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 1-8 of 8 results.

A279445 Triangle read by rows: T(n, k) is the number of ways to place k points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

1, 1, 1, 4, 6, 4, 1, 1, 9, 36, 78, 90, 45, 6, 1, 16, 120, 528, 1428, 2304, 2040, 816, 90, 1, 25, 300, 2200, 10600, 34020, 71400, 93000, 67950, 22650, 2040, 1, 36, 630, 6900, 51525, 270720, 1005720, 2602800, 4531950, 4987800, 3110940, 888840, 67950, 1, 49, 1176, 17934
Offset: 1

Views

Author

Heinrich Ludwig, Dec 17 2016

Keywords

Comments

Length of n-th row is A272651(n) + 1, where A272651(n) is the maximal number of points to be placed under the condition mentioned.
Rotations and reflections of placements are counted. If they are to be ignored, see A279453.
For condition "no more than 2 points on a straight line at any angle", see A194193 (but that one is read by antidiagonals).

Examples

			The table begins with T(1, 0):
1  1
1  4   6    4     1
1  9  36   78    90    45     6
1 16 120  528  1428  2304  2040   816    90
1 25 300 2200 10600 34020 71400 93000 67950 22650 2040
...
T(3, 2) = 36 because there are 36 ways to place 2 points on a 3 X 3 square grid so that no more than 2 points are on a vertical or horizontal straight line.

Links

Crossrefs

Row sums give A197458.
Columns 2..10: A000290, A083374, A279437, A279438, A279439, A279440, A279441, A279442, A279443.
Diagonal T(n, n) is A279444.
Cf. A279453, A194193.

A279438 Number of ways to place 4 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 1, 90, 1428, 10600, 51525, 190806, 584080, 1552608, 3701025, 8088850, 16470036, 31616520, 57743413, 101055150, 170433600, 278290816, 441610785, 683206218, 1033218100, 1530887400, 2226630021, 3184447750, 4484709648, 6227340000, 8535450625, 11559457026, 15481719540
Offset: 1

Views

Author

Heinrich Ludwig, Dec 12 2016

Keywords

Comments

Column 5 of triangle A279445.
Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279448.
For condition "no more than 2 points on straight lines at any angle", see A175383.

Links

Crossrefs

Cf. A175383, A249448, A279444, A279445, A197458.
Same problem but 2,3,5..9 points: A083374, A279437, A279439, A279440, A279441, A279442, A279443.

Programs

  • Mathematica
    Table[(n^8 - 14 n^6 + 30 n^5 - 17 n^4 - 6 n^3 + 6 n^2)/24, {n, 28}] (* Michael De Vlieger, Dec 12 2016 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,1,90,1428,10600,51525,190806,584080,1552608},30] (* Harvey P. Dale, Sep 05 2024 *)
  • PARI
    concat(0, Vec(x^2*(1 + 81*x + 654*x^2 + 904*x^3 + 99*x^4 - 57*x^5 - 2*x^6) / (1 - x)^9 + O(x^30))) \\ Colin Barker, Dec 13 2016
  • PARI
    a(n) = (n^6 - 14*n^4 + 30*n^3 - 17*n^2 - 6*n + 6)*n^2/24 \\ Charles R Greathouse IV, Dec 13 2016

Formula

a(n) = (n^8 - 14*n^6 + 30*n^5 - 17*n^4 - 6*n^3 + 6*n^2)/24.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
G.f.: x^2*(1 + 81*x + 654*x^2 + 904*x^3 + 99*x^4 - 57*x^5 - 2*x^6) / (1 - x)^9. - Colin Barker, Dec 13 2016

A279439 Number of ways to place 5 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 45, 2304, 34020, 270720, 1475145, 6209280, 21654864, 65422080, 176467005, 434206080, 990140580, 2117816064, 4288771305, 8284308480, 15355471680, 27446584320, 47501098029, 79872376320, 130866406020, 209448328320, 328150139625, 504222960384, 761083938000
Offset: 1

Views

Author

Heinrich Ludwig, Dec 21 2016

Keywords

Comments

Column 6 of triangle A279445.
Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279449.
For condition "no more than 2 points on straight lines at any angle", see A194190.

Links

Crossrefs

Cf. A194190, A279449, A279444, A279445, A197458.
Same problem but 2,3,4,6..9 points: A083374, A279437, A279438, A279440, A279441, A279442, A279443.

