A144679 -id:A144679 - cp's OEIS frontend

A144677 Related to enumeration of quantum states (see reference for precise definition).

1, 2, 3, 6, 9, 12, 18, 24, 30, 40, 50, 60, 75, 90, 105, 126, 147, 168, 196, 224, 252, 288, 324, 360, 405, 450, 495, 550, 605, 660, 726, 792, 858, 936, 1014, 1092, 1183, 1274, 1365, 1470, 1575, 1680, 1800, 1920, 2040, 2176, 2312, 2448, 2601, 2754, 2907, 3078, 3249, 3420

Offset: 0

N. J. A. Sloane, Feb 06 2009

Equals (1, 2, 3, ...) convolved with (1, 0, 0, 2, 0, 0, 3, ...) = (1 + 2*x + 3*x^2 + ...) * (1 + 2*x^3 + 3*x^6 + ...). - Gary W. Adamson, Feb 23 2010

The Ca2 and Ze4 triangle sums, see A180662 for their definitions, of the Connell-Pol triangle A159797 are linear sums of shifted versions of the sequence given above, e.g., Ca2(n) = a(n-1) + 2*a(n-2) + 3*a(n-3) + a(n-4). - Johannes W. Meijer, May 20 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
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=3]
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A006918, A144678, A144679.

Cf. A000292, A190717. [Johannes W. Meijer, May 20 2011]

I:=[1,2,3,6,9,12,18,24]; [n le 8 select I[n] else 2*Self(n-1)-Self(n-2)+2*Self(n-3)-4*Self(n-4)+2*Self(n-5)-Self(n-6)+2*Self(n-7)-Self(n-8): n in [1..60]]; // Vincenzo Librandi, Mar 28 2015

n:=80; lambda:=3; S10b:=[];
for ii from 0 to n do
x:=floor(ii/lambda);
snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3);
S10b:=[op(S10b),snc];
od:
S10b;
A144677 := proc(n) option remember; local k1; sum(A190717(n-k1),k1=0..2) end: A190717:= proc(n) option remember; A190717(n):= binomial(floor(n/3)+3,3) end: seq(A144677(n), n=0..53); # Johannes W. Meijer, May 20 2011

CoefficientList[Series[1/((x - 1)^4*(x^2 + x + 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Mar 28 2015 *)
LinearRecurrence[{2, -1, 2, -4, 2, -1, 2, -1}, {1, 2, 3, 6, 9, 12, 18, 24}, 60 ] (* Vincenzo Librandi, Mar 28 2015 *)

@CachedFunction
def a(n): return sum( ((j+3)//3)*((n-j+3)//3) for j in (0..n) )
[a(n) for n in (0..60)] # G. C. Greubel, Oct 18 2021

From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A190717(n-2) + A190717(n-1) + A190717(n).
a(n-2) + a(n-1) + a(n) = A014125(n).
G.f.: 1/((1-x)^4*(1+x+x^2)^2). (End)
From Wesley Ivan Hurt, Mar 28 2015: (Start)
a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -4*a(n-4) +2*a(n-5) -a(n-6) +2*a(n-7) -a(n-8).
a(n) = ((2 + floor(n/3))^3 - floor((n+4)/3) + floor((n+4)/3)^3 - floor((n+5)/3) + floor((n+5)/3)^3 - floor((n+6)/3))/6. (End)
a(n) = Sum_{j=0..n} floor((j+3)/3)*floor((n-j+3)/3). - G. C. Greubel, Oct 18 2021
a(n) = (132+129*n+36*n^2+3*n^3+6*(n+5)*cos(2*n*Pi/3)+2*sqrt(3)*(3*n+11)*sin(2*n*Pi/3))/162. - Wesley Ivan Hurt, Sep 01 2025

A122047 Degree of the polynomial P(n,x), defined by a Somos-6 type sequence: P(n,x)=(x^(n-1)P(n-1,x)P(n-5,x) + P(n-2,x)*P(n-4,x))/P(n-6,x), initialized with P(n,x)=1 at n<0.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 22, 31, 42, 55, 70, 88, 109, 133, 160, 190, 224, 262, 304, 350, 400, 455, 515, 580, 650, 725, 806, 893, 986, 1085, 1190, 1302, 1421, 1547, 1680, 1820, 1968, 2124, 2288, 2460, 2640, 2829, 3027

Offset: 0

Roger L. Bagula, Sep 13 2006

nonn

Maximum Wiener index of all maximal 5-degenerate graphs with n vertices. (A maximal 5-degenerate graph can be constructed from a 5-clique by iteratively adding a new 5-leaf (vertex of degree 5) adjacent to 5 existing vertices.) The extremal graphs are 5th powers of paths, so the bound also applies to 5-trees. - Allan Bickle, Sep 15 2022

Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004; arXiv:0807.2538 [nlin.SI], 2008. [Set a(n)=d(n+3) on p. 8]
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231 [quant-ph], 2008. [Eq 10a, lambda=5]

Cf. A014125, A122046.

