements with one element on level 1, three elements on level 2 and 2 elements on level 3.
"E.g. for n=3 one has a(3) = 11 possible hierarchies: l_1***, l_1**l_2*, l_1*l_2**, l_2**l_1*, l_2*l_1**, l_1*l_2*l_3*, l_3*l_1*l_2*, l_2*l_3*l_1*, l_1*l_3*l_2*, l_2*l_1*l_3*, l_3*l_2*l_1*. See A064618 for the number of hierarchies with labeled elements and labeled levels." (End)
Polynomials in A010027 evaluated at 2.
Also the permanent of any n X n cofactor of an n+1 X n+1 version of J+I other than an n X n version of J + I (that is, a (1, 2) matrix with n - 1 2s, at most one per row and column). - D. G. Rogers, Aug 27 2006
a(n) = number of partitions of [n+1] into lists of sets that are both non-nesting and non-crossing. Non-nesting means that no set is contained in the span (interval from min to max) of another. For example, a(1) counts 12, 1-2, 2-1 and a(2) counts 123, 1-23, 23-1, 3-12, 12-3, 1-2-3, 1-3-2, 2-1-3, 2-3-1, 3-1-2, 3-2-1. - David Callan, Sep 20 2007
Row sums of triangle A137594. - Gary W. Adamson, Jan 28 2008
From Peter Bala, Jul 10 2008: (Start)
a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). See A000522 for further properties of difference divisibility sequences.
a(n) equals the sum of the lengths of the paths between a pair of distinct vertices of the complete graph K_(n + 2) on n + 2 vertices [Hassani]. For example, for the complete graph K_4 with vertex set {A,B,C,D} the 5 paths between A and B are AB of length 1, ACB and ADB, both of length 2 and ACDB and ADCB, both of length 3. The sum of the lengths is 1 + 2 + 2 + 3 + 3 = 11 = a(2).
The number of paths between 2 distinct vertices of K_n is equal to A000522(n - 2); the number of simple cycles through a vertex of K_n equals A038154(n - 1).
Recurrence relation: a(0) = 1, a(1) = 3, a(n) = (n+2)*a(n - 1) - (n - 1)*a(n - 2) for n >= 2. The sequence b(n) := n*n! = A001563(n) satisfies the same recurrence with the initial conditions b(0) = 0, b(1) = 1. This leads to the finite continued fraction expansion a(n)/b(n) = 3 - 1/(4 - 2/(5 - 3/(6 - ... - (n - 1)/(n + 2)))), n >= 1.
Limit_{n->oo} a(n)/b(n) = e = 3 - 1/(4 - 2/(5 - 3/(6 - ... - n/((n + 3) - ...)))).
For n >= 1, a(n) = b(n)*(3 - Sum_{k=2..n} 1/(k!*(k - 1)*k)) (see the formula by Deutsch) since the rhs satisfies the above recurrence with the same initial conditions. Hence e = 3 - Sum_{k>=2} 1/(k!*(k - 1)*k).
For sequences satisfying the more general recurrence a(n) = (n + 1 + r)*a(n-1) - (n-1)*a(n-2), which yield series acceleration formulas for e/r! that involve the Poisson-Charlier polynomials c_r(-n; -1), refer to A000522 (r=0), A082030 (r=2), A095000 (r=3) and A095177 (r=4). (End)
Binomial transform of n! Offset 1. a(3) = 11. - Al Hakanson (hawkuu(AT)gmail.com), May 18 2009
Equals eigensequence of a triangle with (1, 2, 3, ...) as the right border and the rest 1's; equivalent to a(n) = [n terms of the sequence (1, 1, 1, ...) followed by (n + 1)] dot [(n + 1) terms of the sequence (1, 1, 3, 11, 245, ...)]. Example: 261 = a(4) = (1, 1, 1, 1, 5) dot (1, 1, 3, 11, 49) = 1 + 1 + 3 + 11 + 245 = 261. - Gary W. Adamson, Jul 24 2010
Number of nonnegative integers which use each digit at most once in base n+1. - Franklin T. Adams-Watters, Oct 02 2011
a(n) is the number of permutations of {1,2,...,n+2} in which there is an increasing contiguous subsequence (ascending run) beginning with 1 and ending with n+2. The number of such permutations with exactly k, 0<=k<=n, elements between the 1 and (n+2) is given by A132159(n,k) whose row sums equal this sequence. See example. - Geoffrey Critzer, Feb 15 2013

