A014409 -id:A014409 - cp's OEIS frontend

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

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

Heinrich Ludwig, Oct 28 2013

			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 .

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1)

Cf. A001399, A014409, A082966, A227327.

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 *)

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

Heinrich Ludwig, May 10 2014

Heinrich Ludwig, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1).

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

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 *)

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

Heinrich Ludwig, May 11 2014

Heinrich Ludwig, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

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

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 *)

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

A242709 Nonequivalent ways to place two different markers (e.g., a pair of Go stones, black and white) on an n X n grid.

Original entry on oeis.org

0, 2, 12, 33, 85, 165, 315, 518, 846, 1260, 1870, 2607, 3627, 4823, 6405, 8220, 10540, 13158, 16416, 20045, 24465, 29337, 35167, 41538, 49050, 57200, 66690, 76923, 88711, 101355, 115785, 131192, 148632, 167178, 188020, 210105, 234765, 260813, 289731, 320190

Offset: 1

James Stein, May 21 2014

We say two placements are equivalent if one can be obtained from the other by rotating or reflecting the grid.

The formula was derived by categorizing and counting grid cells into four exclusive categories: central cell (if any); other diagonal cells, other horizontal and vertical midline cells (if any), and all others (in eight triangular regions) (if any); then determining for each category, how many ways a white stone could be placed in each category, given the category in which the black stone was placed prior. The sequence was verified by another program which generated all positions, removed reflections and rotations, and tallied the residue.

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A014409 (with indistinguishable checkers)

[n*(n^3 + n*3^(n mod 2) - 2*2^(n mod 2))/8: n in [1..50]]; // Wesley Ivan Hurt, May 21 2014

A242709:=n->n*(n^3 + n*3^(n mod 2) - 2*2^(n mod 2))/8; seq(A242709(n), n=1..50); # Wesley Ivan Hurt, May 21 2014

f[n_] := If[OddQ[n], (n^3 + 3 n - 4), (n^3 + n - 2)] n/8;
Table[f[n], {n, 1, 40}]

concat(0, Vec(-x^2*(x^5+x^4+7*x^3+5*x^2+8*x+2)/((x-1)^5*(x+1)^3) + O(x^100))) \\ Colin Barker, May 21 2014

For odd n, a(n) = n*(n^3 + 3*n - 4)/8.
For even n, a(n) = n*(n^3 + n - 2)/8.
G.f.: -x^2*(x^5+x^4+7*x^3+5*x^2+8*x+2) / ((x-1)^5*(x+1)^3). - Colin Barker, May 21 2014
a(n) = n*(n^3 + n*3^(n mod 2) - 2*2^(n mod 2))/8. - Wesley Ivan Hurt, May 21 2014

A296999 Number of nonequivalent (mod D_8) ways to place 4 points on an n X n point grid so that no point is equally distant from two other points on the same row or the same column.

Original entry on oeis.org

0, 1, 17, 226, 1550, 7221, 26120, 78484, 206242, 486640, 1056377, 2137506, 4085167, 7430276, 12964014, 21801632, 35520743, 56249658, 86880957, 131186720, 194133425, 282024809, 402949496, 566950056, 786640454, 1077397347, 1458190435, 1951789266, 2585856152, 3393157995

Offset: 1

Heinrich Ludwig, Jan 21 2018

Rotations and reflections of placements are not counted. If they are to be counted see A296998.

The condition of placements is also known as "no 3-term arithmetic progressions".

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-4,4,-4,5,1,-5,6,-10,8,-8,10,-6,5,-1,-5,4,-4,4,1,-3,1).

Cf. A014409, A296996, A296998.

Array[(#^8 - 6 #^6 - 12 #^5 + 64 #^4 + 8 #^3 - 136 #^2 + Boole[OddQ@ #] (14 #^4 - 96 #^3 + 162 #^2 - 92 # + 93))/192 + Boole[Mod[#, 6] == 2] #/6 + Boole[Mod[#, 4] == 2] #/4 + Boole[Mod[#, 6] == 5] (# + 1)/6 &, 30] (* Michael De Vlieger, Jan 21 2018 *)

a(n) = (n^8 - 6*n^6 - 12*n^5 + 64*n^4 + 8*n^3 - 136*n^2 + (n == 1 (mod 2))*(14*n^4 - 96*n^3 + 162*n^2 - 92*n + 93))/192 + (n == 2 (mod 6))*n/6 + (n == 2 (mod 4))*n/4 + (n == 5 (mod 6))*(n + 1)/6.
a(n) = (n^8 - 6*n^6 - 12*n^5)/192 + b(n) + c(n), where
  b(n) = (64*n^4 + 8*n^3 - 136*n^2)/192  for n even,
  b(n) = (78*n^4 - 88*n^3 + 26*n^2 - 92*n + 93)/192  for n odd,
  c(n) = 0          for n == 0, 1, 3, 4, 7, 9  (mod 12),
  c(n) = n/4        for n == 6, 10             (mod 12),
  c(n) = n/6        for n == 8                 (mod 12),
  c(n) = 5/12*n     for n == 2                 (mod 12),
  c(n) = (n + 1)/6  for n == 5, 11             (mod 12).
Conjectures from Colin Barker, Jan 21 2018: (Start)
G.f.: x^2*(1 + 14*x + 176*x^2 + 893*x^3 + 2861*x^4 + 6847*x^5 + 12704*x^6 + 20412*x^7 + 27052*x^8 + 33142*x^9 + 33910*x^10 + 33289*x^11 + 26586*x^12 + 20709*x^13 + 12212*x^14 + 7178*x^15 + 2639*x^16 + 1094*x^17 + 134*x^18 + 68*x^19 - 3*x^20 + 2*x^21) / ((1 - x)^9*(1 + x)^5*(1 - x + x^2)*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = 3*a(n-1) - a(n-2) - 4*a(n-3) + 4*a(n-4) - 4*a(n-5) + 5*a(n-6) + a(n-7) - 5*a(n-8) + 6*a(n-9) - 10*a(n-10) + 8*a(n-11) - 8*a(n-13) + 10*a(n-14) - 6*a(n-15) + 5*a(n-16) - a(n-17) - 5*a(n-18) + 4*a(n-19) - 4*a(n-20) + 4*a(n-21) + a(n-22) - 3*a(n-23) + a(n-24) for n>24.
(End)

A173799 Partial sums of A019318.

Original entry on oeis.org

1, 3, 19, 271, 7085, 251429, 10997806, 564316854, 33175912910, 2196968168590, 161790768056642, 13114202824936638, 1160158996141467678, 111226473580172327222, 11486922450679555836573

Offset: 1

Jonathan Vos Post, Feb 25 2010

nonn

Partial sums of number of inequivalent ways of choosing n squares from an n X n board, considering rotations and reflections to be the same.The subsequence of primes in this partial sum (unexpectedly dense at first) begins: 3, 19, 271, 251429, no more through a(20) yet 4 of the first 5 values after a(1).

			a(6) = 1 + 2 + 16 + 252 + 6814 + 244344 = 251429 is prime.

Cf. A019318, A054252, A014409.

a(n) = SUM[i=1..n] A019318(i) = SUM[i=1..n] {number of inequivalent ways of choosing i squares from an i X i board, considering rotations and reflections to be the same}.

cp's OEIS Frontend

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

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

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

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A242709 Nonequivalent ways to place two different markers (e.g., a pair of Go stones, black and white) on an n X n grid.

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A296999 Number of nonequivalent (mod D_8) ways to place 4 points on an n X n point grid so that no point is equally distant from two other points on the same row or the same column.

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A173799 Partial sums of A019318.

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