Richard M. Low - cp's OEIS frontend

A287120 Number of non-attacking bishop positions on a 3 X n chessboard.

1, 8, 25, 70, 225, 748, 2401, 7668, 24649, 79344, 255025, 819494, 2634129, 8467464, 27217089, 87483296, 281199361, 903867144, 2905317801, 9338615022, 30017295025, 96485195716, 310134268609, 996870677460, 3204261102025, 10299519778080, 33105949765729, 106413107836334

Offset: 0

Richard M. Low, May 20 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 1.
Index entries for linear recurrences with constant coefficients, signature (3,0,2,4,-10,-2,0,-1,1).

Cf. A287168, A287169.

CoefficientList[Series[(-1 - 5 x - x^2 + 7 x^3 + 5 x^4 - x^5 + x^6 - x^7)/(-1 + 3 x + 2 x^3 + 4 x^4 - 10 x^5 - 2 x^6 - x^8 + x^9), {x, 0, 27}], x] (* Michael De Vlieger, May 20 2017 *)

Vec((-1-5*x-x^2+7*x^3+5*x^4-x^5+x^6-x^7)/ (-1+3*x+2*x^3 +4*x^4-10*x^5-2*x^6-x^8+x^9) + O(x^30)) \\ Michel Marcus, May 20 2017

G.f.: (-1-5*x-x^2+7*x^3+5*x^4-x^5+x^6-x^7) / (-1+3*x +2*x^3 +4*x^4 -10*x^5 -2*x^6 -x^8 +x^9).

A287168 Number of non-attacking bishop positions on a cylindrical 3 X 2n chessboard.

Original entry on oeis.org

1, 16, 144, 1156, 11664, 118336, 1218816, 12574116, 129868816, 1341610384, 13860823824, 143206237476, 1479580304400, 15286786268224, 157940749232704, 1631820172890436, 16859722986240016, 174192150898142224, 1799727414404326416, 18594516209802790084

Offset: 0

Richard M. Low, May 20 2017

nonn

Ray Chandler, Table of n, a(n) for n = 0..986 (terms to 1000 digits)
Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 11.
Index entries for linear recurrences with constant coefficients, signature (14, -35, -48, 198, -112, -78, 72, -5, -6, 1).

Cf. A286810, A287169.

CoefficientList[Series[(-1 - 2 x + 45 x^2 + 252 x^3 - 1090 x^4 + 644 x^5 + 802 x^6 - 740 x^7 + 35 x^8 + 86 x^9 - 15 x^10) / (-1 + 14 x - 35 x^2 - 48 x^3 + 198 x^4 - 112 x^5 - 78 x^6 + 72 x^7 - 5 x^8 - 6 x^9 + x^10), {x, 0, 986}], x] (* Michael De Vlieger, May 21 2017; simplified by Georg Fischer, May 23 2019 *)

G.f.: (-1 - 2 x + 45 x^2 + 252 x^3 - 1090 x^4 + 644 x^5 + 802 x^6 - 740 x^7 + 35 x^8 + 86 x^9 - 15 x^10) / (-1 + 14 x - 35 x^2 - 48 x^3 + 198 x^4 - 112 x^5 - 78 x^6 + 72 x^7 - 5 x^8 - 6 x^9 + x^10). [Corrected by Georg Fischer, May 23 2019]

More terms from Michael De Vlieger, May 21 2017

A287169 Number of non-attacking king positions on a cylindrical 3 X 2n chessboard.

Original entry on oeis.org

1, 11, 67, 503, 3939, 31111, 246163, 1948503, 15424707, 122107175, 966645747, 7652334327, 60578794275, 479564842183, 3796418256467, 30053895424663, 237918103255427, 1883450483360871, 14910112659965107, 118034140795537975, 934403294669416419, 7397093003809879047

Offset: 0

Richard M. Low, May 21 2017

nonn

Ray Chandler, Table of n, a(n) for n = 0..1000
Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 15.
Index entries for linear recurrences with constant coefficients, signature (11, -27, 21, -4).

Cf. A287120, A287168.

CoefficientList[Series[(1-27*x^2+42*x^3-12*x^4)/(1-11*x+27*x^2-21*x^3+4*x^4), {x, 0, 44}], x] (* Michael De Vlieger, May 21 2017; simplified by Georg Fischer, May 23 2019 *)

G.f.: (1-27*x^2+42*x^3-12*x^4)/(1-11*x+27*x^2-21*x^3+4*x^4). [Corrected by Georg Fischer, May 23 2019]

A286810 Number of non-attacking bishop positions on a cylindrical 2 X 2n chessboard.

Original entry on oeis.org

1, 9, 49, 324, 2209, 15129, 103684, 710649, 4870849, 33385284, 228826129, 1568397609, 10749957124, 73681302249, 505019158609, 3461452808004, 23725150497409, 162614600673849, 1114577054219524, 7639424778862809, 52361396397820129, 358890350005878084, 2459871053643326449, 16860207025497407049

Offset: 0

Richard M. Low, May 20 2017

Essentially the same as A081069. - R. J. Mathar, May 25 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 9.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Vec((1 + x - 15*x^2 + 3*x^3) / ((1 - x)*(1 - 7*x + x^2)) + O(x^30)) \\ Colin Barker, May 21 2017

G.f.: (1+x^2-15*x^4+3*x^6) / (1-8*x^2+8*x^4-x^6).

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>3. - Colin Barker, May 21 2017

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User: Richard M. Low

A287120 Number of non-attacking bishop positions on a 3 X n chessboard.

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A287168 Number of non-attacking bishop positions on a cylindrical 3 X 2n chessboard.

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A287169 Number of non-attacking king positions on a cylindrical 3 X 2n chessboard.

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A286810 Number of non-attacking bishop positions on a cylindrical 2 X 2n chessboard.

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