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 1-5 of 5 results.

A014409 Number of inequivalent ways (mod D_4) a pair of checkers can be placed on an n X n board.

Original entry on oeis.org

0, 2, 8, 21, 49, 93, 171, 278, 446, 660, 970, 1347, 1863, 2471, 3269, 4188, 5356, 6678, 8316, 10145, 12365, 14817, 17743, 20946, 24714, 28808, 33566, 38703, 44611, 50955, 58185, 65912, 74648, 83946, 94384, 105453, 117801, 130853, 145331, 160590, 177430, 195132
Offset: 1

Views

Author

Borghard, William (bogey(AT)hostare.att.com)

Keywords

References

  • Computed by Fred Hallden.

Links

Crossrefs

Cf. A054252, A019318, A082966, A242279, A242358, A054247.

Programs

  • Magma
    [(2*n^4+14*n^2-12*n-1-(-1)^n*(2*n^2-4*n-1))/32 : n in [1..60]]; // Wesley Ivan Hurt, Dec 30 2023
  • Mathematica
    LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 2, 8, 21, 49, 93, 171, 278}, 40]
CoefficientList[Series[- x (x^5 + x^4 + 3 x^3 + x^2 + 4 x + 2)/((x - 1)^5 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2013 *)
  • PARI
    a(n)=if(n%2, n^4 + 8*n^2 - 8*n - 1, n^4 + 6*n^2 - 4*n)/16  \\ Charles R Greathouse IV, Feb 09 2017

Formula

a(2*n) = n/2*(2*n^3 + 3*n - 1); a(2*n+1) = n/2*(2*n^3 + 4*n^2 + 7*n + 3).
a(0)=0, a(1)=2, a(2)=8, a(3)=21, a(4)=49, a(5)=93, a(6)=171, a(7)=278, a(n)=2*a(n-1)+2*a(n-2)-6*a(n-3)+0*a(n-4)+6*a(n-5)-2*a(n-6)- 2*a(n-7)+ a(n-8). - Harvey P. Dale, May 06 2012
G.f.: -x^2*(x^5+x^4+3*x^3+x^2+4*x+2) / ((x-1)^5*(x+1)^3). - Colin Barker, Jul 11 2013
From James Stein, May 22 2014: (Start)
For odd n: a(n) = (n^4 + 8*n^2 - 8*n - 1)/16;
For even n: a(n) = n*(n^3 + 6*n - 4)/16. (End)
a(n) = A054252(n, 2), n >= 0. - Wolfdieter Lang, Oct 03 2016
E.g.f.: (x*(1 + 13*x + 6*x^2 + x^3)*cosh(x) + (-1 + 3*x + 15*x^2 + 6*x^3 + x^4)*sinh(x))/16. - Stefano Spezia, Apr 14 2022
a(n) = (2*n^4+14*n^2-12*n-1-(-1)^n*(2*n^2-4*n-1))/32. - Wesley Ivan Hurt, Dec 30 2023

Extensions

More terms and formula from Hugo van der Sanden
More terms from Colin Barker, Jul 11 2013

A094170 Number of quasi-triominoes in an n X n bounding box.

Original entry on oeis.org

0, 0, 1, 10, 33, 88, 187, 360, 625, 1024, 1581, 2350, 3361, 4680, 6343, 8428, 10977, 14080, 17785, 22194, 27361, 33400, 40371, 48400, 57553, 67968, 79717, 92950, 107745, 124264, 142591, 162900, 185281, 209920, 236913, 266458, 298657, 333720, 371755, 412984, 457521
Offset: 0

Views

Author

Jon Wild, May 07 2004

Keywords

Comments

A quasi-polyomino is a polyomino whose cells are not necessarily connected. For all m > 1 there are an infinite number of quasi-m-ominoes; a(n) counts the quasi-triomino (quasi-3-omino) equivalence classes (under translation, rotation by 90 degrees and vertical and horizontal symmetry) whose members fit into an n X n bounding box.
This is different from A082966 because that sequence considers these two (for example) as different ways of placing 3 counters on a 3 X 3 checkerboard:
---
-X-
X-X
and
-X-
X-X
---
whereas here they are the same quasi-polyomino.
a(n) can also be interpreted as the number of non-equivalent Game of Life patterns on an n X n board that have exactly 3 live cells, etc.

Examples

			Illustration of a(3), the 10 quasi-triominoes that fit into a 3 X 3 bounding box:
XXX -XX XX- X-X X-X XX- X-X X-X X-- X--
--- -X- --X X-- -X- --- --- --- -X- --X
--- --- --- --- --- --X X-- -X- --X -X-

Links

Crossrefs

Cf. A094171, A094172.

Programs

  • Mathematica
    CoefficientList[Series[x^2 (x^5 + x^4 + 4 x^3 + 4 x^2 + 7 x + 1)/((1 - x)^5 (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 17 2014 *)
  • PARI
    Vec(-x^2*(x^5+x^4+4*x^3+4*x^2+7*x+1)/((x-1)^5*(x+1)^2) + O(x^100)) \\ Colin Barker, Feb 16 2014

Formula

a(n) = (1/32)*(6*n^4 - 12*n^3 + 32*n^2 - 58*n + 29 - (6*n-3)*(-1)^n). - Ralf Stephan, Dec 03 2004
G.f.: -x^2*(x^5+x^4+4*x^3+4*x^2+7*x+1) / ((x-1)^5*(x+1)^2). - Colin Barker, Feb 15 2014

Extensions

Corrected and extended by Jon Wild, May 11 2004
More terms from Colin Barker, Feb 16 2014

A230723 Number of non-equivalent ways to choose three points in an equilateral triangle grid of side n.