The maximum Wiener index of all maximal k-degenerate graphs for k=1..6 are given in A000292, A002623, A014125, A122046, A122047 (this sequence), A175724, respectively.

p[n_] := p[n] = Cancel[Simplify[(x^(n - 1)p[n - 1]p[n - 5] + p[n - 2]*p[n - 4])/p[n - 6]]];p[ -6] = 1; p[ -5] = 1; p[ -4] = 1; p[ -3] = 1; p[ -2] = 1; p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]

Conjectures from R. J. Mathar, Jul 15 2008: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8);
o.g.f.: x^2/((x^4+x^3+x^2+x+1)(x-1)^4). (End)
Conjecture: a(n) = (A000292(n+1) - n - 2 - (-1)^floor((n-1)/5)*A099443(n+1))/5. - R. J. Mathar, Jul 15 2008
a(n+2) = A144679(n) + A144679(n-1) + A144679(n-2) + A144679(n-3) + A144679(n-4). - Johannes W. Meijer, May 20 2011
a(n) = floor((n^3 + 6*n^2 + 5*n)/30). - Allan Bickle, Sep 15 2022

Edited by N. J. A. Sloane, Jul 15 2008

a(22)-a(43) from R. J. Mathar, Jul 15 2008

A144678 Related to enumeration of quantum states (see reference for precise definition).

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 13, 16, 22, 28, 34, 40, 50, 60, 70, 80, 95, 110, 125, 140, 161, 182, 203, 224, 252, 280, 308, 336, 372, 408, 444, 480, 525, 570, 615, 660, 715, 770, 825, 880, 946, 1012, 1078, 1144, 1222, 1300, 1378, 1456, 1547, 1638, 1729, 1820, 1925, 2030, 2135

Offset: 0

N. J. A. Sloane, Feb 06 2009

The Gi2 triangle sums of the triangle A159797 are linear sums of shifted versions of the sequence given above, i.e., Gi2(n) = a(n-1) + 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5). For the definitions of the Gi2 and other triangle sums see A180662. [Johannes W. Meijer, May 20 2011]
Partial sums of 1,1,1,1, 3,3,3,3, 6,6,6,6,..., the quadruplicated A000217. - R. J. Mathar, Aug 25 2013
Number of partitions of n into two different parts of size 4 and two different parts of size 1. a(4) = 7: 4, 4', 1111, 1111', 111'1', 11'1'1', 1'1'1'1'. - Alois P. Heinz, Dec 22 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
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=4]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,2,-4,2,0,-1,2,-1).

Cf. A000292, A006918, A144677, A144679, A190718.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^4))^2 )); // G. C. Greubel, Oct 18 2021

n:=80; lambda:=4; S10b:=[];
for ii from 0 to n do
x:=floor(ii/lambda);
snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3);
S10b:=[op(S10b),snc];
od:
S10b;
A144678 := proc(n) option remember;
   local k;
   sum(A190718(n-k),k=0..3)
end:
A190718:= proc(n)
   binomial(floor(n/4)+3,3)
end:
seq(A144678(n),n=0..54); # Johannes W. Meijer, May 20 2011

a[n_] = (r = Mod[n, 4]; (4+n-r)(8+n-r)(3+n+2r)/96); Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Sep 02 2011 *)
LinearRecurrence[{2,-1,0,2,-4,2,0,-1,2,-1}, {1,2,3,4,7,10,13,16,22,28}, 60] (* G. C. Greubel, Oct 18 2021 *)

Vec(1/(x-1)^4/(x^3+x^2+x+1)^2+O(x^99)) \\ Charles R Greathouse IV, Jun 20 2013

def A144678_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)*(1-x^4))^2 ).list()
A144678_list(60) # G. C. Greubel, Oct 18 2021