Programs

  • Mathematica
    Table[(n^10 - 30 n^8 + 90 n^7 - 27 n^6 - 270 n^5 + 500 n^4 - 360 n^3 + 96 n^2)/120, {n, 25}] (* or *)
Rest@ CoefficientList[Series[9 x^3*(5 + 201 x + 1239 x^2 + 1755 x^3 + 335 x^4 - 165 x^5 - 11 x^6 + x^7)/(1 - x)^11, {x, 0, 25}], x] (* Michael De Vlieger, Dec 22 2016 *)
  • PARI
    concat(vector(2), Vec(9*x^3*(5 +201*x +1239*x^2 +1755*x^3 +335*x^4 -165*x^5 -11*x^6 +x^7) / (1 -x)^11 + O(x^30))) \\ Colin Barker, Dec 22 2016

Formula

a(n) = (n^10 -30*n^8 +90*n^7 -27*n^6 -270*n^5 +500*n^4 -360*n^3 +96*n^2)/120.
a(n) = 11*a(n-1) -55*a(n-2) +165*a(n-3) -330*a(n-4) +462*a(n-5) -462*a(n-6) +330*a(n-7) -165*a(n-8) +55*a(n-9) -11*a(n-10) +*a(n-11).
G.f.: 9*x^3*(5 +201*x +1239*x^2 +1755*x^3 +335*x^4 -165*x^5 -11*x^6 +x^7) / (1 -x)^11. - Colin Barker, Dec 22 2016

A279440 Number of ways to place 6 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 6, 2040, 71400, 1005720, 8421630, 50092896, 233483040, 905925600, 3045791430, 9125544120, 24868110696, 62593429080, 147255640350, 326843422080, 689604309120, 1391614736256, 2699616160710, 5055848825400, 9173923662120, 16177675640280, 27798546316926, 46651469520480
Offset: 1

Views

Author

Heinrich Ludwig, Dec 22 2016

Keywords

Comments

Column 7 of triangle A279445.
Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279450.
For condition "no more than 2 points on straight lines at any angle", see A194191.

Links

Crossrefs

Cf. A194191, A279450, A279444, A279445, A197458.
Same problem but 2..5,7,8,9 points: A083374, A279437, A279438, A279439, A279441, A279442, A279443.

Programs

  • Mathematica
    Table[(n^12 - 55 n^10 + 210 n^9 + 93 n^8 - 2220 n^7 + 5855 n^6 - 7350 n^5 + 4786 n^4 - 1440 n^3 + 120 n^2)/720, {n, 24}] (* or *)
Rest@ CoefficientList[Series[6 x^3 (1 + 327 x + 7558 x^2 + 39154 x^3 + 56220 x^4 + 14724 x^5 - 6262 x^6 - 978 x^7 + 131 x^8 + 5 x^9)/(1 - x)^13, {x, 0, 24}], x] (* Michael De Vlieger, Dec 22 2016 *)
  • PARI
    concat(vector(2), Vec(6*x^3*(1 +327*x +7558*x^2 +39154*x^3 +56220*x^4 +14724*x^5 -6262*x^6 -978*x^7 +131*x^8 +5*x^9) / (1 -x)^13 + O(x^30))) \\ Colin Barker, Dec 22 2016

Formula

a(n) = (n^12 -55*n^10 +210*n^9 +93*n^8 -2220*n^7 +5855*n^6 -7350*n^5 +4786*n^4 -1440*n^3 +120*n^2)/720.
a(n) = 13*a(n-1) -78*a(n-2) +286*a(n-3) -715*a(n-4) +1287*a(n-5) -1716*a(n-6) +1716*a(n-7) -1287*a(n-8) +715*a(n-9) -286*a(n-10) +78*a(n-11) -13*a(n-12) +*a(n-13).
G.f.: 6*x^3*(1 +327*x +7558*x^2 +39154*x^3 +56220*x^4 +14724*x^5 -6262*x^6 -978*x^7 +131*x^8 +5*x^9) / (1 -x)^13. - Colin Barker, Dec 22 2016

A279441 Number of ways to place 7 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 0, 816, 93000, 2602800, 35526120, 309328320, 1972234656, 9989784000, 42369069600, 155993500080, 511660972680, 1524225598896, 4185197289000, 10715254368000, 25817751281280, 58981960615680, 128554066935936, 268691201838000, 540886175310600, 1052558059827120
Offset: 1

Views

Author

Heinrich Ludwig, Dec 22 2016

Keywords

Comments

Column 8 of triangle A279445.
Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279451.
For condition "no more than 2 points on straight lines at any angle", see A194192.

Links

Crossrefs

Cf. A194192, A279451, A279444, A279445, A197458.
Same problem but 2..6,8,9 points: A083374, A279437, A279438, A279439, A279440, A279442, A279443.

