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A006918 a(n) = binomial(n+3, 3)/4 for odd n, n*(n+2)*(n+4)/24 for even n.
(Formerly M1349)
102
0, 1, 2, 5, 8, 14, 20, 30, 40, 55, 70, 91, 112, 140, 168, 204, 240, 285, 330, 385, 440, 506, 572, 650, 728, 819, 910, 1015, 1120, 1240, 1360, 1496, 1632, 1785, 1938, 2109, 2280, 2470, 2660, 2870, 3080, 3311, 3542, 3795, 4048, 4324, 4600, 4900, 5200, 5525, 5850, 6201, 6552, 6930 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Maximal number of inconsistent triples in a tournament on n+2 nodes [Kac]. - corrected by Leen Droogendijk, Nov 10 2014
a(n-4) is the number of aperiodic necklaces (Lyndon words) with 4 black beads and n-4 white beads.
a(n-3) is the maximum number of squares that can be formed from n lines, for n>=3. - Erich Friedman; corrected by Leen Droogendijk, Nov 10 2014
Number of trees with diameter 4 where at most 2 vertices 1 away from the graph center have degree > 2. - Jon Perry, Jul 11 2003
a(n+1) is the number of partitions of n into parts of two kinds, with at most two parts of each kind. Also a(n-3) is the number of partitions of n with Durfee square of size 2. - Franklin T. Adams-Watters, Jan 27 2006
Factoring the g.f. as x/(1-x)^2 times 1/(1-x^2)^2 we find that the sequence equals (1, 2, 3, 4, ...) convolved with (1, 0, 2, 0, 3, 0, 4, ...), A000027 convolved with its aerated variant. - Gary W. Adamson, May 01 2009
Starting with "1" = triangle A171238 * [1,2,3,...]. - Gary W. Adamson, Dec 05 2009
The Kn21, Kn22, Kn23, Fi2 and Ze2 triangle sums, see A180662 for their definitions, of the Connell-Pol triangle A159797 are linear sums of shifted versions of this sequence, e.g., Kn22(n) = a(n+1) + a(n) + 2*a(n-1) + a(n-2) and Fi2(n) = a(n) + 4*a(n-1) + a(n-2). - Johannes W. Meijer, May 20 2011
For n>3, a(n-4) is the number of (w,x,y,z) having all terms in {1,...,n} and w+x+y+z=|x-y|+|y-z|. - Clark Kimberling, May 23 2012
a(n) is the number of (w,x,y) having all terms in {0,...,n} and w+x+y < |w-x|+|x-y|. - Clark Kimberling, Jun 13 2012
For n>0 number of inequivalent (n-1) X 2 binary matrices, where equivalence means permutations of rows or columns or the symbol set. - Alois P. Heinz, Aug 17 2014
Number of partitions p of n+5 such that p[3] = 2. Examples: a(1)=1 because we have (2,2,2); a(2)=2 because we have (2,2,2,1) and (3,2,2); a(3)=5 because we have (2,2,2,1,1), (2,2,2,2), (3,2,2,1), (3,3,2), and (4,2,2). See the R. P. Stanley reference. - Emeric Deutsch, Oct 28 2014
Sum over each antidiagonal of A243866. - Christopher Hunt Gribble, Apr 02 2015
Number of nonisomorphic outer planar graphs of order n>=3, size n+2, and maximum degree 3. - Christian Barrientos and Sarah Minion, Feb 27 2018
a(n) is the number of 2413-avoiding odd Grassmannian permutations of size n+1. - Juan B. Gil, Mar 09 2023
REFERENCES
J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 147.
M. Kac, An example of "counting without counting", Philips Res. Reports, 30 (1975), 20*-22* [Special issue in honour of C. J. Bouwkamp].
E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, 2004.
K. B. Reid and L. W. Beineke "Tournaments", pp. 169-204 in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, p. 186, Theorem 6.11.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 2nd ed., 2012, Exercise 4.16, pp. 530, 552.
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.
LINKS
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020.
Y. Choliy and A. V. Sills, A formula for the partition function that “counts”, Preprint 2015.
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
Steven Edwards and William Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
B. G. Eke, Monotonic triads, Discrete Math., Vol. 9, No. 4 (1974), pp. 359-363. MR0354390 (50 #6869)
Irene Erazo, John López, and Carlos Trujillo, A combinatorial problem that arose in integer B_3 Sets, Revista Colombiana de Matemáticas, Vol. 53, No. 2 (2019), pp. 195-203.
Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
John Golden and Marcus Spradlin, Collinear and Soft Limits of Multi-Loop Integrands in N= 4 Yang-Mills, arXiv preprint arXiv:1203.1915 [hep-th], 2012. - From N. J. A. Sloane, Sep 14 2012
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10b, lambda=2]
FORMULA
G.f.: x/((1-x)^2*(1-x^2)^2) = x/((1+x)^2*(1-x)^4).
0, 0, 0, 1, 2, 5, 8, 14, ... has a(n) = (Sum_{k=0..n} floor(k(n-k)/2))/2. - Paul Barry, Sep 14 2003
0, 0, 0, 0, 0, 1, 2, 5, 8, 14, 20, 30, 40, 55, ... has a(n) = binomial(floor(1/2 n), 3) + binomial(floor(1/2 n + 1/2), 3) [Eke]. - N. J. A. Sloane, May 12 2012
a(0)=0, a(1)=1, a(n) = (2/(n-1))*a(n-1) + ((n+3)/(n-1))*a(n-2). - Benoit Cloitre, Jun 28 2004
a(n) = floor(binomial(n+4, 4)/(n+4)) - floor((n+2)/8)(1+(-1)^n)/2. - Paul Barry, Jan 01 2005
a(n+1) = a(n) + binomial(floor(n/2)+2,2), i.e., first differences are A008805. Convolution of A008619 with itself, then shifted right (or A004526 with itself, shifted left by 3). - Franklin T. Adams-Watters, Jan 27 2006
a(n+1) = (A027656(n) + A003451(n+5))/2 with a(1)=0. - Yosu Yurramendi, Sep 12 2008
Linear recurrence: a(n) = 2a(n-1) + a(n-2) - 4a(n-3) + a(n-4) + 2a(n-5) - a(n-6). - Jaume Oliver Lafont, Dec 05 2008
Euler transform of length 2 sequence [2, 2]. - Michael Somos, Aug 15 2009
a(n) = -a(-4-n) for all n in Z.