			G.f. = 1 + 3*x + 11*x^2 + 49*x^3 + 261*x^4 + 1631*x^5 + 11743*x^6 + 95901*x^7 + ...
a(2) = 11: {1, 12, 21, 13, 31, 123, 132, 213, 231, 312, 321}.
a(2) = 11 because we have 11 permutations of {1,2,3,4} (written in one line notation) that have an increasing subsequence beginning with 1 and ending with 4: 1,2,3,4; 1,2,4,3; 1,3,4,2; 1,4,2,3; 1,4,3,2; 2,1,3,4; 2,1,4,3; 2,3,1,4; 3,1,2,4; 3,1,4,2; 3,2,1,4. - _Geoffrey Critzer_, Feb 15 2013

A. Hordijk, Markov Decision Chains, pp. 97-103 in Images of SMC Research, 1996, Stichting Mathematisch Centrum, Amsterdam, Netherlands, 1996. See p. 103.
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).
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 56, ex. 232.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas, Freeman, San Francisco, 1974. [Annotated scans of pages 69 and 70 only]
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7.
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
E. Biondi, L. Divieti and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35.
Philip Feinsilver and John McSorley, Zeons, Permanents, the Johnson Scheme, and Generalized Derangements, International Journal of Combinatorics, Volume 2011, Article ID 539030, 29 pages.
Loïc Foissy and Frédéric Patras, Natural endomorphisms of shuffle algebras, arXiv preprint arXiv:1205.2986 [math.RA], 2012.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
M. Hassani, Counting and computing by e, arXiv:math/0606613 [math.CO], 2006.
F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 400
M. J. Knight and W. O. Egerland, Solution to Problem 5911, Amer. Math. Monthly 81 (1974) 675-676.
Zhentao Lu, Elementary proofs of generalized continued fraction formulae for e, arXiv:1907.05563 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Poisson-Charlier polynomial
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.

Cf. A001563, A064618, A089900, A137594.
a(n) = A000522(n+1) - A000522(n).
First differences of A000522, A007526, A026243, A073591.
Equals (1/2)*A006183(n-2).
Equals A036918(n+1) + 1.
Leftmost column of A276588.
Cf. also A136104.

A001339:=List([0..20],n-> Sum([0..n], k-> Factorial(k+1)*Binomial(n,k))); # Muniru A Asiru, Feb 17 2018

[Factorial(n)*(&+[(n-k+1)/Factorial(k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jul 15 2019

a:=proc(n) options operator, arrow: factorial(n)*n*(3-(sum(1/(j*(j-1)*factorial(j)), j=2..n))) end proc: 1, seq(a(n),n=1..20); # Emeric Deutsch, Apr 12 2008
a := n -> hypergeom([2, -n], [], -1); seq(simplify(a(n)), n=0..18); # Peter Luschny, Sep 20 2014

a[n_] := n!*Sum[(k+1)/(n-k)!, {k, 0, n}]; a /@ Range[0, 20] (* Jean-François Alcover, Jul 13 2011 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[x] / (1 - x)^2, {x, 0, n}]] (* Michael Somos, Oct 20 2011 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp(x + x * O(x^n)) / (1 - x)^2, n))} /* Michael Somos, Mar 04 2004 */

vector(20, n, n--; n!*sum(k=0,n,(n-k+1)/k!)) \\ G. C. Greubel, Jul 15 2019

[factorial(n)*sum((n-k+1)/factorial(k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jul 15 2019

