A144678 -id:A144678 - cp's OEIS frontend

A159797 Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n-1.

0, 1, 1, 2, 3, 4, 3, 5, 7, 9, 4, 7, 10, 13, 16, 5, 9, 13, 17, 21, 25, 6, 11, 16, 21, 26, 31, 36, 7, 13, 19, 25, 31, 37, 43, 49, 8, 15, 22, 29, 36, 43, 50, 57, 64, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101

Offset: 0

Omar E. Pol, Jul 09 2009

Note that the last term of the n-th row is the n-th square A000290(n).
See also A162611, A162614 and A162622.
The triangle sums, see A180662 for their definitions, link the triangle A159797 with eleven sequences, see the crossrefs. - Johannes W. Meijer, May 20 2011
T(n,k) is the number of distinct sums in the direct sum of {1, 2, ... n} with itself k times for 1 <= k <= n+1, e.g., T(5,3) = the number of distinct sums in the direct sum {1,2,3,4,5} + {1,2,3,4,5} + {1,2,3,4,5}. The sums range from 1+1+1=3 to 5+5+5=15. So there are 13 distinct sums. - Derek Orr, Nov 26 2014

			Triangle begins:
0;
1, 1;
2, 3, 4;
3, 5, 7, 9;
4, 7,10,13,16;
5, 9,13,17,21,25;
6,11,16,21,26,31,36;

Harvey P. Dale, Table of n, a(n) for n = 0..1000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A000290, A001477, A081493, A159798, A162609, A162610, A162611, A162614, A162622.
Cf.: A006002 (row sums). - R. J. Mathar, Jul 17 2009
Cf. A163282, A163283, A163284, A163285. - Omar E. Pol, Nov 18 2009
From Johannes W. Meijer, May 20 2011: (Start)
Triangle sums (see the comments): A006002 (Row1), A050187 (Row2), A058187 (Related to Kn11, Kn12, Kn13, Fi1 and Ze1), A006918 (Related to Kn21, Kn22, Kn23, Fi2 and Ze2), A000330 (Kn3), A016061 (Kn4), A190717 (Related to Ca1 and Ze3), A144677 (Related to Ca2 and Ze4), A000292 (Related to Ca3, Ca4, Gi3 and Gi4) A190718 (Related to Gi1) and A144678 (Related to Gi2). (End)

A159797:=proc(n) local m: m := floor( (sqrt(8*n+1)-1)/2 ): A159797(n):= m + (n - m*(m+1)/2)*(m-1) end: seq(A159797(n),n=0..75); # Johannes W. Meijer, May 20 2011

Flatten[Table[NestList[#+n-1&,n,n],{n,0,12}]] (* Harvey P. Dale, Aug 04 2014 *)

Given m = floor( (sqrt(8*n+1)-1)/2 ), then a(n) = m + (n - m*(m+1)/2)*(m-1). - Carl R. White, Jul 24 2010

Edited by Omar E. Pol, Jul 18 2009
More terms from Omar E. Pol, Nov 18 2009
More terms from Carl R. White, Jul 24 2010

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

Original entry on oeis.org

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

A122046 Partial sums of floor(n^2/8).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 10, 16, 24, 34, 46, 61, 79, 100, 124, 152, 184, 220, 260, 305, 355, 410, 470, 536, 608, 686, 770, 861, 959, 1064, 1176, 1296, 1424, 1560, 1704, 1857, 2019, 2190, 2370, 2560, 2760, 2970, 3190, 3421, 3663, 3916, 4180, 4456, 4744, 5044, 5356, 5681, 6019, 6370

Offset: 0

Roger L. Bagula, Sep 13 2006

Degree of the polynomial P(n+1,x), defined by P(n,x) = [x^(n-1)*P(n-1,x)*P(n-4,x)+P(n-2,x)*P(n-3,x)]/P(n-5,x) with P(1,x)=P(0,x)=P(-1,x)=P(-2,x)=P(-3,x)=1.
Define the sequence b(n) = 1, 4, 10, 20, 36, 60,... for n>=0 with g.f. 1/((1+x)*(1+x^2)*(1-x)^5). Then a(n+3) = b(n)-b(n-1) and b(n)+b(n+1)+b(n+2)+b(n+3) = A052762(n+7)/24. - J. M. Bergot, Aug 21 2013
Maximum Wiener index of all maximal 4-degenerate graphs with n-1 vertices.  (A maximal 4-degenerate graph can be constructed from a 4-clique by iteratively adding a new 4-leaf (vertex of degree 4) adjacent to four existing vertices.)  The extremal graphs are 4th powers of paths, so the bound also applies to 4-trees. - Allan Bickle, Sep 15 2022

			a(6) = 10 = 0 + 0 + 0 + 1 + 2 + 3 + 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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, Comments on A122046
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 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=4]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-3,3,-1).

Cf. A014125, A052762, A057077, A090294, A122047, A144678, A190718.
Partial sums of A001972.
Maximum Wiener index of all maximal k-degenerate graphs for k=1..6: A000292, A002623, A014125, A122046 (this sequence), A122047, A175724.

