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.

Showing 1-3 of 3 results.

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

A274221 List of quadruples: 3*n*(3*n-1), 3*n*(3*n+1), (3*n+1)^2, (3*n+2)^2.

Original entry on oeis.org

0, 0, 1, 4, 6, 12, 16, 25, 30, 42, 49, 64, 72, 90, 100, 121, 132, 156, 169, 196, 210, 240, 256, 289, 306, 342, 361, 400, 420, 462, 484, 529, 552, 600, 625, 676, 702, 756, 784, 841, 870, 930, 961, 1024, 1056, 1122, 1156, 1225, 1260, 1332, 1369, 1444, 1482
Offset: 0

Views

Author

Luce ETIENNE, Sep 14 2016

Keywords

Comments

For the formulae of the permutations of A152743, A045945, A016778 and A016790, see the link.

Links

Crossrefs

Cf. A152743, A045945, A016778, A016790, A064412, A269064.

Programs

  • Magma
    &cat [[3*n*(3*n-1), 3*n*(3*n+1), (3*n+1)^2, (3*n+2)^2]: n in [0..15]]; // Bruno Berselli, Sep 15 2016
  • Mathematica
    Flatten[Table[{3 n (3 n - 1), 3 n (3 n + 1), (3 n + 1)^2, (3 n + 2)^2}, {n, 0, 15}]] (* Bruno Berselli, Sep 15 2016 *)

Formula

G.f.: x^2*(1+3*x+x^2+3*x^3+x^4)/((1-x)^3*(1+x)^2*(1+x^2)). - Robert Israel, Sep 15 2016
a(n) = (18*n^2-18*n+1-3*(2*n-1)*(-1)^n-4*(-1)^((2*n-1+(-1)^n)/4))/32. Therefore: a(2k) = (18*k^2-12*k+1-(-1)^k)/8, a(2k+1) = (18*k^2+12*k+1-(-1)^k)/8.
a(n) = A064412(n) - A269064(n) for n>0.
E.g.f.: ((9*x^2 - 3*x - 1)*sinh(x) + (9*x^2 + 3*x + 2)*cosh(x) - 2*(sin(x) + cos(x)))/16. - Stefano Spezia, Nov 07 2022

A280304 a(n) = 3*n*(n^2 + 3*n + 4).

Original entry on oeis.org

0, 24, 84, 198, 384, 660, 1044, 1554, 2208, 3024, 4020, 5214, 6624, 8268, 10164, 12330, 14784, 17544, 20628, 24054, 27840, 32004, 36564, 41538, 46944, 52800, 59124, 65934, 73248, 81084, 89460, 98394, 107904, 118008, 128724, 140070, 152064, 164724, 178068, 192114, 206880, 222384, 238644, 255678, 273504, 292140, 311604, 331914, 353088, 375144, 398100
Offset: 0

Views

Author

Luce ETIENNE, Dec 31 2016

Keywords

Comments

Numbers of unit triangles in a certain structure obtained from A006003.

Examples

			a(0) = 6*(1-1) = 0, a(1) = 6*(5-1) = 24, a(2) = 6*(15-1) = 84, a(3) = 6*(34-1) = 198, a(4) = 6*(65-1) = 384.

Links

Crossrefs

Cf. A003215, A005448, A006003, A033428, A213389, A269064.

Programs

  • Magma
    [3*n*(n^2 + 3*n + 4) : n in [0..60]]; // Wesley Ivan Hurt, Dec 31 2016
  • Maple
    A280304:=n->3*n*(n^2 + 3*n + 4): seq(A280304(n), n=0..60); # Wesley Ivan Hurt, Dec 31 2016
  • Mathematica
    Table[3 n (n^2 + 3 n + 4), {n, 0, 50}] (* or *)
CoefficientList[Series[6 x (x^2 - 2 x + 4)/(1 - x)^4, {x, 0, 50}], x] (* Michael De Vlieger, Dec 31 2016 *)
LinearRecurrence[{4,-6,4,-1},{0,24,84,198},60] (* Harvey P. Dale, Feb 08 2023 *)
  • PARI
    concat(0, Vec(6*x*(x^2-2*x+4) / (1-x)^4 + O(x^30))) \\ Colin Barker, Dec 31 2016

Formula

G.f.: 6*x*(x^2-2*x+4) / (1-x)^4.
a(n) = 6*(A006003(n+1)-1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, Dec 31 2016
Showing 1-3 of 3 results.

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