Programs

  • Mathematica
    Table[(n^14 - 91 n^12 + 420 n^11 + 693 n^10 - 10500 n^9 + 33647 n^8 - 45780 n^7 + 5866 n^6 + 65940 n^5 - 89796 n^4 + 50400 n^3 - 10800 n^2)/5040, {n, 23}] (* or *)
Rest@ CoefficientList[Series[24 x^4*(34 + 3365 x + 53895 x^2 + 244910 x^3 + 355390 x^4 + 115542 x^5 - 42490 x^6 - 11570 x^7 + 1500 x^8 + 145 x^9 - x^10)/(1 - x)^15, {x, 0, 23}], x] (* Michael De Vlieger, Dec 22 2016 *)
  • PARI
    concat(vector(3), Vec(24*x^4*(34 +3365*x +53895*x^2 +244910*x^3 +355390*x^4 +115542*x^5 -42490*x^6 -11570*x^7 +1500*x^8 +145*x^9 -x^10) / (1 -x)^15 + O(x^30))) \\ Colin Barker, Dec 22 2016
  • PARI
    a(n) = (n^12 -91*n^10 +420*n^9 +693*n^8 -10500*n^7 +33647*n^6 -45780*n^5 +5866*n^4 +65940*n^3 -89796*n^2 +50400*n -10800)*n^2/5040 \\ Charles R Greathouse IV, Dec 22 2016

Formula

a(n) = (n^14 -91*n^12 +420*n^11 +693*n^10 -10500*n^9 +33647*n^8 -45780*n^7 +5866*n^6 +65940*n^5 -89796*n^4 +50400*n^3 -10800*n^2)/5040.
a(n) = 15*a(n-1) -105*a(n-2) +455*a(n-3) -1365*a(n-4) +3003*a(n-5) -5005*a(n-6) +6435*a(n-7) -6435*a(n-8) +5005*a(n-9) -3003*a(n-10) +1365*a(n-11) -455*a(n-12) +105*a(n-13) -15*a(n-14) +a(n-15).
G.f.: 24*x^4*(34 +3365*x +53895*x^2 +244910*x^3 +355390*x^4 +115542*x^5 -42490*x^6 -11570*x^7 +1500*x^8 +145*x^9 -x^10) / (1 -x)^15. - Colin Barker, Dec 22 2016

A279442 Number of ways to place 8 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 0, 90, 67950, 4531950, 109425330, 1460297160, 13112872920, 88456195800, 480149029800, 2196080372970, 8743233946590, 31033043111070, 99992483914050, 296626638016800, 819218054279520, 2125440234303840, 5218743585428640, 12201529135725450, 27304286810701950
Offset: 1

Views

Author

Heinrich Ludwig, Dec 22 2016

Keywords

Comments

Column 9 of triangle A279445.
Rotations and reflections of placements are counted.

Links

Crossrefs

Cf. A279444, A279445, A197458.
Same problem but 2..7,9 points: A083374, A279437, A279438, A279439, A279440, A279441, A279443.

Programs

  • Mathematica
    Table[n^2*(n - 1)^2*(n - 2)^2*(n - 3)^2*(n^8 + 12 n^7 - 54 n^6 - 444 n^5 + 1845 n^4 + 1392 n^3 - 11332 n^2 + 9660 n + 1260)/8!, {n, 21}] (* Michael De Vlieger, Dec 22 2016 *)
  • PARI
    concat(vector(3), Vec(90*x^4*(1 +738*x +37656*x^2 +461802*x^3 +1893555*x^4 +2767824*x^5 +1058064*x^6 -331920*x^7 -140913*x^8 +15950*x^9 +3096*x^10 -90*x^11 -3*x^12) / (1 -x)^17 + O(x^30))) \\ Colin Barker, Dec 23 2016

Formula

a(n) = (n^16 -140*n^14 +756*n^13 +2506*n^12 -36540*n^11 +130940*n^10 -117432*n^9 -559615*n^8 +2186100*n^7 -3622360*n^6 +3228876*n^5 -1439892*n^4 +181440*n^3 +45360*n^2)/40320; factorized: a(n) = n^2*(n-1)^2*(n-2)^2*(n-3)^2*(n^8 +12*n^7 -54*n^6 -444*n^5 +1845*n^4 +1392*n^3 -11332*n^2 +9660*n +1260)/8!.
a(n) = SUM(1<=j<=17, C(17,j)*(-1)^(j-1)*a(n-j)).
G.f.: 90*x^4*(1 +738*x +37656*x^2 +461802*x^3 +1893555*x^4 +2767824*x^5 +1058064*x^6 -331920*x^7 -140913*x^8 +15950*x^9 +3096*x^10 -90*x^11 -3*x^12) / (1 -x)^17. - Colin Barker, Dec 23 2016

A279443 Number of ways to place 9 points on an n X n board so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 0, 0, 22650, 4987800, 240023070, 5219088000, 68483325960, 630486309600, 4456523194200, 25647802519680, 125166919041450, 533442526857240, 2029603476250350, 7011735609715200, 22291042191643680, 65914292362262400, 182880685655641440, 479548000781222400
Offset: 1

Views

Author

Heinrich Ludwig, Dec 24 2016

Keywords

Comments

Column 9 of triangle A279445.
Rotations and reflections of placements are counted.