[Round((2*n^3+3*n^2-8*n)/48): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011

A122046 := proc(n) round((2*n^3+3*n^2-8*n)/48) ; end proc: # Mircea Merca

p[n_] := p[n] = Cancel[Simplify[ (x^(n - 1)p[n - 1]p[n - 4] + p[n - 2]*p[n - 3])/p[n - 5]]]; p[ -5] = 1;p[ -4] = 1;p[ -3] = 1;p[ -2] = 1;p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]
Accumulate[Floor[Range[0,60]^2/8]] (* or *) LinearRecurrence[{3,-3,1,1,-3,3,-1},{0,0,0,1,3,6,10},60] (* Harvey P. Dale, Dec 23 2019 *)

a(n)=(2*n^3+3*n^2-8*n+3)\48 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = Sum_{k=0..n} floor(k^2/8).
a(n) = round((2*n^3 + 3*n^2 - 8*n)/48) = round((4*n^3 + 6*n^2 - 16*n - 9)/96) = floor((2*n^3 + 3*n^2 - 8*n + 3)/48) = ceiling((2*n^3 + 3*n^2 - 8*n - 12)/48). - Mircea Merca
a(n) = a(n-8) + (n-4)^2 + n, n > 8. - Mircea Merca
From Andrew Hone, Jul 15 2008: (Start)
a(n+1) = cos((2*n+1)*Pi/4)/(4*sqrt(2)) + (2*n+3)*(2*n^2 + 6*n - 5)/96 + (-1)^n/32.
a(n+1) = A057077(n+1)/8 + A090294(n-1)/32 + (-1)^n/32.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) - 3*a(n-5) + 3*a(n-6) - a(n-7). (End)
O.g.f.: x^3 / ( (1+x)*(x^2+1)*(x-1)^4 ). - R. J. Mathar, Jul 15 2008
From Johannes W. Meijer, May 20 2011: (Start)
a(n+3) = A144678(n) + A144678(n-1) + A144678(n-2) + A144678(n-3);
a(n+3) = Sum_{k=0..6} min(6-k+1,k+1)* A190718(n+k-6). (End)
a(n) = (4*n^3 + 6*n^2 - 16*n - 9 - 3*(-1)^n + 12*(-1)^((2*n - 1 + (-1)^n)/4))/96. - Luce ETIENNE, Mar 21 2014
E.g.f.: ((2*x^3 + 9*x^2 - 3*x - 6)*cosh(x) + 6*(cos(x) + sin(x)) + (2*x^3 + 9*x^2 - 3*x - 3)*sinh(x))/48. - Stefano Spezia, Apr 05 2023

Edited by N. J. A. Sloane, Sep 17 2006, Jul 11 2008, Jul 12 2008

More formulas and better name from Mircea Merca, Nov 19 2010

A190718 Quadruplicated tetrahedral numbers A000292.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 4, 4, 10, 10, 10, 10, 20, 20, 20, 20, 35, 35, 35, 35, 56, 56, 56, 56, 84, 84, 84, 84, 120, 120, 120, 120, 165, 165, 165, 165, 220, 220, 220, 220, 286, 286, 286, 286, 364, 364, 364, 364, 455, 455, 455, 455

Offset: 0

Johannes W. Meijer, May 18 2011

The Gi1 triangle sums, for the definitions of these and other triangle sums see A180662, of the triangle A159797 are linear sums of shifted versions of the quadruplicated tetrahedral numbers A000292, i.e., Gi1(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4) + a(n-8).

The Gi1 and Gi2 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,3,-3,0,0,-3,3,0,0,1,-1).

Cf. A000292 (tetrahedral numbers), A058187 (duplicated), A190717 (triplicated).

A190718:= proc(n) binomial(floor(n/4)+3,3) end:
seq(A190718(n),n=0..52);

LinearRecurrence[{1,0,0,3,-3,0,0,-3,3,0,0,1,-1},{1,1,1,1,4,4,4,4,10,10,10,10,20},60] (* Harvey P. Dale, Oct 20 2012 *)

a(n) = binomial(floor(n/4)+3,3).
a(n-3) + a(n-2) + a(n-1) + a(n) = A144678(n).
a(n) = +a(n-1) +3*a(n-4) -3*a(n-5) -3*a(n-8) +3*a(n-9) +a(n-12) -a(n-13).
G.f.: 1 / ( (1+x)^3*(1+x^2)^3*(x-1)^4 ).
Sum_{n>=0} 1/a(n) = 6. - Amiram Eldar, Aug 18 2022

A144679 a(n) = [n/5 + 1][n/5 + 2](3n - 10[n/5] + 3)/6, where [.] = floor.

Original entry on oeis.org

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, 1232, 1298, 1364, 1430, 1508, 1586, 1664

Offset: 0

N. J. A. Sloane, Feb 06 2009

Related to enumeration of quantum states: this is S_c defined in eq.(10b) of the O'Sullivan and Busch reference, with lambda = 5.

