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 A178722 Number of ways to place 6 nonattacking queens on an n X n toroidal board. 3
 0, 0, 0, 0, 0, 0, 196, 3072, 42768, 550000, 3573856, 25009344, 102800672, 454967744, 1441238400, 4811118592, 12616778208, 34692705648, 79514466480, 189770459200, 392908083876, 842040318416, 1610365515264, 3172863442176, 5692888800000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS Previous recurrence (order 142) was right, but Artem M. Karavaev and his team found (Jun 19 2011) another recurrence with smaller order (124). LINKS Artem M. Karavaev, Zealint blog (in Russian) V. Kotesovec, Non-attacking chess pieces, 6ed, 2013 Vaclav Kotesovec, G.f. FORMULA Explicit formula (Artem M. Karavaev, after values computed by Andrey Khalyavin, Jun 19 2011): n^2/6*(n^10/120-5*n^9/8+125*n^8/6-3275*n^7/8+316073*n^6/60-371219*n^5/8+282695*n^4-4676911*n^3/4+15512322*n^2/5-4626944*n+2452536 +(n^8/4-14*n^7+1411*n^6/4-5227*n^5+199399*n^4/4-313302*n^3+2530255*n^2/2-2984844*n+3117968)*Floor[n/2] +(24*n^4-864*n^3+12852*n^2-95112*n+309128)*Floor[n/3]+(12*n^4-432*n^3+6180*n^2-42384*n+117584)*Floor[(n+1)/3] +(27*n^4-1044*n^3+16044*n^2-118296*n+350388)*Floor[n/4]+(27*n^4-1044*n^3+16044*n^2-118296*n+360348)*Floor[(n+1)/4] +(96*n^2-1920*n+22248)*Floor[n/5]+(48*n^2-960*n+10224)*Floor[(n+1)/5]+(48*n^2-960*n+12024)*Floor[(n+2)/5]+(48*n^2-960*n+10224)*Floor[(n+3)/5] +(492*n^2-10344*n+73960)*Floor[(n+1)/6] +1968*Floor[n/7]+984*Floor[(n+1)/7]+984*Floor[(n+2)/7]+984*Floor[(n+3)/7]+984*Floor[(n+4)/7]+984*Floor[(n+5)/7] +9960*Floor[n/8]+9960*Floor[(n+3)/8] +1800*Floor[(n+1)/10]-1800*Floor[(n+2)/10]+1800*Floor[(n+3)/10]). Alternative formula (Vaclav Kotesovec, after values computed by Andrey Khalyavin, Jun 20 2011): a(n) = n^2*(n^10/720-n^9/12+661*n^8/288-153*n^7/4+615887*n^6/1440-80581*n^5/24+1801697*n^4/96-295355*n^3/4+9389033*n^2/48-626899*n/2+142789469/630 +(n^8/96-7*n^7/12+1411*n^6/96-5227*n^5/24+199399*n^4/96-52217*n^3/4+843309*n^2/16-745349*n/6+2315441/18)*(-1)^n +(9*n^4/4-87*n^3+1337*n^2-9858*n+29614)*Cos[Pi*n/2] +2*(123*n^2-2586*n+18490)*Cos[Pi*n/3]/9+2*(6*n^4-216*n^3+3213*n^2-23778*n+77282)*Cos[2*Pi*n/3]/9 +415*(Cos[Pi*n/4]+Cos[3*Pi*n/4]) +8/5*Cos[Pi*n/5]*(75*Cos[2*Pi*n/5]+(927-80*n+4*n^2)*Cos[3*Pi*n/5]) +328/7*(Cos[2*Pi*n/7]+Cos[4*Pi*n/7]+Cos[6*Pi*n/7])). Recurrence: a(n) = a(n-124) + 5a(n-123) + 19a(n-122) + 53a(n-121) + 126a(n-120) + 256a(n-119) + 460a(n-118) + 731a(n-117) + 1024a(n-116) + 1234a(n-115) + 1180a(n-114) + 631a(n-113) - 677a(n-112) - 2917a(n-111) - 6108a(n-110) - 