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.

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

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

A279448 Number of nonequivalent 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, 17, 202, 1397, 6582, 24185, 73496, 195086, 463875, 1013505, 2061426, 3956947, 7222992, 12640817, 21312992, 34801420, 55215621, 85424721, 129174250, 191397185, 278361226, 398108777, 560635032, 778491962, 1066995527, 1445034305, 1935301746, 2565356031, 3367870500
Offset: 1

Views

Author

Heinrich Ludwig, Dec 18 2016

Keywords

Comments

Column 5 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279438.
For condition "no more than 2 points on straight lines at any angle", see A235455.

Links

Crossrefs

Cf. A235455, A279438, A279452, A279453, A279454.
Same problem but 2,3,5,6,7 points: A014409, A279447, A279449, A279450, A279451.

Programs

  • PARI
    concat(0, Vec(x^2*(1 +13*x +135*x^2 +622*x^3 +1449*x^4 +2143*x^5 +1557*x^6 +781*x^7 +34*x^8 -8*x^9 -8*x^10 +x^11) / ((1 -x)^9*(1 +x)^5) + O(x^40))) \\ Colin Barker, Dec 18 2016

Formula

a(n) = (n^8 - 14*n^6 + 30*n^5 + 12*n^4 - 60*n^3 + 40*n^2)/192 + IF(MOD(n, 2) = 1, 4*n^4 - 20*n^3 + 22*n^2 - 2*n - 7)/64.
a(n) = 4*a(n-1) - a(n-2) - 16*a(n-3) + 19*a(n-4) + 20*a(n-5) - 45*a(n-6) + 45*a(n-8) - 20*a(n-9) - 19*a(n-10) + 16*a(n-11) + a(n-12) - 4*a(n-13) + a(n-14).
G.f.: x^2*(1 +13*x +135*x^2 +622*x^3 +1449*x^4 +2143*x^5 +1557*x^6 +781*x^7 +34*x^8 -8*x^9 -8*x^10 +x^11) / ((1 -x)^9*(1 +x)^5). - Colin Barker, Dec 18 2016

A279450 Number of nonequivalent 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, 2, 285, 9110, 126396, 1055025, 6266614, 29198740, 113262680, 380775248, 1140764611, 3108667306, 7824370092, 18407341855, 40855872764, 86201399496, 173952773328, 337453762782, 631982899545, 1146743732126, 2022212701212, 3474824082125, 5831439251154, 9576836632860
Offset: 1

Views

Author

Heinrich Ludwig, Dec 18 2016

Keywords

Comments

Column 7 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279440.
For condition "no more than 2 points on straight lines at any angle", see A235457.

Links

Crossrefs

Cf. A235457, A279440, A279452, A279453, A279454.
Same problem but 2,3,4,5,7 points: A014409, A279447, A279448, A279449, A279451.

Programs

  • PARI
    concat(vector(2), Vec(x^3*(2 +273*x +7416*x^2 +74060*x^3 +375661*x^4 +1128403*x^5 +2194010*x^6 +2815082*x^7 +2424155*x^8 +1294751*x^9 +376028*x^10 -5296*x^11 -32173*x^12 -8195*x^13 +178*x^14 +122*x^15 +3*x^16) / ((1 -x)^13*(1 +x)^7) + O(x^40))) \\ Colin Barker, Dec 18 2016

Formula

a(n) = (n^12 - 55*n^10 + 210*n^9 + 93*n^8 - 2220*n^7 + 6052*n^6 - 8040*n^5 + 4236*n^4 + 3240*n^3 - 5872*n^2 + 2400*n)/5760 + IF(MOD(n, 2) = 1, 2*n^6 - 18*n^5 + 53*n^4 - 64*n^3 + 33*n^2 - 12*n + 5)/128.
a(n) = 6*a(n-1) - 8*a(n-2) - 22*a(n-3) + 69*a(n-4) - 8*a(n-5) - 176*a(n-6) + 168*a(n-7) + 182*a(n-8) - 364*a(n-9) + 364*a(n-11) - 182*a(n-12) - 168*a(n-13) + 176*a(n-14) + 8*a(n-15) - 69*a(n-16) + 22*a(n-17) + 8*a(n-18) - 6*a(n-19) + *a(n-20).
G.f.: x^3*(2 +273*x +7416*x^2 +74060*x^3 +375661*x^4 +1128403*x^5 +2194010*x^6 +2815082*x^7 +2424155*x^8 +1294751*x^9 +376028*x^10 -5296*x^11 -32173*x^12 -8195*x^13 +178*x^14 +122*x^15 +3*x^16) / ((1 -x)^13*(1 +x)^7). - Colin Barker, Dec 18 2016