E.g.f.: exp(x)/(1-x)^2.
a(n) = round(evalf(exp(1)*(n-1)*(n-1)!)) (n>1).
a(n) = floor(n*n!*e) + 1. - Melvin J. Knight (knightmj(AT)juno.com), May 30 2001
a(n) = {e*n*n!} for n > 0, where {x} denotes the nearest integer part. Proposed by Simon Plouffe, March 1993.
The n-th row of array A089900 is the n-th binomial transform of this sequence. The (n+1)-th term of the n-th binomial transform is (n+1)^(n+1), for n >= 0. E.g., the 5th term of the 4th binomial transform is 5^5: [1, 7, 51, 389, 3125, ...]. - Paul D. Hanna, Nov 14 2003
G.f.: Sum_{k>=0} k! * (x / (1 - x))^k. - Michael Somos, Mar 04 2004
a(n) = Sum_{k = 0..n} A046716(n, k)*2^(n-k). - Philippe Deléham, Jun 12 2004
(n-1)*a(n) = n^2*a(n-1)-1. - Vladeta Jovovic, Sep 04 2004
a(n) = Sum_{k=0..n} P(n, k)*(k+1). - Ross La Haye, Aug 28 2005
a(n) = n!*n*(3 - Sum_{j=2..n} 1/(j*(j-1)*j!)) for n>=1. - Emeric Deutsch, Apr 12 2008
a(n) = (a(n-1)^2 + 2 * a(n-2)^2 + a(n-2) * a(n-3) - 4 * a(n-1) * a(n-3)) / (a(n-2) - a(n-3)) if n>1. - Michael Somos, Oct 20 2011
E.g.f.:1/Q(0); Q(k) = 1 - 2*x/(1+x/(2-x-2/(1-x*(k+1)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2011
G.f.: 1/Q(0), where Q(k) = 1 - x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: Q(0)/x - 1/x, where Q(k) = 1 + (2*k + 1)*x/( 1 - x - 2*x*(1-x)*(k+1)/(2*x*(k+1) + (1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 09 2013
G.f.: (2/x)/G(0) - 1/x, where G(k) = 1 + 1/(1 - x*(2*k+2)/(x*(2*k+3) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
G.f.: Q(0)/(2*x) - 1/x, where Q(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1-x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 08 2013
G.f.: W(0)/x - 1/x, where W(k) = 1 - x*(k+1)/( x*(k+2) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013
a(n) = hypergeometric([2, -n], [], -1). - Peter Luschny, Sep 20 2014
Upper and bottom right terms of the infinite 2 X 2 matrix product_{N=1,2,3,...} [(1,1); (1,N)]. - Gary W. Adamson, Jul 28 2016
a(n) = R(n,n+1,n) where R(x,y,z) is defined to be R(x+1,y,z+1) = R(y,x,x) + R(z,y,z), R(0,y,z+1) = R(z,y,z), R(x+1,y,0) = R(y,x,x), and R(0,y,0) = y. - David M. Cerna, Feb 16 2018
a(n) = (n + 1)!*hypergeom([-n], [-n-1], 1). - Peter Luschny, Nov 02 2018
a(n) = Integral_{x=0..1} (-LambertW(-1,-x/e))^n dx. - Gleb Koloskov, Jul 25 2021
a(n) = KummerU(-n, -n-1, 1). - Peter Luschny, May 10 2022

Typo in description in 1995 Encyclopedia of Integer Sequences corrected Mar 15 1997

Link updated by Susanne Wienand, Nov 19 2011

A248727 A046802(x,y) --> A046802(x,y+1), transform of e.g.f. for the graded number of positroids of the totally nonnegative Grassmannians G+(k,n); enumerates faces of the stellahedra.

Original entry on oeis.org

1, 2, 1, 5, 5, 1, 16, 24, 10, 1, 65, 130, 84, 19, 1, 326, 815, 720, 265, 36, 1, 1957, 5871, 6605, 3425, 803, 69, 1, 13700, 47950, 65646, 44240, 15106, 2394, 134, 1, 109601, 438404, 707840, 589106, 267134, 63896, 7094, 263, 1

Offset: 0

Tom Copeland, Oct 12 2014

This is a transform of A046802 treating it as an array of h-vectors, so y is replaced by (y+1) in the e.g.f. for A046802.
An e.g.f. for the reversed row polynomials with signs is given by exp(a.(0;t)x) = [e^{(1+t)x} [1+t(1-e^(-x))]]^(-1) = 1 - (1+2t)x + (1+5t+5t^2)x^2/2! + ... . The reciprocal is an e.g.f. for the reversed face polynomials of the simplices A074909, i.e., exp(b.(0;t)x) = e^{(1+t)x} [1+t(1-e^(-x))] = 1 + (1+2t)x +(1+3t+3t^2) x^2/2! + ... , so the relations of A133314 apply between the two sets of polynomials. In particular, umbrally [a.(0;t)+b.(0;t)]^n vanishes except for n=0 for which it's unity, implying the two sets of Appell polynomials formed from the two bases, a_n(z;t) = (a.(0;t)+z)^n and b_n(z;t) = (b.(0;t) + z)^n, are an umbral compositional inverse pair, i.e., b_n(a.(x;t);t)= x^n = a_n(b.(x;t);t). Raising operators for these Appell polynomials are related to the polynomials of A028246, whose reverse polynomials are given by A123125 * A007318. Compare: A248727 = A007318 * A123125 * A007318 and A046802 = A007318 * A123125. See A074909 for definitions and related links. - Tom Copeland, Jan 21 2015
The o.g.f. for the umbral inverses is Og(x) = x / (1 - x b.(0;t)) = x / [(1-tx)(1-(1+t)x)] = x + (1+2t) x^2 + (1+3t+3t^2) x^3 + ... . Its compositional inverse is an o.g.f for signed A033282, the reverse f-polynomials for the simplicial duals of the Stasheff polytopes, or associahedra of type A, Oginv(x) =[1+(1+2t)x-sqrt[1+2(1+2t)x+x^2]] / (2t(1+t)x) = x - (1+2t) x^2 + (1+5t+5t^2) x^3 + ... . Contrast this with the o.g.f.s related to the corresponding h-polynomials in A046802. - Tom Copeland, Jan 24 2015
Face vectors, or coefficients of the face polynomials, of the stellahedra, or stellohedra. See p. 59 of Buchstaber and Panov. - Tom Copeland, Nov 08 2016
See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra and stellahedra. - Tom Copeland, Nov 14 2016

