A033437 -id:A033437 - cp's OEIS frontend

A033440 Number of edges in 8-partite Turán graph of order n.

0, 0, 1, 3, 6, 10, 15, 21, 28, 35, 43, 52, 62, 73, 85, 98, 112, 126, 141, 157, 174, 192, 211, 231, 252, 273, 295, 318, 342, 367, 393, 420, 448, 476, 505, 535, 566, 598, 631, 665, 700, 735, 771, 808, 846, 885, 925

Offset: 0

N. J. A. Sloane

Graham et al., Handbook of Combinatorics, Vol. 2, p. 1234.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Turán Graph [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033437, A033438, A033439, A033441, A033442, A033443, A033444. [Reinhard Zumkeller, Nov 30 2009]

CoefficientList[Series[- x^2 (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)/((x - 1)^3 (x + 1) (x^2 + 1) (x^4 + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 19 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1},{0,0,1,3,6,10,15,21,28,35},50] (* Harvey P. Dale, Mar 23 2015 *)

a(n) = round( (7/16)*n(n-2) ) +0 or -1 depending on n: if there is k such 8k+4<=n<=8k+6 then a(n) = floor( (7/16)*n*(n-2)) otherwise a(n) = round( (7/16)*n(n-2)). E.g. because 8*2+4<=21<=8*2+6 a(n) = floor((7/16)*21*19) = floor(174, 5625)=174. - Benoit Cloitre, Jan 17 2002
a(n) = Sum_{k=0..n} A168181(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]
G.f.: -x^2*(x^6+x^5+x^4+x^3+x^2+x+1)/((x-1)^3*(x+1)*(x^2+1)*(x^4+1)). [Colin Barker, Aug 09 2012]
a(n) = Sum_{i=1..n} floor(7*i/8). - Wesley Ivan Hurt, Sep 12 2017

A033442 Number of edges in 10-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 54, 64, 75, 87, 100, 114, 129, 145, 162, 180, 198, 217, 237, 258, 280, 303, 327, 352, 378, 405, 432, 460, 489, 519, 550, 582, 615, 649, 684, 720, 756, 793, 831, 870, 910, 951, 993, 1036, 1080, 1125, 1170, 1216, 1263

Offset: 0

N. J. A. Sloane

Graham et al., Handbook of Combinatorics, Vol. 2, p. 1234.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Turán Graph [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033437, A033438, A033439, A033440, A033441, A033443, A033444. [Reinhard Zumkeller, Nov 30 2009]

CoefficientList[Series[- x^2 (x^2 + x + 1) (x^6 + x^3 + 1)/((x - 1)^3 (x + 1) (x^4 - x^3 + x^2 - x + 1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 20 2013 *)

a(n) = Sum_{k=0..n} A168184(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]
G.f.: -x^2*(x^2+x+1)*(x^6+x^3+1)/((x-1)^3*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)). [Colin Barker, Aug 09 2012]
a(n) = Sum_{i=1..n} floor(9*i/10). - Wesley Ivan Hurt, Sep 12 2017

More terms from Vincenzo Librandi, Oct 20 2013

A033443 Number of edges in 11-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 65, 76, 88, 101, 115, 130, 146, 163, 181, 200, 220, 240, 261, 283, 306, 330, 355, 381, 408, 436, 465, 495, 525, 556, 588, 621, 655, 690, 726, 763, 801, 840, 880, 920, 961, 1003, 1046, 1090, 1135, 1181, 1228, 1276

Offset: 0

N. J. A. Sloane

Graham et al., Handbook of Combinatorics, Vol. 2, p. 1234.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Turán Graph [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033437, A033438, A033439, A033440, A033441, A033442, A033444. [Reinhard Zumkeller, Nov 30 2009]

CoefficientList[Series[- x^2 (x + 1) (x^4 - x^3 + x^2 - x + 1) (x^4 + x^3 + x^2 + x + 1)/((x - 1)^3 (x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 20 2013 *)

a(n) = Sum_{k=0..n} A145568(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]
G.f.: -x^2*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)/((x-1)^3*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)). [Colin Barker, Aug 09 2012]
a(n) = Sum_{i=1..n} floor(10*i/11). - Wesley Ivan Hurt, Sep 12 2017

More terms from Vincenzo Librandi, Oct 20 2013

A279169 a(n) = floor( 4*n^2/5 ).