a(n+1) + a(n) = A002623(n). - Johannes W. Meijer, May 20 2011
a(n) = (n+2)*(2*n*(n+4)-3*(-1)^n+3)/48. - Bruno Berselli, May 21 2011
a(2n) = A007290(n+2). - Jon Perry, Nov 10 2014
G.f.: (1/(1-x)^4-1/(1-x^2)^2)/4. - Herbert Kociemba, Oct 23 2016
E.g.f.: (x*(18 + 9*x + x^2)*cosh(x) + (6 + 15*x + 9*x^2 + x^3)*sinh(x))/24. - Stefano Spezia, Dec 07 2021
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=1} 1/a(n) = 75/4 - 24*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 69/4 - 24*log(2). (End)
EXAMPLE
G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 14*x^5 + 20*x^6 + 30*x^7 + 40*x^8 + 55*x^9 + ...
From Gus Wiseman, Apr 06 2019: (Start)
The a(4 - 3) = 1 through a(8 - 3) = 14 integer partitions with Durfee square of length 2 are the following (see Franklin T. Adams-Watters's second comment). The Heinz numbers of these partitions are given by A325164.
(22) (32) (33) (43) (44)
(221) (42) (52) (53)
(222) (322) (62)
(321) (331) (332)
(2211) (421) (422)
(2221) (431)
(3211) (521)
(22111) (2222)
(3221)
(3311)
(4211)
(22211)
(32111)
(221111)
The a(0 + 1) = 1 through a(4 + 1) = 14 integer partitions of n into parts of two kinds with at most two parts of each kind are the following (see Franklin T. Adams-Watters's first comment).
()() ()(1) ()(2) ()(3) ()(4)
(1)() (2)() (3)() (4)()
()(11) (1)(2) (1)(3)
(1)(1) ()(21) ()(22)
(11)() (2)(1) (2)(2)
(21)() (22)()
(1)(11) ()(31)
(11)(1) (3)(1)
(31)()
(11)(2)
(1)(21)
(2)(11)
(21)(1)
(11)(11)
The a(6 - 5) = 1 through a(10 - 5) = 14 integer partitions whose third part is 2 are the following (see Emeric Deutsch's comment). The Heinz numbers of these partitions are given by A307373.
(222) (322) (332) (432) (442)
(2221) (422) (522) (532)
(2222) (3222) (622)
(3221) (3321) (3322)
(22211) (4221) (4222)
(22221) (4321)
(32211) (5221)
(222111) (22222)
(32221)
(33211)
(42211)
(222211)
(322111)
(2221111)
(End)
MAPLE
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=r), U=Sequence(Z, card>=3)}, unlabeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m), m=11..58) ; # Zerinvary Lajos, Mar 09 2007
A006918 := proc(n)
if type(n, 'even') then
n*(n+2)*(n+4)/24 ;
else
binomial(n+3, 3)/4 ;
fi ;
end proc: # R. J. Mathar, May 17 2016
MATHEMATICA
f[n_]:=If[EvenQ[n], (n(n+2)(n+4))/24, Binomial[n+3, 3]/4]; Join[{0}, Array[f, 60]] (* Harvey P. Dale, Apr 20 2011 *)
durf[ptn_]:=Length[Select[Range[Length[ptn]], ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n], durf[#]==2&]], {n, 0, 30}] (* Gus Wiseman, Apr 06 2019 *)
PROG
(PARI) { parttrees(n)=local(pt, k, nk); if (n%2==0, pt=(n/2+1)^2, pt=ceil(n/2)*(ceil(n/2)+1)); pt+=floor(n/2); for (x=1, floor(n/2), pt+=floor(x/2)+floor((n-x)/2)); if (n%2==0 && n>2, pt-=floor(n/4)); k=1; while (3*k<=n, for (x=k, floor((n-k)/2), pt+=floor(k/2); if (x!=k, pt+=floor(x/2)); if ((n-x-k)!=k && (n-x-k)!=x, pt+=floor((n-x-k)/2))); k++); pt }
(PARI) {a(n) = n += 2; (n^3 - n * (2-n%2)^2) / 24}; /* Michael Somos, Aug 15 2009 */
(Haskell)
a006918 n = a006918_list !! n
a006918_list = scanl (+) 0 a008805_list
-- Reinhard Zumkeller, Feb 01 2013
(Magma) [Floor(Binomial(n+4, 4)/(n+4))-Floor((n+2)/8)*(1+(-1)^n)/2: n in [0..60]]; // Vincenzo Librandi, Nov 10 2014
CROSSREFS
Cf. A000031, A001037, A028723, A051168. a(n) = T(n,4), array T as in A051168.
Cf. A000094.
Cf. A171238. - Gary W. Adamson, Dec 05 2009
Row sums of A173997. - Gary W. Adamson, Mar 05 2010
Column k=2 of A242093. Column k=2 of A115720 and A115994.
Sequence in context: A095348 A215725 A022907 * A274523 A165189 A358055
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved

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Last modified May 19 14:45 EDT 2024. Contains 372698 sequences. (Running on oeis4.)