From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A190718(n-3) + A190718(n-2) + A190718(n-1) + A190718(n).
a(n-3) + a(n-2) + a(n-1) + a(n) = A122046(n+3).
G.f.: 1/((x-1)^4*(x^3+x^2+x+1)^2). (End)
a(n) = A009531(n+5)/16 + (n+5)*(2*n^2+20*n+33+3*(-1)^n)/192 . - R. J. Mathar, Jun 20 2013
a(n) = Sum_{i=1..n+8} floor(i/4) * floor((n+8-i)/4). - Wesley Ivan Hurt, Jul 21 2014
From Alois P. Heinz, Dec 22 2021: (Start)
G.f.: 1/((1-x)*(1-x^4))^2.
a(n) = Sum_{j=0..floor(n/4)} (j+1)*(n-4*j+1). (End)

A343608 a(n) = [n/5][n/5 - 1](3n - 10[n/5 + 1])/6, where [.] = floor: upper bound for minimum number of monochromatic triangles in a 3-edge-colored complete graph K_n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 8, 11, 14, 17, 20, 26, 32, 38, 44, 50, 60, 70, 80, 90, 100, 115, 130, 145, 160, 175, 196, 217, 238, 259, 280, 308, 336, 364, 392, 420, 456, 492, 528, 564, 600, 645, 690, 735, 780, 825, 880, 935, 990, 1045, 1100, 1166

Offset: 0

M. F. Hasler, Jun 25 2021

nonn

Consider the complete graph K_n whose n(n-1)/2 edges are colored with 3 distinct colors as to minimize the number of monochromatic triangles, i.e., subgraphs isomorphic to K_3 having all edges of the same color.
Cummings et al. show that this minimum number of monochromatic triangles M_3(K_3,n) is given by a(n) for sufficiently large n.
For the actual minimum numbers, we currently only know M_3(K_3,n) = 0 for n < 17, and M_3(K_3, 17) = 5.
Known examples for n>17, where the actual minimum value is below this upper bound: M_3(K_3, 18) <= 10 < 14 = a(18), M_3(K_3, 19) <= 15 < 17 = a(19), M_3(K_3, 21) <= 25 < 26 = a(21). For n>21, the current best known minimum values match the upper bound. - Dmitry Kamenetsky, Jul 01 2021

			a(17) = [17/5]*[17/5 - 1]*(3*17 - 10[17/5 + 1])/6 = 3*2*11/6 = 11, which is the number of monochromatic triangles obtained by coloring the edges of the complete graph K_17 as follows: edges (i,j) with mod(j-i,5) in {1,4} get color 1, edges (i,j) with mod(j-i,5) in {2,3} get color 2, and edges (i,j) with mod(j-i,5) = 0 get color 3.  The minimum number of monochromatic triangles in an edge-coloring of K_17 with 3 colors turns out to be 5 <= 11.

James Cummings, Daniel Král', Florian Pfender, Konrad Sperfeld, Andrew Treglown, Michael Young, Monochromatic triangles in three-coloured graphs, Journal of Combinatorial Theory B 103 no. 4 (2013) 489-503 (also: arXiv:1206.1987).
Al Zimmermann, Programming Contest: Monochromatic Triangles, June 2021.

apply( {A343608(n)=n\5*(n\5-1)*(n-10+n%5*2)/6}, [1..55])

a(n) = A144679(n-11) = r*A000292(q-1) + (5-r)*A000292(q-2) = q(q - 1)(n + 2r - 10)/6, where A000292(q-2) = binomial(q,3), r = n mod 5 and q = (n-r)/5, for sufficiently large n, according to Cummings et al., Corollary 3.

Definition corrected following an observation by Rob Pratt, Jun 28 2021

cp's OEIS Frontend

A144677 Related to enumeration of quantum states (see reference for precise definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A122047 Degree of the polynomial P(n,x), defined by a Somos-6 type sequence: P(n,x)=(x^(n-1)*P(n-1,x)*P(n-5,x) + P(n-2,x)*P(n-4,x))/P(n-6,x), initialized with P(n,x)=1 at n<0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A144678 Related to enumeration of quantum states (see reference for precise definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A343608 a(n) = [n/5]*[n/5 - 1]*(3n - 10[n/5 + 1])/6, where [.] = floor: upper bound for minimum number of monochromatic triangles in a 3-edge-colored complete graph K_n.

Views

Author

Keywords

Comments

Examples

Links

Programs

PARI

Formula

Extensions

A122047 Degree of the polynomial P(n,x), defined by a Somos-6 type sequence: P(n,x)=(x^(n-1)P(n-1,x)P(n-5,x) + P(n-2,x)*P(n-4,x))/P(n-6,x), initialized with P(n,x)=1 at n<0.

A343608 a(n) = [n/5][n/5 - 1](3n - 10[n/5 + 1])/6, where [.] = floor: upper bound for minimum number of monochromatic triangles in a 3-edge-colored complete graph K_n.