A178717 -id:A178717 - cp's OEIS frontend

A348129 Number T(n,k) of ways to place k nonattacking queens on an n X n board; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 1, 4, 0, 1, 9, 8, 0, 1, 16, 44, 24, 2, 1, 25, 140, 204, 82, 10, 1, 36, 340, 1024, 982, 248, 4, 1, 49, 700, 3628, 7002, 4618, 832, 40, 1, 64, 1288, 10320, 34568, 46736, 22708, 3192, 92, 1, 81, 2184, 25096, 131248, 310496, 312956, 119180, 13848, 352, 1, 100, 3480, 54400, 412596, 1535440, 2716096, 2119176, 636524, 56832, 724

Offset: 0

Alois P. Heinz, Oct 01 2021

			T(3,2) = 8:
  .-----. .-----. .-----. .-----. .-----. .-----. .-----. .-----.
  |Q . .| |Q . .| |. . Q| |. . Q| |. . .| |. Q .| |. Q .| |. . .|
  |. . Q| |. . .| |. . .| |Q . .| |Q . .| |. . .| |. . .| |. . Q|
  |. . .| |. Q .| |. Q .| |. . .| |. . Q| |. . Q| |Q . .| |Q . .|
  `-----´ `-----´ `-----´ `-----´ `-----´ `-----´ `-----´ `-----´.
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,    0;
  1,  9,    8,     0;
  1, 16,   44,    24,      2;
  1, 25,  140,   204,     82,     10;
  1, 36,  340,  1024,    982,    248,      4;
  1, 49,  700,  3628,   7002,   4618,    832,     40;
  1, 64, 1288, 10320,  34568,  46736,  22708,   3192,    92;
  1, 81, 2184, 25096, 131248, 310496, 312956, 119180, 13848, 352;
  ...

Columns k=0-8 give: A000012, A000290, A036464, A047659, A061994, A108792, A176186, A178721, A252593.
Main diagonal gives A000170.
Row sums give A287227.
T(2n,n) gives A348130.
Cf. A090642, A144084, A178717.

A178721 Number of ways to place 7 nonattacking queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 40, 3192, 119180, 2119176, 23636352, 186506000, 1131544008, 5613017128, 23670094984, 87463182432, 289367715488, 872345119896, 2427609997716, 6305272324272

Offset: 1

Vaclav Kotesovec, Jun 07 2010

nonn

S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-Queens Problem, I. General theory, Jan 26 2013, updated Feb 21 2014
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013
Index entries for linear recurrences with constant coefficients, order 197.

Cf. A036464, A047659, A061994, A108792, A176186, A178717.

Column k=7 of A348129.

(* General formulas (denominator and recurrence) for k nonattacking queens on an n X n board: *) inversef[j_]:=(m=2;While[j>Fibonacci[m],m=m+1];m); denom[k_]:=(x-1)^(2k+1)*Product[Cyclotomic[j,x]^(2*(k-inversef[j]+1)),{j,2,Fibonacci[k]}]; Table[denom[k],{k,1,7}]//TraditionalForm Table[Sum[Coefficient[Expand[denom[k]],x,i]*Subscript[a,n-i],{i,0,Exponent[denom[k],x]}],{k,1,7}]//TraditionalForm

Denominator of G.f.: (x-1)^15*(x+1)^10*(x^2+x+1)^8*(x^2+1)^6*(x^4+x^3+x^2+x+1)^6*(x^2-x+1)^4*(x^6+x^5+x^4+x^3+x^2+x+1)^4*(x^4+1)^4*(x^6+x^3+1)^2*(x^4-x^3+x^2-x+1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)^2*(x^4-x^2+1)^2*(x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)^2.

