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A178721 Number of ways to place 7 nonattacking queens on an n X n board 5
0, 0, 0, 0, 0, 0, 40, 3192, 119180, 2119176, 23636352, 186506000, 1131544008, 5613017128, 23670094984, 87463182432, 289367715488, 872345119896, 2427609997716, 6305272324272 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

LINKS

Table of n, a(n) for n=1..20.

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

FORMULA

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).

MATHEMATICA

(* 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

CROSSREFS

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

Sequence in context: A188154 A268151 A269824 * A059948 A229604 A045502

Adjacent sequences:  A178718 A178719 A178720 * A178722 A178723 A178724

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Jun 07 2010

EXTENSIONS

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

STATUS

approved

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Last modified October 20 17:39 EDT 2017. Contains 293648 sequences.