A279451 Number of nonequivalent 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, 115, 11810, 326190, 4444935, 38675954, 246563232, 1248782460, 5296300670, 19499431941, 63958228738, 190528987506, 523151460045, 1339408935540, 3227223506896, 7372750196952, 16069268866908, 33586411339335, 67610793877650, 131569779776182, 248290280743571
Offset: 1

Views

Author

Heinrich Ludwig, Dec 22 2016

Keywords

Comments

Column 8 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279441.
For condition "no more than 2 points on straight lines at any angle", see A235458.

Links

Crossrefs

Cf. A235458, A279441, A279452, A279453, A279454.
Same problem but 2..6 points: A014409, A279447, A279448, A279449, A279450.

Programs

  • Mathematica
    Table[(n^14 - 91 n^12 + 420 n^11 + 693 n^10 - 10500 n^9 + 33647 n^8 - 45316 n^7 + 3682 n^6 + 62300 n^5 - 51996 n^4 - 28504 n^3 + 54384 n^2 - 18720 n)/40320 + Boole[OddQ@ n] (2 n^7 - 17 n^6 + 50 n^5 - 59 n^4 + 38 n^3 - 71 n^2 + 102 n - 45)/384, {n, 23}] (* or *)
Rest@ CoefficientList[Series[x^4*(115 + 11005 x + 245015 x^2 + 2317550 x^3 + 12037814 x^4 + 39232894 x^5 + 85494738 x^6 + 129182670 x^7 + 135873108 x^8 + 97856368 x^9 + 44499480 x^10 + 9709722 x^11 - 1359254 x^12 - 1352974 x^13 - 257282 x^14 + 13866 x^15 + 7705 x^16 + 419 x^17 + x^18)/((1 - x)^15*(1 + x)^8), {x, 0, 23}], x] (* Michael De Vlieger, Dec 23 2016 *)
  • PARI
    concat(vector(3), Vec(x^4*(115 +11005*x +245015*x^2 +2317550*x^3 +12037814*x^4 +39232894*x^5 +85494738*x^6 +129182670*x^7 +135873108*x^8 +97856368*x^9 +44499480*x^10 +9709722*x^11 -1359254*x^12 -1352974*x^13 -257282*x^14 +13866*x^15 +7705*x^16 +419*x^17 +x^18) / ((1 -x)^15*(1 +x)^8) + O(x^30))) \\ Colin Barker, Dec 23 2016

Formula

a(n) = (n^14 -91*n^12 +420*n^11 +693*n^10 -10500*n^9 +33647*n^8 -45316*n^7 +3682*n^6 +62300*n^5 -51996*n^4 -28504*n^3 +54384*n^2 -18720*n)/40320 + IF(MOD(n, 2) = 1, 2*n^7 -17*n^6 +50*n^5 -59*n^4 +38*n^3 -71*n^2 +102*n -45)/384.
G.f.: x^4*(115 +11005*x +245015*x^2 +2317550*x^3 +12037814*x^4 +39232894*x^5 +85494738*x^6 +129182670*x^7 +135873108*x^8 +97856368*x^9 +44499480*x^10 +9709722*x^11 -1359254*x^12 -1352974*x^13 -257282*x^14 +13866*x^15 +7705*x^16 +419*x^17 +x^18) / ((1 -x)^15*(1 +x)^8). - Colin Barker, Dec 23 2016
Showing 1-5 of 5 results.

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