Recurrence: a(n) = a(n-197) + 11a(n-196) + 66a(n-195) + 284a(n-194) + 979a(n-193) + 2867a(n-192) + 7391a(n-191) + 17167a(n-190) + 36502a(n-189) + 71854a(n-188) + 132001a(n-187) + 227579a(n-186) + 369573a(n-185) + 566345a(n-184) + 818910a(n-183) + 1114468a(n-182) + 1418684a(n-181) + 1667858a(n-180) + 1762862a(n-179) + 1567406a(n-178) + 913631a(n-177) - 382005a(n-176) - 2490306a(n-175) - 5527702a(n-174) - 9503162a(n-173) - 14258598a(n-172) - 19411273a(n-171) - 24310113a(n-170) - 28020291a(n-169) - 29351159a(n-168) - 26940769a(n-167) - 19405263a(n-166) - 5553140a(n-165) + 15346812a(n-164) + 43268288a(n-163) + 77138720a(n-162) + 114608227a(n-161) + 151932369a(n-160) + 184024666a(n-159) + 204725598a(n-158) + 207315406a(n-157) + 185268748a(n-156) + 133212155a(n-155) + 48004017a(n-154) - 70183102a(n-153) - 216930246a(n-152) - 382960078a(n-151) - 554012366a(n-150) - 711346353a(n-149) - 832955143a(n-148) - 895498622a(n-147) - 876864666a(n-146) - 759163548a(n-145) - 531860790a(n-144) - 194674273a(n-143) + 240182841a(n-142) + 746828188a(n-141) + 1285960424a(n-140) + 1806771216a(n-139) + 2250587298a(n-138) + 2556103772a(n-137) + 2665846492a(n-136) + 2533288725a(n-135) + 2129874995a(n-134) + 1451101463a(n-133) + 520790749a(n-132) - 607206046a(n-131) - 1850443990a(n-130) - 3102719461a(n-129) - 4242198625a(n-128) - 5142328327a(n-127) - 5684628585a(n-126) - 5772140029a(n-125) - 5342085203a(n-124) - 4376237801a(n-123) - 2907601789a(n-122) - 1022286568a(n-121) + 1144093134a(n-120) + 3415602536a(n-119) + 5590244180a(n-118) + 7458159648a(n-117) + 8822115392a(n-116) + 9518231826a(n-115) + 9434741790a(n-114) + 8526633540a(n-113) + 6824351658a(n-112) + 4435274433a(n-111) + 1537407289a(n-110) - 1634445881a(n-109) - 4808938651a(n-108) - 7703022656a(n- 107) - 10048957558a(n-106) - 11620750186a(n-105) - 12257251526a(n-104) - 11879415820a(n-103) - 10499785534a(n-102) - 8223052813a(n-101) - 5237477687a(n-100) - 1797913038a(n-99) + 1797913038a(n-98) + 5237477687a(n-97) + 8223052813a(n-96) + 10499785534a(n-95) + 11879415820a(n-94) + 12257251526a(n-93) + 11620750186a(n-92) + 10048957558a(n-91) + 7703022656a(n-90) + 4808938651a(n-89) + 1634445881a(n-88) - 1537407289a(n-87) - 4435274433a(n-86) - 6824351658a(n-85) - 8526633540a(n-84) - 9434741790a(n-83) - 9518231826a(n-82) - 8822115392a(n-81) - 7458159648a(n-80) - 5590244180a(n-79) - 3415602536a(n-78) - 1144093134a(n-77) + 1022286568a(n-76) + 2907601789a(n-75) + 4376237801a(n-74) + 5342085203a(n-73) + 5772140029a(n-72) + 5684628585a(n-71) + 5142328327a(n-70) + 4242198625a(n-69) + 3102719461a(n-68) + 1850443990a(n-67) + 607206046a(n-66) - 520790749a(n-65) - 1451101463a(n-64) - 2129874995a(n-63) - 2533288725a(n-62) - 2665846492a(n-61) - 2556103772a(n-60) - 2250587298a(n-59) - 1806771216a(n-58) - 1285960424a(n-57) - 746828188a(n-56) - 240182841a(n-55) + 194674273a(n-54) + 531860790a(n-53) + 759163548a(n-52) + 876864666a(n-51) + 895498622a(n-50) + 832955143a(n-49) + 711346353a(n-48) + 554012366a(n-47) + 382960078a(n-46) + 216930246a(n-45) + 70183102a(n-44) - 48004017a(n-43) - 133212155a(n-42) - 185268748a(n-41) - 207315406a(n-40) - 204725598a(n-39) - 184024666a(n-38) - 151932369a(n-37) - 114608227a(n-36) - 77138720a(n-35) - 43268288a(n-34) - 15346812a(n-33) + 5553140a(n-32) + 19405263a(n-31) + 26940769a(n-30) + 29351159a(n-29) + 28020291a(n-28) + 24310113a(n-27) + 19411273a(n-26) + 14258598a(n-25) + 9503162a(n-24) + 5527702a(n-23) + 2490306a(n-22) + 382005a(n-21) - 913631a(n-20) - 1567406a(n-19) - 1762862a(n-18) - 1667858a(n-17) - 1418684a(n-16) - 1114468a(n-15) - 818910a(n-14) - 566345a(n-13) - 369573a(n-12) - 227579a(n-11) - 132001a(n-10) - 71854a(n-9) - 36502a(n-8) - 17167a(n-7) - 7391a(n-6) - 2867a(n-5) - 979a(n-4) - 284a(n-3) - 66a(n-2) - 11a(n-1).