9923a(n-109) - 13657a(n-108) - 16137a(n-107) - 15876a(n-106) - 11304a(n-105) - 1172a(n-104) + 14879a(n-103) + 35916a(n-102) + 59190a(n-101) + 80301a(n-100) + 93334a(n-99) + 92030a(n-98) + 70850a(n-97) + 26815a(n-96) - 39130a(n-95) - 120942a(n-94) - 207185a(n-93) - 282105a(n-92) - 327419a(n-91) - 326009a(n-90) - 265142a(n-89) - 140929a(n-88) + 39571a(n-87) + 256518a(n-86) + 479114a(n-85) + 668872a(n-84) + 785798a(n-83) + 795775a(n-82) + 677688a(n-81) + 430187a(n-80) + 74064a(n-79) - 347112a(n-78) - 773130a(n-77) - 1134433a(n-76) - 1364780a(n-75) - 1412189a(n-74) - 1250448a(n-73) - 885628a(n-72) - 357906a(n-71) + 262286a(n-70) + 885029a(n-69) + 1413752a(n-68) + 1762777a(n-67) + 1870496a(n-66) + 1712484a(n-65) + 1305033a(n-64) + 705009a(n-63) - 705009a(n-61) - 1305033a(n-60) - 1712484a(n-59) - 1870496a(n-58) - 1762777a(n-57) - 1413752a(n-56) - 885029a(n-55) - 262286a(n-54) + 357906a(n-53) + 885628a(n-52) + 1250448a(n-51) + 1412189a(n-50) + 1364780a(n-49) + 1134433a(n-48) + 773130a(n-47) + 347112a(n-46) - 74064a(n-45) - 430187a(n-44) - 677688a(n-43) - 795775a(n-42) - 785798a(n-41) - 668872a(n-40) - 479114a(n-39) - 256518a(n-38) - 39571a(n-37) + 140929a(n-36) + 265142a(n-35) + 326009a(n-34) + 327419a(n-33) + 282105a(n-32) + 207185a(n-31) + 120942a(n-30) + 39130a(n-29) - 26815a(n-28) - 70850a(n-27) - 92030a(n-26) - 93334a(n-25) - 80301a(n-24) - 59190a(n-23) - 35916a(n-22) - 14879a(n-21) + 1172a(n-20) + 11304a(n-19) + 15876a(n-18) + 16137a(n-17) + 13657a(n-16) + 9923a(n-15) + 6108a(n-14) + 2917a(n-13) + 677a(n-12) - 631a(n-11) - 1180a(n-10) - 1234a(n-9) - 1024a(n-8) - 731a(n-7) - 460a(n-6) - 256a(n-5) - 126a(n-4) - 53a(n-3) - 19a(n-2) - 5a(n-1). MATHEMATICA (* General formulas (denominator and recurrence) for k nonattacking queens on an n X n toroidal board: *) inversef[j_]:=(m=2; While[j>2*Fibonacci[m-1], m=m+1]; m); denomt[k_, par_]:=(x-1)^(2k+1)*Product[Cyclotomic[j, x]^(2*(k-inversef[j]+1)+par), {j, 2, 2*Fibonacci[k-1]}]; Table[denomt[k, 1], {k, 1, 7}]//TraditionalForm Table[Sum[Coefficient[Expand[denomt[k, 1]], x, i]*Subscript[a, n-i], {i, 0, Exponent[denomt[k, 1], x]}], {k, 1, 7}]//TraditionalForm CROSSREFS Cf. A172517, A172518, A172519, A173775, A178720. Sequence in context: A251308 A251301 A277792 * A061622 A128990 A061619 Adjacent sequences:  A178719 A178720 A178721 * A178723 A178724 A178725 KEYWORD nonn AUTHOR Vaclav Kotesovec, Jun 07 2010 STATUS approved

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Last modified June 24 01:48 EDT 2021. Contains 345404 sequences. (Running on oeis4.)