			The triangle T(n, k) starts:
n\k    0     1     2     3     4    5   6  7 ...
1:     1
2:     2     1
3:     5     5     1
4:    16    24    10     1
5:    65   130    84    19     1
6:   326   815   720   265    36    1
7:  1957  5871  6605  3425   803   69   1
8: 13700 47950 65646 44240 15106 2394 134  1
... reformatted, _Wolfdieter Lang_, Mar 27 2015

P. Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
L. Berry, S. Forcey, M. Ronco, and P. Showers, Polytopes and Hopf algebras of painted trees: Fan graphs and Stellohedra, arXiv:1608.08546 [math.CO], 2018.
L. Berry, S. Forcey, M. Ronco, and P. Showers, Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes, arXiv:1608.08546 [math.CO], 2019.
V. Buchstaber and T. Panov, Toric Topology, arXiv:1210.2368v3 [math.AT], 2014.
R. Da Rosa, D. Jensen, and D. Ranganathan, Toric graph associahedra and compactifications of M_(0,n), arXiv:1411.0537 [math.AG], 2015.
S. Forcey, M. Ronco, and P. Showers, Polytopes and algebras of grafted trees: Stellohedra, arXiv:1608.08546v2 [math.CO], 2016.
Stefan Forcey, The Hedra Zoo
I. Limonchenko, Moment-angle manifolds, 2-truncated cubes and Massey operations, arXiv:1510.07778 [math.AT], 2017.
M. Lin, Graph Cohomology, 2016, (Fig. 2.5 is a stellahedron).
T. Manneville and V. Pilaud, Compatibility fans for graphical nested complexes, arXiv:1501.07152v3 [math.CO], 2015-2016.
MathOverflow, Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions, an MO question posed by T. Copeland, 2017. (See Buchstaber references therein.)
V. Pilaud, The Associahedron and its Friends, presentation for Séminaire Lotharingien de Combinatoire, April 4-6, 2016. [From _Tom Copeland_, Jun 26 2018]

Cf. A046802, A007047, A122045, A000522, A036918, A001339, A052944.
Cf. A074909, A028246, A133314, A007318, A123125, A033282.
Cf. A008279, A008292, A090582, A097808, A130850.
Cf. A019538, A119879.

(* t = A046802 *) t[, 1] = 1; t[n, n_] = 1; t[n_, 2] = 2^(n - 1) - 1; t[n_, k_] = Sum[((i - k + 1)^i*(k - i)^(n - i - 1) - (i - k + 2)^i*(k - i - 1)^(n - i - 1))*Binomial[n - 1, i], {i, 0, k - 1}]; T[n_, j_] := Sum[Binomial[k, j]*t[n + 1, k + 1], {k, j, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 23 2015, after Tom Copeland *)