a(19)-a(20) from Vaclav Kotesovec, Jun 16 2010

A180402 a(n) = lcm(1,...,Fibonacci(n)).

Original entry on oeis.org

1, 1, 2, 6, 60, 840, 360360, 232792560, 144403552893600, 164249358725037825439200, 718766754945489455304472257065075294400, 33312720618553145840562713089120360606823375590405920630576000

Offset: 1

Vaclav Kotesovec, Sep 02 2010

nonn

Also least period for number of ways of placing k non-attacking queens on an n X n chessboard. [conjectured by Kotesovec; proved for n <= 5. - Thomas Zaslavsky, Jun 24 2018]

Alois P. Heinz, Table of n, a(n) for n = 1..17
Christopher R. H. Hanusa, T. Zaslavsky, S. Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016. See Table 8.1.
V. Kotesovec, Non-attacking chess pieces, 6ed, p.31, 2013

Cf. A178717, A178721, A078491, A218492, A035105, A062954.

a:= n-> ilcm($1..(<<0|1>, <1|1>>^n)[1,2]):
seq(a(n), n=1..14);  # Alois P. Heinz, Aug 12 2017

Table[Apply[LCM, Range[Fibonacci[k]]], {k, 1, 10}]
Array[LCM @@ Range@Fibonacci@# &, 12] (* Robert G. Wilson v, Sep 05 2010 *)

a(n) = lcm([1..fibonacci(n)]); \\ Michel Marcus, Jun 24 2018

a(11) onwards from Robert G. Wilson v, Sep 05 2010

A178720 Degree of denominator of GF for number of ways to place k nonattacking queens on an n X n toroidal board.

Original entry on oeis.org

3, 8, 12, 28, 58, 142, 350, 906, 2320, 6056, 15778, 41024, 107132, 280184, 732998, 1918354, 5019810, 13141378, 34398686, 90045424, 235729374, 617126438, 1615633560, 4229774958, 11073514332, 28990794770, 75898640094, 198704554772

Offset: 1

Vaclav Kotesovec, Jun 07 2010

nonn

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013

A172517, A172518, A172519, A173775, A000010, A000045, A178717.

Table[If[k > 1, 4*k + Sum[ Sum[(2*j + 1)*EulerPhi[i], {i, 2*Fibonacci[k - j - 1] + 1, 2*Fibonacci[k - j]}], {j, 1, k - 2}], 3], {k, 1, 20}]

Explicit formula (Vaclav Kotesovec, Jun 05 2010), for k>2 : t(k) = 4*k+Sum[Sum[(2*j+1)*EulerPhi[i],{i,2*Fibonacci[k-j-1]+1,2*Fibonacci[k-j]}],{j,1,k-2}], Asymptotic formula: t(k) ~ 12/(5*Pi^2)*((1+Sqrt[5])/2)^(2*k+1) or t(k) ~ 6*(1+Sqrt[5])/Pi^2*Fibonacci[k]^2

cp's OEIS Frontend

A348129 Number T(n,k) of ways to place k nonattacking queens on an n X n board; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A178721 Number of ways to place 7 nonattacking queens on an n X n board.

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A180402 a(n) = lcm(1,...,Fibonacci(n)).

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A178720 Degree of denominator of GF for number of ways to place k nonattacking queens on an n X n toroidal board.

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