Original entry on oeis.org

0, 1, 6, 25, 87, 238, 575, 1228, 2425, 4446, 7734, 12806, 20422, 31444, 47072, 68639, 97929, 136893, 188061, 254170, 338679, 445297, 578616, 743524, 945968, 1192243, 1489894, 1846869, 2272575, 2776880, 3371335, 4068016, 4880921, 5824640, 6915942, 8172258, 9613470
Offset: 1

Views

Author

Heinrich Ludwig, Oct 28 2013

Keywords

Examples

			for n = 3 there are the following a(3) = 6 choices of 3 points (=X) (rotations and reflections ignored):
    X         .         .         X         .         X
   . .       X X       . .       X X       . X       X .
  X . X     . X .     X X X     . . .     X . X     . X .

Links

Crossrefs

Cf. A001399, A014409, A082966, A227327.

Programs

  • Mathematica
    LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{0,1,6,25,87,238,575,1228,2425,4446,7734,12806},40] (* Harvey P. Dale, Oct 24 2020 *)

Formula

a(n) = (n^6 + 3*n^5 - 3*n^4 + 10*n^3 + B + C)/288
where
B = 27*n^2 + 3*n - 9 if n odd
B = 48*n otherwise
and
C = -32 if n == 1 (mod 3)
C = 0 otherwise
G.f.: x^2*(1 + 3*x + 7*x^2 + 19*x^3 + 16*x^4 + 12*x^5 + x^6 + 2*x^7 - x^8)/((1-x^3) * (1-x^2)^3 * (1-x)^3). - Ralf Stephan, Nov 03 2013

A242279 Number of inequivalent (mod D_4) ways four checkers can be placed on an n X n board.

Original entry on oeis.org

1, 23, 252, 1666, 7509, 26865, 79920, 209096, 491425, 1064575, 2150076, 4104738, 7458437, 13005041, 21857984, 35598880, 56353185, 87019191, 131364700, 194364050, 282314901, 403316353, 567402672, 787201416, 1078078209, 1459020095, 1952782300, 2587048786, 3394568325
Offset: 2

Views

Author

Heinrich Ludwig, May 10 2014

Keywords

Links

Crossrefs

Cf. A054252, A008805, A014409, A082966, A242358, A019318, A054772.

Programs

  • Mathematica
    CoefficientList[Series[x^2*(1 + 19*x + 161*x^2 + 697*x^3 + 1446*x^4 + 2070*x^5 + 1422*x^6 + 766*x^7 + 105*x^8 + 31*x^9 + x^10 + x^11) / ((1-x)^9 * (1+x)^5), {x, 0, 20}], x] (* Vaclav Kotesovec, May 10 2014 *)
LinearRecurrence[{4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1},{0,0,1,23,252,1666,7509,26865,79920,209096,491425,1064575,2150076,4104738},40] (* Harvey P. Dale, May 06 2018 *)

Formula

a(n) = (n^8 - 6*n^6 + 40*n^4 - 48*n^3 + 16*n^2 + IF(MOD(n, 2) = 1)*(14*n^4 - 48*n^3 + 34*n^2 - 3))/192.
G.f.: x^2*(1 + 19*x + 161*x^2 + 697*x^3 + 1446*x^4 + 2070*x^5 + 1422*x^6 + 766*x^7 + 105*x^8 + 31*x^9 + x^10 + x^11) / ((1-x)^9 * (1+x)^5). - Vaclav Kotesovec, May 10 2014
a(n) = A054772(n, 4), n >= 2. - Wolfdieter Lang, Oct 03 2016

A242358 Number of inequivalent (mod D_4) ways five checkers can be placed on an n X n board.

Original entry on oeis.org

23, 567, 6814, 47358, 239511, 954226, 3207212, 9414828, 24862239, 60136329, 135311658, 286229762, 574460495, 1101240084, 2028333848, 3605765688, 6211552455, 10402472811, 16984387958, 27099325638, 42342870823, 64905898662, 97761436356, 144885584740, 211543443215
Offset: 3

Views

Author

Heinrich Ludwig, May 11 2014

Keywords

Links

Crossrefs

Cf. A054252, A008805, A014409, A082966, A242279, A054772.

Programs

  • Mathematica
    Drop[CoefficientList[Series[x^3*(-23 - 452*x - 4071*x^2 - 16016*x^3 - 40397*x^4 - 59335*x^5 - 61954*x^6 - 38236*x^7 - 17221*x^8 - 3614*x^9 - 623*x^10 + 20*x^11 + x^12 + x^13)/((x-1)^11*(x+1)^6), {x, 0, 20}], x],3] (* Vaclav Kotesovec, May 11 2014 *)

Formula

a(n) = (n^10 - 10*n^8 + 35*n^6 + 52*n^5 - 210*n^4 + 140*n^3 - 56*n^2 + 48*n + IF(MOD(n, 2) = 1)*(52*n^5 - 145*n^4 + 140*n^3 - 80*n^2 + 48*n - 15))/960.
G.f.: x^3*(-23 - 452*x - 4071*x^2 - 16016*x^3 - 40397*x^4 - 59335*x^5 - 61954*x^6 - 38236*x^7 - 17221*x^8 - 3614*x^9 - 623*x^10 + 20*x^11 + x^12 + x^13)/((x-1)^11*(x+1)^6). - Vaclav Kotesovec, May 11 2014
a(n) = A054772(n, 5), n >=3. - Wolfdieter Lang, Oct 03 2016
Showing 1-5 of 5 results.

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

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