Let M(n,k)= sum{i=0,..,k-1, C(n,i)[(i-k)^i*(k-i+1)^(n-i)- (i-k+1)^i*(k-i)^(n-i)]} with M(n,0)=1. Then M(n,k)= A046802(n,k), and T(n,j)= sum(k=j,..,n, C(k,j)*M(n,k)) for j>0 with T(n,0)= 1 + sum(k=1,..,n, M(n,k)) for n>0 and T(0,0)=1.
E.g.f: y * exp[x*(y+1)]/[y+1-exp(x*y)].
Row sums are A007047. Row polynomials evaluated at -1 are unity. Row polynomials evaluated at -2 are A122045.
First column is A000522. Second column appears to be A036918/2 = (A001339-1)/2 = n*A000522(n)/2.
Second diagonal is A052944. (Changed from conjecture to fact on Nov 08 2016.)
The raising operator for the reverse row polynomials with row signs is R = x - (1+t) - t e^(-D) / [1 + t(1-e^(-D))] evaluated at x = 0, with D = d/dx. Also R = x - d/dD log[exp(a.(0;t)D], or R = - d/dz log[e^(-xz) exp(a.(0;t)z)] = - d/dz log[exp(a.(-x;t)z)] with the e.g.f. defined in the comments and z replaced by D. Note that t e ^(-D) / [1+t(1-e^(-D))] = t - (t+t^2) D + (t+3t^2+2t^3) D^2/2! - ... is an e.g.f. for the signed reverse row polynomials of A028246. - Tom Copeland, Jan 23 2015
Equals A007318*(padded A090582)*A007318*A097808 = A007318*(padded (A008292*A007318))*A007318*A097808 = A007318*A130850 = A007318*(mirror of A028246). Padded means in the same way that A097805 is padded A007318. - Tom Copeland, Nov 14 2016
Umbrally, the row polynomials are p_n(x) = (1 + q.(x))^n, where (q.(x))^k = q_k(x) are the row polynomials of A130850. - Tom Copeland, Nov 16 2016
From the previous umbral statement, OP(x,d/dy) y^n =  (y + q.(x))^n, where OP(x,y) = exp[y * q.(x)] = x/((1+x)*exp(-x*y) - 1), the e.g.f. of A130850, so OP(x,d/dy) y^n evaluated at y = 1 is  p_n(x), the n-th row polynomial of this entry, with offset 0. - Tom Copeland, Jun 25 2018
Consolidating some formulas in this entry and A046082, in umbral notation for concision, with all offsets 0: Let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of this entry (A248727, the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of the Worpitsky triangle (A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - Tom Copeland, Jan 24 2020

Title expanded by Tom Copeland, Nov 08 2016

A072842 Largest m such that we can partition the set {1,2,...,m} into n subsets with the property that we never have a+b=c for any distinct elements a, b, c in one subset.

Original entry on oeis.org

2, 8, 23, 66

Offset: 1

Tor G. J. Myklebust (pi(AT)flyingteapot.bnr.usu.edu), Jul 24 2002

The fourth term is at least 66 (Ernst Munter), from { 24 26 27 28 29 30 31 32 33 36 37 38 39 41 42 44 45 46 47 48 49 } { 9 10 12 13 14 15 17 18 20 54 55 56 57 58 59 60 61 62 } { 1 2 4 8 11 16 22 25 40 43 53 66 } { 3 5 6 7 19 21 23 34 35 50 51 52 63 64 65 }
Another set of subsets can be described with this sequence of digits (among 8238): 112122213313333333232124144444144422244144441444412223333333331222 (where each digit represents a subset) The fifth term is at least 195 and can be built with the previous sequence, 515, then 66 digits 5 and finally the sequence 122133333333312224144441444222441444444441422213333133331222. I'd like to see a 196-digit sequence. [Julien de Prabere]
Actually a(5)=196 was given by Walker without proof. But Eliahou et al. give an example of such a partition, so a(5) >= 196. And Robilliard et al. give an example for n=6 with [1..574], so a(6) >= 574. - Michel Marcus, Mar 26 2013
To clarify: a(1)-a(4) are known. a(5) = 196 was claimed by Walker but no proof is known, though the value seems likely to be correct. - Charles R Greathouse IV, Jun 13 2013
The best known lower bounds for the next terms: a(6) >= 582, a(7) >= 1740, a(8) >= 5201, a(9) >= 15596. See link to Eliahou's 2017 article. - Dmitry Kamenetsky, Oct 20 2019
From Fred Rowley, Aug 29 2025: (Start)
New lower bounds a(6) >= 642, a(7) >= 2146, a(8) >= 6976 and beyond were established by Rowley in 2021 (see link). Improved lower bounds a(6) >= 646, a(9) >= 22536 and a(10) >= 71256 and beyond were established in 2022 by Ageron et al (see link).
The following coloring demonstrates that a(5) >= 207, confirming this number remains an open problem:
 1 2 1 3 1 2 1 3 1 2   1 4 1 2 1 3 3 2 1 4   1 3 3 4 4 4 4 4 4 4   4 3 4 4 4 2 3 1 5 2
 2 3 3 2 2 1 5 2 2 1   5 2 2 1 5 2 2 1 5 2   2 1 5 2 2 1 5 2 2 1   5 2 2 1 5 3 3 4 4 4
 4 4 4 4 4 4 4 4 4 1   5 5 5 1 5 3 3 1 5 5   5 1 5 5 5 1 5 5 5 1   5 5 5 1 5 5 5 1 5 5
 5 1 5 5 5 1 5 5 5 1   3 3 5 1 5 5 5 1 4 4   4 4 4 4 4 4 4 4 4 4   3 3 5 1 2 2 5 1 2 2
 5 1 2 2 5 1 2 2 5 1   2 2 5 1 2 2 5 1 2 2   5 1 2 2 3 3 2 2 5 1   3 3 4 4 4 4 4 4 4 4
 4 4 4 4 3 3 1. (End)

