cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

A033444 Number of edges in 12-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 77, 89, 102, 116, 131, 147, 164, 182, 201, 221, 242, 264, 286, 309, 333, 358, 384, 411, 439, 468, 498, 529, 561, 594, 627, 661, 696, 732, 769, 807, 846, 886, 927, 969, 1012, 1056, 1100, 1145, 1191, 1238, 1286
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A002620, A000212, A033436, A033437, A033438, A033439, A033440, A033441, A033442, A033443. [Reinhard Zumkeller, Nov 30 2009]

Programs

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

Formula

a(n) = Sum_{k=0..n} A168185(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]
G.f.: -x^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)/((x-1)^3*(x+1)*(x^2-x+1)*(x^2+1)*(x^2+x+1)*(x^4-x^2+1)). [Colin Barker, Aug 09 2012]
a(n) = Sum_{i=1..n} floor(11*i/12). - Wesley Ivan Hurt, Sep 12 2017

Extensions

More terms from Vincenzo Librandi, Oct 20 2013

A033441 Number of edges in 9-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 44, 53, 63, 74, 86, 99, 113, 128, 144, 160, 177, 195, 214, 234, 255, 277, 300, 324, 348, 373, 399, 426, 454, 483, 513, 544, 576, 608, 641, 675, 710, 746, 783, 821, 860, 900, 940, 981, 1023, 1066, 1110, 1155, 1201, 1248, 1296
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A002620, A000212, A033436 - A033444. - Reinhard Zumkeller, Nov 30 2009

Programs

  • Mathematica
    CoefficientList[Series[- x^2 (x + 1) (x^2 + 1) (x^4 + 1)/((x - 1)^3 (x^2 + x + 1) (x^6 + x^3 + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 20 2013 *)
LinearRecurrence[{2, -1, 0, 0, 0, 0, 0, 0, 1, -2, 1},{0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 44},55] (* Ray Chandler, Aug 04 2015 *)

Formula

G.f.: x*(1/(1-x) - 1/(1-x^9))/(1-x)^2. - Ralf Stephan, Mar 05 2004
a(n) = Sum_{k=0..n} A168182(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009
G.f.: -x^2*(x+1)*(x^2+1)*(x^4+1)/((x-1)^3*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Aug 09 2012
a(n) = Sum_{i=1..n} floor(8*i/9). - Wesley Ivan Hurt, Sep 12 2017

Extensions

More terms from Vincenzo Librandi, Oct 20 2013

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

More terms from Vincenzo Librandi, Oct 20 2013

A282513 a(n) = floor((3*n + 2)^2/24 + 1/3).

Original entry on oeis.org

0, 1, 3, 5, 8, 12, 17, 22, 28, 35, 43, 51, 60, 70, 81, 92, 104, 117, 131, 145, 160, 176, 193, 210, 228, 247, 267, 287, 308, 330, 353, 376, 400, 425, 451, 477, 504, 532, 561, 590, 620, 651, 683, 715, 748, 782, 817, 852, 888, 925, 963
Offset: 0

Views

Author

Luce ETIENNE, Feb 17 2017

Keywords

Comments

List of quadruples: 2*n*(3*n+1), (2*n+1)*(3*n+1), 6*n^2+8*n+3, (n+1)*(6*n+5). These terms belong to the sequences A033580, A033570, A126587 and A049452, respectively. See links for all the permutations.
After 0, subsequence of A025767.
It seems that a(n) is the smallest number of cells that need to be painted in a (n+1) X (n+1) grid, such that it has no unpainted hexominoes (see link to Kamenetsky and Pratt). - Rob Pratt, Dmitry Kamenetsky, Aug 30 2020

Examples

			Rectangular array with four columns:
.   0,   1,   3,   5;
.   8,  12,  17,  22;
.  28,  35,  43,  51;
.  60,  70,  81,  92;
. 104, 117, 131, 145, etc.
From _Rob Pratt_, Aug 30 2020: (Start)
For n = 3, painting only 2 cells would leave an unpainted hexomino, but painting the following 3 cells avoids all unpainted hexominoes:
    . . .
    . . X
    X X .
(End)

Links

Crossrefs

Cf. A033436: floor((3*n)^2/24 + 1/3).
Cf. A000326, A000567, A025767, A033570, A033580, A049452, A064412, A126587, A222017, A269064, A274221.
Cf. A130519.
Minimum number of painted cells in other n-ominoes: A337501, A337502, A337503.

Programs

  • Magma
    [(3*n^2+4*n+4) div 8: n in [0..50]]; // Bruno Berselli, Feb 17 2017
  • Mathematica
    Table[Floor[(3 n + 2)^2/24 + 1/3], {n, 0, 50}] (* or *) CoefficientList[Series[x (1 + x + x^3)/((1 + x) (1 + x^2) (1 - x)^3), {x, 0, 50}], x] (* or *) Table[(6 n^2 + 8 n + 3 + Cos[n Pi] - 4 Cos[n Pi/2])/16, {n, 0, 50}] (* or *) Table[(3 n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n) (1 + 2 I^((n + 1) (n + 2))))/16, {n, 0, 50}] (* Michael De Vlieger, Feb 17 2017 *)
LinearRecurrence[{2,-1,0,1,-2,1},{0,1,3,5,8,12},60] (* Harvey P. Dale, Aug 10 2024 *)
  • PARI
    a(n)=(3*n^2 + 4*n + 4)\8 \\ Charles R Greathouse IV, Feb 17 2017

Formula

G.f.: x*(1 + x + x^3)/((1 + x)*(1 + x^2)*(1 - x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>5.
a(n) = floor((3*n + 2)^2/24 + 2/3).
a(n) = (6*n^2 + 8*n + 3 + (-1)^n - 2*((-1)^((2*n - 1 + (-1)^n)/4) + (-1)^((2*n + 1 - (-1)^n)/4)))/16. Therefore:
a(2*k) = (6*k^2 + 4*k + 1 - (-1)^k)/4,
a(2*k+1) = (k + 1)*(3*k + 2)/2.
a(n) = (6*n^2 + 8*n + 3 + cos(n*Pi) - 4*cos(n*Pi/2))/16.
a(n) = (3*n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n)*(1 + 2*i^((n+1)*(n+2))))/16, where i=sqrt(-1).
a(n) = A130519(n+3)+A130519(n+2)+A130519(n). - R. J. Mathar, Jun 23 2021

Extensions

Corrected and extended by Bruno Berselli, Feb 17 2017
Previous Showing 11-16 of 16 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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