Links

Crossrefs

Cf. A279444, A279445, A197458.
Same problem but 2..8 points: A083374, A279437, A279438, A279439, A279440, A279441, A279442.

Programs

  • PARI
    concat(vector(4), Vec(30*x^5*(755 +151915*x +4970934*x^2 +49653854*x^3 +187307071*x^4 +275138271*x^5 +119386656*x^6 -31251744*x^7 -19595619*x^8 +1706821*x^9 +667466*x^10 -26334*x^11 -2543*x^12 +17*x^13) / (1 -x)^19 + O(x^30))) \\ Colin Barker, Dec 24 2016

Formula

a(n) = (n^18 -204*n^16 +1260*n^15 +6846*n^14 -104076*n^13 +394504*n^12 +128520*n^11 -6237075*n^10 +24018372*n^9 -43820196*n^8 +30400020*n^7 +34251148*n^6 -99199296*n^5 +98504496*n^4 -47779200*n^3 +9434880*n^2)/362880; factorized: a(n) = n^2*(n-1)^2*(n-2)^2*(n-3)^2*(n-4)^2*(n^8 +20*n^7 +26*n^6 -820*n^5 -247*n^4 +9704*n^3 -9104*n^2 -14700*n +16380)/9!.
a(n) = SUM(1<=j<=19, C(19,j)*(-1)^(j-1)*a(n-j)).
G.f.: 30*x^5*(755 +151915*x +4970934*x^2 +49653854*x^3 +187307071*x^4 +275138271*x^5 +119386656*x^6 -31251744*x^7 -19595619*x^8 +1706821*x^9 +667466*x^10 -26334*x^11 -2543*x^12 +17*x^13) / (1 -x)^19. - Colin Barker, Dec 24 2016

A279447 Number of nonequivalent ways to place 3 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 1, 14, 73, 301, 890, 2321, 5166, 10654, 20055, 35880, 60511, 98419, 153608, 233331, 343820, 496076, 699261, 969234, 1318885, 1770185, 2340646, 3059749, 3950618, 5051786, 6393075, 8023756, 9981531, 12328239, 15110740, 18405415, 22269656, 26796504, 32055353, 38158166
Offset: 1

Views

Author

Heinrich Ludwig, Dec 17 2016

Keywords

Comments

Column 4 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279437.
For condition "no more than 2 points on straight lines at any angle", see A235454.

Links

Crossrefs

Cf. A235454, A279437, A279452, A279453, A279454.
Same problem but 2, 4..7 points: A014409, A279448, A279449, A279450, A279451.

Programs

  • Magma
    I:=[0,1,14,73,301,890,2321,5166,10654,20055,35880]; [n le 11 select I[n] else 3*Self(n-1)+Self(n-2)-11*Self(n-3)+ 6*Self(n-4)+14*Self(n-5)-14*Self(n-6)-6*Self(n-7)+11*Self(n-8)-Self(n-9)-3*Self(n-10)+Self(n-11): n in [1..40]]; // Vincenzo Librandi, Dec 17 2016
  • Mathematica
    LinearRecurrence[{3, 1, -11, 6, 14, -14, -6, 11, -1, -3, 1},{0, 1, 14, 73, 301, 890, 2321, 5166, 10654, 20055, 35880}, 35] (* Vincenzo Librandi Dec 17 2016 *)
  • PARI
    concat(0, Vec(x^2*(1 + 11*x + 30*x^2 + 79*x^3 + 62*x^4 + 55*x^5 + 4*x^6 - x^7 - x^8) / ((1 - x)^7*(1 + x)^4) + O(x^30))) \\ Colin Barker, Dec 17 2016

Formula

a(n) = (n^6 - 5*n^4 + 14*n^3 - 14*n^2 + 4*n)/48 + IF(MOD(n, 2) = 1, 2*n^3 - 3*n^2 + 1)/16.
a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11).
G.f.: x^2*(1 + 11*x + 30*x^2 + 79*x^3 + 62*x^4 + 55*x^5 + 4*x^6 - x^7 - x^8) / ((1 - x)^7*(1 + x)^4). - Colin Barker, Dec 17 2016
Showing 1-8 of 8 results.

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