			a(2) = 8 because we may partition the set {1, 2, ..., 8} into {1, 2, 4, 8} and {3, 5, 6, 7} with the desired property, and this is the unique solution; attempting to add 9 to either will produce a set with the property that a+b=c for some a,b,c (1+8=9 or 3+6=9). [Corrected by Julien de Prabere, Dec 17 2009]

EFNet #math, Jul 23 2002 (can we replace this with a link? - N. J. A. Sloane)

R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel and J. Tomasik, New lower bounds for Schur and weak Schur numbers, arXiv:2112.03175 [math.CO], 2021-2022.
P. Blanchard, F. Harary and R. Reis, Partitions into sum-free sets, Integers: electronic journal of combinatorial number theory, 6. 2006.
P. Bornsztein, An extension of a theorem of Schur, Acta Arithmetica, 101.4 (2001), pp. 395-399.
Dr. Dobb's Journal, Solutions to the "Monopoles" Problem, Dec 01 1999.
S. Eliahou, J. M. Marín, M. P. Revuelta, M. I. Sanz, Weak Schur numbers and the search for G.W. Walker’s lost partitions, Computers & Mathematics with Applications, Volume 63, Issue 1, January 2012, Pages 175-182.
Shalom Eliahou, Les extraordinaires prédictions du Révérend Walker, Images des Mathématiques, CNRS, 2017 (in French).
Gordon Hamilton, Grade 2 $1,000,000 Unsolved Problem (2011).
Robert W. Irving, An extension of Schur's theorem on sum-free partitions, Acta Arithmetica 25 (1973/74), pp. 55-64. (see p. 59ff)
Dmitry Kamenetsky, Paint numbers from 1 to 8 with two colours, Puzzling StackExchange, 2019.
Dmitry Kamenetsky, Paint numbers from 1 to 23 with three colours, Puzzling StackExchange, 2019.
MathEnJeans, Les tiroirs anti-sommes, 2010-2011 (in French).
D. Robilliard, C. Fonlupt, V. Marion-Poty, Amine Boumaza, A multilevel Tabu Search with Backtracking for Exploring Weak Schur Numbers, Artificial Evolution, Lecture Notes in Computer Science, Volume 7401, 2012, pp 109-119.
Fred Rowley, New lower bounds for weak Schur partitions, Integers, vol. 21, p. #A59, 2021.
G. W. Walker, Solution to the problem E985, American Mathematical Monthly, Vol. 59 (1952), p. 253.

The requirement that a not equal b is the only difference between these numbers and the Schur numbers A045652.

Cf. A036918, A118771, A297023.

It is known that 315^((n-1)/5) <= a(n) <= floor(n!*n*e). - Pierre Bornsztein (bornsztein(AT)voila.fr), Sep 02 2003

a(n) < A118771(n), and also a(n) <= A036918(n+1). - Michel Marcus, Mar 26 2013

Additional comments from Rob Pratt and Brendan McKay, Nov 02 2002
More terms from Pierre Bornsztein (bornsztein(AT)voila.fr), Sep 02 2003
Minor additions from Julien de Prabere (jdpbr(AT)aliceadsl.fr), Feb 25 2010
Term a(5) = 196 removed by Fred Rowley, Aug 29 2025

cp's OEIS Frontend

A036919 A036918/2.

Views

Author

Keywords

A001339 a(n) = Sum_{k=0..n} (k+1)! binomial(n,k).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A248727 A046802(x,y) --> A046802(x,y+1), transform of e.g.f. for the graded number of positroids of the totally nonnegative Grassmannians G+(k,n); enumerates faces of the stellahedra.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A072842 Largest m such that we can partition the set {1,2,...,m} into n subsets with the property that we never have a+b=c for any distinct elements a, b, c in one subset.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions