A000537 -id:A000537 - cp's OEIS frontend

A159065 Number of crossings in a regular drawing of the complete bipartite graph K(n,n).

0, 1, 7, 27, 65, 147, 261, 461, 737, 1143, 1637, 2349, 3217, 4401, 5769, 7457, 9433, 11945, 14753, 18235, 22173, 26771, 31801, 37813, 44449, 52161, 60489, 69955, 80289, 92203, 104941, 119493, 135261, 152705, 171205, 191649, 213473, 237877

Offset: 1

Stéphane Legendre, Apr 04 2009, Jul 11 2009

			For n = 3 draw vertically 3 points regularly spaced on the right, and 3 points regularly spaced on the left. Join the left and right points by straight lines. These lines cross at c(3) = 7 points.

Umberto Eco, Foucault's Pendulum. San Diego: Harcourt Brace Jovanovich, p. 473, 1989.
Athanasius Kircher (1601-1680). Ars Magna Sciendi, In XII Libros Digesta, qua nova et universali Methodo Per Artificiosum Combinationum contextum de omni re proposita plurimis et prope infinitis rationibus disputari, omniumque summaria quaedam cognitio comparari potest, Amstelodami, Apud Joannem Janssonium a Waesberge, et Viduam Elizei Weyerstraet, 1669, fol., pp. 482 (altra ed.: Amstelodami.(ut supra), 1671).

N. J. A. Sloane, Table of n, a(n) for n = 1..1000 [First 500 terms from Indranil Ghosh]
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5, Lemma 2.
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Cf. A000537, A115004, A114999, A290131, A331755.

A159065 := proc(n)
    local a,b,c ;
    c := 0 ;
    for a from 1 to n-1 do
    for b from 1 to n-1 do
        if igcd(a,b) = 1 then
            c := c+(n-a)*(n-b) ;
            if 2*a< n and 2*b < n then
                c := c-(n-2*a)*(n-2*b) ;
            end if;
        end if;
    end do:
    end do:
    c ;
end proc:
seq(A159065(n),n=1..30); # R. J. Mathar, Jul 20 2017

a[n_] := Module[{x, y, s1 = 0, s2 = 0}, For[x = 1, x <= n-1, x++, For[y = 1, y <= n-1, y++, If[GCD[x, y] == 1, s1 += (n-x)*(n-y); If[2*x <= n-1 && 2*y <= n-1, s2 += (n-2*x)*(n-2*y)]]]]; s1-s2]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 10 2014, translated from Joerg Arndt's PARI code *)

a(n) = {
    my(s1=0, s2=0);
    for (x=1, n-1,
        for (y=1, n-1,
            if ( gcd(x, y)==1,
                s1 += (n-x) * (n-y);
                if ( ( 2*x<=n-1) && (2*y<=n-1),
                    s2 += (n-2*x) * (n-2*y); );
             );
        );
    );
    return( s1 - s2 );
}
\\ Joerg Arndt, Oct 13 2013

s1:=0; s2:=0;
for a:=1 to n-1 do
   for b:=1 to n-1 do
      if gcd(a, b)=1 then
      begin
         s1:=s1+(n-a)*(n-b);
         if (2*a<=n-1) and (2*b<=n-1) then
            s2:=s2+(n-2*a)*(n-2*b);
      end;
a:=s1-s2;

from math import gcd
def a159065(n):
    c=0
    for a in range(1, n):
        for b in range(1, n):
            if gcd(a, b)==1:
                c+=(n - a)*(n - b)
                if 2*aIndranil Ghosh, Jul 20 2017

from sympy import totient
def A159065(n): return n-1 if n <= 2 else 2*n-3+3*sum(totient(i)*(n-i)*i for i in range(2,(n+1)//2)) + sum(totient(i)*(n-i)*(2*n-i) for i in range((n+1)//2,n)) # Chai Wah Wu, Aug 16 2021

a(n) = Sum((n-a)*(n-b); 1<=a

a(n) = (9/(8*Pi^2))*n^4 + O(n^3 log(n)). Asymptotic to (9/(2*Pi^2))*A000537(n-1).

For n > 2: a(n) = A115004(n-1)-(n-2)^2-2*Sum{n=2..floor((n-1)/2)} (n-2i)*(n-i)*phi(i) = 2n-3+3*Sum{n=2..floor((n-1)/2)}(n-i)*i*phi(i) + Sum_{n=floor((n+1)/2)..n-1} (n-i)*(2n-i)*phi(i). - Chai Wah Wu, Aug 16 2021

A224012 T(n,k)=Number of nXk 0..2 arrays with rows nondecreasing and antidiagonals unimodal.

Original entry on oeis.org

3, 6, 9, 10, 36, 27, 15, 100, 216, 81, 21, 225, 868, 1296, 243, 28, 441, 2661, 7378, 7776, 729, 36, 784, 6815, 28541, 62764, 46656, 2187, 45, 1296, 15340, 90051, 297859, 534352, 279936, 6561, 55, 2025, 31324, 245055, 1108969, 3094127, 4549684, 1679616

Offset: 1

R. H. Hardin Mar 30 2013

Table starts
.....3........6.........10..........15...........21............28............36
.....9.......36........100.........225..........441...........784..........1296
....27......216........868........2661.........6815.........15340.........31324
....81.....1296.......7378.......28541........90051........245055........595822
...243.....7776......62764......297859......1108969.......3516324.......9866389
...729....46656.....534352.....3094127.....13275381......47735665.....150787422
..2187...279936....4549684....32148473....157347899.....630339756....2200064042
..6561..1679616...38737252...334179881...1859567103....8213689391...31256208954
.19683.10077696..329817976..3474343713..21962353421..106375878027..437370837827
.59049.60466176.2808146488.36122604265.259365424097.1373916879120.6067995150599

			Some solutions for n=3 k=4
..1..1..2..2....0..1..1..1....0..0..0..0....0..0..0..1....0..2..2..2
..0..2..2..2....0..1..1..1....0..1..2..2....2..2..2..2....0..0..1..1
..1..1..2..2....0..2..2..2....1..2..2..2....2..2..2..2....0..0..2..2

R. H. Hardin, Table of n, a(n) for n = 1..449

Column 1 is A000244
Column 2 is A000400
Row 1 is A000217(n+1)
Row 2 is A000537(n+1)

Empirical: columns k=1..7 have recurrences of order 1,1,5,7,11,14,19

Empirical: rows n=1..7 are polynomials of degree 2*n for k>0,0,1,2,3,4,5

A224353 T(n,k)=Number of nXk 0..2 arrays with diagonals and antidiagonals unimodal and rows nondecreasing.

Original entry on oeis.org

3, 6, 9, 10, 36, 27, 15, 100, 216, 81, 21, 225, 788, 1296, 243, 28, 441, 2321, 5880, 7776, 729, 36, 784, 5840, 19608, 45064, 46656, 2187, 45, 1296, 13052, 57387, 160362, 349280, 279936, 6561, 55, 2025, 26610, 151010, 495985, 1351748, 2710892, 1679616

Offset: 1

R. H. Hardin Apr 04 2013

Table starts
.....3........6.........10.........15..........21..........28...........36
.....9.......36........100........225.........441.........784.........1296
....27......216........788.......2321........5840.......13052........26610
....81.....1296.......5880......19608.......57387......151010.......363392
...243.....7776......45064.....160362......495985.....1421762......3816783
...729....46656.....349280....1351748.....4231138....12340932.....34697869
..2187...279936....2710892...11704964....37433596...107694133....300892325
..6561..1679616...21021916..102319662...342170839...977742699...2654062881
.19683.10077696..163012744..895494806..3178789749..9202126546..24422915139
.59049.60466176.1264202660.7833508842.29672959682.88363107023.233364588801

			Some solutions for n=3 k=4
..1..1..1..2....0..0..1..1....0..0..2..2....1..2..2..2....0..2..2..2
..1..1..1..2....0..0..2..2....0..1..1..2....0..1..2..2....1..1..2..2
..0..0..1..2....0..1..1..1....0..1..1..2....1..2..2..2....1..1..1..2

R. H. Hardin, Table of n, a(n) for n = 1..264

Column 1 is A000244
Column 2 is A000400
Row 1 is A000217(n+1)
Row 2 is A000537(n+1)

Empirical: columns k=1..6 have recurrences of order 1,1,10,26,56,98

Empirical: rows n=1..7 are polynomials of degree 2*n for k>0,0,1,3,5,7,9

A060300 a(n) = (2n(n+1))^2.

Original entry on oeis.org

0, 16, 144, 576, 1600, 3600, 7056, 12544, 20736, 32400, 48400, 69696, 97344, 132496, 176400, 230400, 295936, 374544, 467856, 577600, 705600, 853776, 1024144, 1218816, 1440000, 1690000, 1971216, 2286144, 2637376, 3027600, 3459600, 3936256, 4460544, 5035536, 5664400

Offset: 0

Jason Earls, Mar 25 2001

Arises from middle column 4^2, 12^2, 24^2, ... of following triangle: :
                        3^2 +  4^2 = 5^2
                10^2 + 11^2 + 12^2 = 13^2 + 14^2
         21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2
  36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2
...

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, pp. 90-92.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000537, A002378, A035287, A046092, A163102, A254371.

[(2*n*(n+1))^2: n in [0..30]]; // Vincenzo Librandi, Nov 18 2016

CoefficientList[Series[16 x (1 + 4 x + x^2) / (1 - x)^5, {x, 0, 33}], x] (* Vincenzo Librandi, Nov 18 2016 *)
Table[(2n(n+1))^2,{n,0,30}] (* Harvey P. Dale, Jan 19 2019 *)

a(n) = { (2*n*(n + 1))^2 } \\ Harry J. Smith, Jul 03 2009

G.f.: 16*x*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Apr 22 2012
a(n) = 4*A035287(n+1) = 4*A002378(n)^2. - Michel Marcus, May 24 2016
a(n) = 16*A000537(n) = 16*(n*(n+1)/2)^2 = 16*A000217(n)^2 = A046092(n)^2. - Bruce J. Nicholson, Jun 05 2017
a(n) = Integral_{x=1..2*n+1} (x^3-x) dx. - César Aguilera, Jun 27 2020
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: 4*x*(2 + x)*(2 + 6*x + x^2)*exp(x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 2*A254371(n) = 8*A163102(n). (End)

Name corrected by Harry J. Smith, Jul 03 2009

A085582 The number of rectangles (orthogonal or not) with corners on an n X n grid of points.

Original entry on oeis.org

0, 1, 10, 44, 130, 313, 640, 1192, 2044, 3305, 5078, 7524, 10750, 14993, 20388, 27128, 35448, 45665, 57922, 72636, 89970, 110297, 133976, 161440, 192860, 228857, 269758, 316012, 367974, 426417, 491468, 564120, 644640, 733633, 831674, 939292

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 06 2003

nonn

			a(3) = 10 because on the 3 X 3 grid there are four 1 X 1 rectangles, two 1 X 2s, two 2 X 1's, one 2 X 2 and one 45-degree rectangle, sqrt(2) X sqrt(2).

David Radcliffe, Table of n, a(n) for n = 1..1000
David Radcliffe, Python script to calculate a(n)

Cf. A000537, A002415, A113751 (diagonal rectangles on an n X n grid).

a(n) = A000537(n-1) + A113751(n). - T. D. Noe, Nov 09 2005 [corrected by David Radcliffe, Feb 06 2020]

a(n) = n*(n-1)^2*(2n-1)/6 + 2*Sum_{a,b>0, 0David Radcliffe, Feb 06 2020

Edited by Don Reble, Nov 05 2005

A094414 Triangle T read by rows: dot product <1,2,...,r> * .