This coincides with the formula for an upper bound on the minimum number of monochromatic triangles in the complete graph K_{n+11} with 3-colored edges given by Cummings et al. (2013) in Corollary 3. (The paper claims that this bound is sharp only for all sufficiently large n.) - M. F. Hasler, Jun 25 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
James Cummings, Daniel Král', Florian Pfender, Konrad Sperfeld, Andrew Treglown, and Michael Young, Monochromatic triangles in three-coloured graphs, Journal of Combinatorial Theory B 103 no. 4 (2013) 489-503 (also: arXiv:1206.1987).
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=5]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,2,-4,2,0,0,-1,2,-1).

Cf. A000292, A006918, A122047, A144678, A144677.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^5))^2 )); // G. C. Greubel, Oct 18 2021

n:=80; lambda:=5; 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;
A144679 := proc(n) option remember; local k; sum(THN5(n-k),k=0..4) end: THN5:= proc(n) option remember; THN5(n):= binomial(floor(n/5)+3,3) end: seq(A144679(n), n=0..57); # Johannes W. Meijer, May 20 2011

LinearRecurrence[{2,-1,0,0,2,-4,2,0,0,-1,2,-1}, {1,2,3,4,5,8,11,14,17,20,26,32}, 60] (* Jean-François Alcover, Nov 22 2017 *)
CoefficientList[Series[1/((x-1)^4(x^4+x^3+x^2+x+1)^2),{x,0,100}],x] (* Harvey P. Dale, Aug 29 2021 *)

apply( {A144679(n)=(3*n+3-10*n\=5)*(n+1)*(n+2)\6}, [0..55]) \\ M. F. Hasler, Jun 25 2021

def A144679_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)*(1-x^5))^2 ).list()
A144679_list(60) # G. C. Greubel, Oct 18 2021

From Johannes W. Meijer, May 20 2011: (Start)
a(n-4) + a(n-3) + a(n-2) + a(n-1) + a(n) = A122047(n+2).
G.f.: 1/((1-x)^4*(1 + x + x^2 + x^3 + x^4)^2). (End)
a(n) = r*A000292(q+1) + (5-r)*A000292(q) = (n + 2r + 1)*(q + 2)*(q + 1)/6, where A000292(q) = binomial(q+2,3), r = (n+1) mod 5, q = (n+1-r)/5. - M. F. Hasler, Jun 25 2021

A189376 Expansion of 1/((1-x)^5*(x^3+x^2+x+1)^2).

Original entry on oeis.org

1, 3, 6, 10, 17, 27, 40, 56, 78, 106, 140, 180, 230, 290, 360, 440, 535, 645, 770, 910, 1071, 1253, 1456, 1680, 1932, 2212, 2520, 2856, 3228, 3636, 4080, 4560, 5085, 5655, 6270, 6930, 7645, 8415, 9240, 10120, 11066, 12078

Offset: 0

Johannes W. Meijer, Apr 29 2011

The Gi2 triangle sums of A139600 lead to the sequence given above, see the formulas. For the definitions of the Gi2 and other triangle sums see A180662.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,2,-6,6,-2,-1,3,-3,1).

Cf. A139600, A189374, A189375.

a:= n-> coeff (series (1/((1-x)^5*(x^3+x^2+x+1)^2), x, n+1), x, n):
seq (a(n), n=0..50);

CoefficientList[Series[1/((1-x)^5(x^3+x^2+x+1)^2),{x,0,50}],x] (* or *) LinearRecurrence[{3,-3,1,2,-6,6,-2,-1,3,-3,1},{1,3,6,10,17,27,40,56,78,106,140},50] (* Harvey P. Dale, Apr 12 2015 *)

a(n) = sum(A144678(n-k), k=0..n).
Gi2(n) = A189376(n-1) - A189376(n-2) - A189376(n-5) + 2*A189376(n-6) with A189376(n)=0 for n <= -1.
a(0)=1, a(1)=3, a(2)=6, a(3)=10, a(4)=17, a(5)=27, a(6)=40, a(7)=56, a(8)=78, a(9)=106, a(10)=140, a(n)=3*a(n-1)-3*a(n-2)+a(n-3)+ 2*a(n-4)- 6*a(n-5)+6*a(n-6)-2*a(n-7)-a(n-8)+3*a(n-9)-3*a(n-10)+a(n-11). - Harvey P. Dale, Apr 12 2015

cp's OEIS Frontend

A159797 Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n-1.

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

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A122046 Partial sums of floor(n^2/8).

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A190718 Quadruplicated tetrahedral numbers A000292.

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A144679 a(n) = [n/5 + 1]*[n/5 + 2]*(3*n - 10*[n/5] + 3)/6, where [.] = floor.

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A189376 Expansion of 1/((1-x)^5*(x^3+x^2+x+1)^2).

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A144679 a(n) = [n/5 + 1][n/5 + 2](3n - 10[n/5] + 3)/6, where [.] = floor.