Original entry on oeis.org

0, 0, 3, 7, 12, 20, 28, 39, 51, 64, 80, 96, 115, 135, 156, 180, 204, 231, 259, 288, 320, 352, 387, 423, 460, 500, 540, 583, 627, 672, 720, 768, 819, 871, 924, 980, 1036, 1095, 1155, 1216, 1280, 1344, 1411, 1479, 1548, 1620, 1692, 1767, 1843, 1920, 2000, 2080, 2163, 2247

Offset: 0

Bruno Berselli, Dec 07 2016

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A090223: floor(4*n/5).
Subsequence of A008728, A014601, A118015, A131242.
Cf. similar sequences with closed form floor(k*n^2/5): A118015 (k=1), A033437 (k=2), A184535 (k=3).

Magma
```
[4*n^2 div 5: n in [0..60]];
```

Table[Floor[4 n^2/5], {n, 0, 60}]
LinearRecurrence[{2,-1,0,0,1,-2,1},{0,0,3,7,12,20,28},60] (* Harvey P. Dale, Nov 07 2020 *)

PARI
```
vector(60, n, n--; floor(4*n^2/5))
```
Python
```
[int(4*n**2/5) for n in range(60)]
```
Sage
```
[floor(4*n^2/5) for n in range(60)]
```

O.g.f.: x^2*(3 + x + x^2 + 3*x^3)/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).
a(5*m+r) = 4*m*(5*m + 2*r) + a(r), where m >= 0 and 0 <= r < 5. Example: for m=4 and r=3, a(5*4+3) = a(23) = 4*4*(5*4 + 2*3) + a(3) = 416 + 7 = 423.
a(n) = A118015(2*n) = A008728(4*n+2) = A131242(4*n+4) = A014601(floor(2*n^2/5)).
Sum_{n>=2} 1/a(n) = Pi^2/120 + sqrt(29 - 62/sqrt(5))*Pi/8 + 5/16. - Amiram Eldar, Sep 26 2022

A227353 Number of lattice points in the closed region bounded by the graphs of y = 3*x/5, x = n, and y = 0, excluding points on the x-axis.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 14, 18, 23, 29, 35, 42, 49, 57, 66, 75, 85, 95, 106, 118, 130, 143, 156, 170, 185, 200, 216, 232, 249, 267, 285, 304, 323, 343, 364, 385, 407, 429, 452, 476, 500, 525, 550, 576, 603, 630, 658, 686, 715, 745, 775, 806, 837, 869, 902, 935

Offset: 1

Clark Kimberling, Jul 08 2013

See A227347.

			a(1) = floor(3/5) = 0; a(2) = floor(6/5) = 1; a(3) = a(2) + floor(9/5) = 2; a(4) = a(3) + floor(12/5) = 4.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A227347, A130520, A011858, A033437.

Cf. A057355 (first differences).

z = 150; r = 3/5; k = 1; a[n_] := Sum[Floor[r*x^k], {x, 1, n}]; t = Table[a[n], {n, 1, z}]

a(n) = (3*n^2-n)\10; \\ Kevin Ryde, Mar 15 2022

a = lambda n: n*(3*n-1)//10 # Gennady Eremin, Mar 20 2022

a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).
G.f.: (x*(1 + x^2 + x^3))/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
According to Wolfram Alpha, a(n) = floor(Re(E(n^2|Pi))) where E(x|m) is the incomplete elliptic integral of the second kind. - Kritsada Moomuang, Jan 28 2022
a(n) = a(n-1) + floor(3*n/5), n > 1. - Gennady Eremin, Mar 15 2022
a(n) = floor(n*(3*n-1)/10). - Kevin Ryde, Mar 15 2022

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A033440 Number of edges in 8-partite Turán graph of order n.

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A033442 Number of edges in 10-partite Turán graph of order n.

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A033443 Number of edges in 11-partite Turán graph of order n.

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A279169 a(n) = floor( 4*n^2/5 ).

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A227353 Number of lattice points in the closed region bounded by the graphs of y = 3*x/5, x = n, and y = 0, excluding points on the x-axis.

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