Original entry on oeis.org

1, 5, 4, 14, 11, 11, 30, 24, 22, 24, 55, 45, 40, 40, 45, 91, 76, 67, 64, 67, 76, 140, 119, 105, 98, 98, 105, 119, 204, 176, 156, 144, 140, 144, 156, 176, 285, 249, 222, 204, 195, 195, 204, 222, 249, 385, 340, 305, 280, 265, 260, 265, 280, 305, 340, 506, 451, 407, 374, 352, 341, 341, 352, 374, 407, 451

Offset: 0

Ralf Stephan, May 02 2004

Offset for r (the rows) is 1, for s (the columns) it is 0.

			Triangle begins as:
   1;
   5,  4;
  14, 11, 11;
  30, 24, 22, 24;
  55, 45, 40, 40, 45;
  91, 76, 67, 64, 67, 76;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Columns 0-6 are A000330, A006527, A026037, A026040, A026043, A026046, A026049.
Row sums are A000537.
See also A094415, A088003.

Flat(List([0..12], n-> List([0..n-1], k-> n*((n+1)*(2*n+1) -3*k*(n-k))/6 ))); # G. C. Greubel, Oct 30 2019

[n*((n+1)*(2*n+1) -3*k*(n-k))/6: k in [0..n-1], n in [0..12]]; // G. C. Greubel, Oct 30 2019

T:=proc(r,s) if s>=r then 0 else r*(2*r^2+3*r+1-3*r*s+3*s^2)/6 fi end: for r from 1 to 11 do seq(T(r,s),s=0..r-1) od; # yields sequence in triangular form # Emeric Deutsch, Nov 27 2006

Table[n*((n+1)*(2*n+1) -3*k*(n-k))/6, {n,0,12}, {k,0,n-1}]//Flatten (* G. C. Greubel, Oct 30 2019 *)

T(n,k) = n*((n+1)*(2*n+1) -3*k*(n-k))/6;
for(n=0,12, for(k=0,n-1, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019

[[n*((n+1)*(2*n+1) -3*k*(n-k))/6 for k in (0..n-1)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

T(r, s) = r*(2*r^2 + 3*r - 3*r*s + 1 + 3*s^2)/6, r >= 1, 0 <= s <= r-1.

More terms from G. C. Greubel, Oct 30 2019

A224190 T(n,k) = Number of n X k 0..2 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

3, 9, 6, 22, 36, 10, 46, 158, 100, 15, 86, 548, 684, 225, 21, 148, 1600, 3526, 2205, 441, 28, 239, 4102, 14751, 15779, 5852, 784, 36, 367, 9503, 52591, 89380, 55438, 13524, 1296, 45, 541, 20299, 165212, 422488, 408222, 163746, 28176, 2025, 55, 771, 40570

Offset: 1

R. H. Hardin Apr 01 2013

Table starts
..3....9.....22......46.......86........148.........239..........367
..6...36....158.....548.....1600.......4102........9503........20299
.10..100....684....3526....14751......52591......165212.......468292
.15..225...2205...15779....89380.....422488.....1727738......6272940
.21..441...5852...55438...408222....2469182....12741432.....57644194
.28..784..13524..163746..1519738...11444292....72710554....400958714
.36.1296..28176..424326..4844576...44435746...340780382...2249643632
.45.2025..54153..992607.13669953..150015321..1366188661..10635858679
.55.3025..97570.2138488.34953776..452158538..4823267213..43724068755
.66.4356.166738.4305730.82399174.1240740774.15322738603.159999462711

			Some solutions for n=3, k=4
..1..2..2..1....0..0..2..0....1..2..1..1....0..1..0..0....1..0..0..0
..1..2..2..1....1..2..2..0....1..2..1..1....0..1..2..0....1..1..0..0
..2..2..2..2....1..2..2..1....2..2..1..1....0..2..2..0....1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..722

Column 1 is A000217(n+1).
Column 2 is A000537(n+1).
Row 1 is A223718.
Row 2 is A223919.
Row 3 is A223865.

Empirical: columns k=1..7 are polynomials of degree 2*k.

Empirical: rows n=1..7 are polynomials of degree 4*n.

A236770 a(n) = n(n + 1)(3n^2 + 3n - 2)/8.

Original entry on oeis.org

0, 1, 12, 51, 145, 330, 651, 1162, 1926, 3015, 4510, 6501, 9087, 12376, 16485, 21540, 27676, 35037, 43776, 54055, 66045, 79926, 95887, 114126, 134850, 158275, 184626, 214137, 247051, 283620, 324105, 368776, 417912, 471801, 530740, 595035, 665001, 740962

Offset: 0

Bruno Berselli, Jan 31 2014

After 0, first trisection of A011779 and right border of A177708.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A004188.
Cf. A000217, A000332, A011779, A177708.
Cf. similar sequences on the polygonal numbers: A002817(n) = A000217(A000217(n)); A000537(n) = A000290(A000217(n)); A037270(n) = A000217(A000290(n)); A062392(n) = A000384(A000217(n)).
Cf. sequences of the form A000217(m)+k*A000332(m+2): A062392 (k=12); A264854 (k=11); A264853 (k=10); this sequence (k=9); A006324 (k=8); A006323 (k=7); A000537 (k=6); A006322 (k=5); A006325 (k=4), A002817 (k=3), A006007 (k=2), A006522 (k=1).
Cf. A232713, A260810.

[n*(n+1)*(3*n^2+3*n-2)/8: n in [0..40]];

Table[n (n + 1) (3 n^2 + 3 n - 2)/8, {n, 0, 40}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,12,51,145},40] (* Harvey P. Dale, Aug 22 2016 *)

for(n=0, 40, print1(n*(n+1)*(3*n^2+3*n-2)/8", "));

G.f.: x*(1 + 7*x + x^2)/(1 - x)^5.
a(n) = a(-n-1) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A000326(A000217(n)).
a(n) = A000217(n) + 9*A000332(n+2).
Sum_{n>=1} 1/a(n) = 2 + 4*sqrt(3/11)*Pi*tan(sqrt(11/3)*Pi/2) = 1.11700627139319... . - Vaclav Kotesovec, Apr 27 2016

A094415 Triangle T read by rows: dot product * .

Original entry on oeis.org

1, 4, 5, 10, 13, 13, 20, 26, 28, 26, 35, 45, 50, 50, 45, 56, 71, 80, 83, 80, 71, 84, 105, 119, 126, 126, 119, 105, 120, 148, 168, 180, 184, 180, 168, 148, 165, 201, 228, 246, 255, 255, 246, 228, 201, 220, 265, 300, 325, 340, 345, 340, 325, 300, 265, 286, 341

Offset: 0

Ralf Stephan, May 02 2004

			Triangle begins as:
   1;
   4,  5;
  10, 13, 13;
  20, 26, 28, 26;
  35, 45, 50, 50, 45;
  56, 71, 80, 83, 80, 71;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Columns 0-6 are A000292, A008778, A026054, A026057, A026060, A026063, A026066.
Half-diagonal is A050410.
Row sums are A000537.
See also A094414, A088003.

Flat(List([0..12], n-> List([0..n], k-> (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 ))); # G. C. Greubel, Oct 30 2019

[(n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6: k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019

seq(seq( (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 , k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019

Table[(n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 30 2019 *)

T(n,k) = (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6;
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019

[[(n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

T(n, k) = n*(n^2 + 3*n*(1+k) + 2 - 3*k^2)/6 for n >= 0, 0 <= k <= n.

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cp's OEIS Frontend

A224262 T(n,k) = number of n X k 0..2 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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A159065 Number of crossings in a regular drawing of the complete bipartite graph K(n,n).

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A224012 T(n,k)=Number of nXk 0..2 arrays with rows nondecreasing and antidiagonals unimodal.

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A224353 T(n,k)=Number of nXk 0..2 arrays with diagonals and antidiagonals unimodal and rows nondecreasing.

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A060300 a(n) = (2*n*(n+1))^2.

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Magma

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A085582 The number of rectangles (orthogonal or not) with corners on an n X n grid of points.

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A094414 Triangle T read by rows: dot product <1,2,...,r> * .

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GAP

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A060300 a(n) = (2n(n+1))^2.

A236770 a(n) = n(n + 1)(3n^2 + 3n - 2)/8.