A047659 -id:A047659 - cp's OEIS frontend

A172202 Number of ways to place 3 nonattacking kings on a 3 X n board.

0, 0, 8, 34, 105, 248, 490, 858, 1379, 2080, 2988, 4130, 5533, 7224, 9230, 11578, 14295, 17408, 20944, 24930, 29393, 34360, 39858, 45914, 52555, 59808, 67700, 76258, 85509, 95480, 106198, 117690, 129983, 143104, 157080, 171938, 187705

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A047659, A061989, A061996.

[0] cat [(n-2)*(9*n^2-45*n+70)/2: n in [2..50]]; // G. C. Greubel, Apr 29 2022

CoefficientList[Series[x^2*(8+2*x+17*x^2)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{4,-6,4,-1},{0,0,8,34,105},40] (* Harvey P. Dale, Oct 07 2023 *)

[(1/8)*(n-2)*(9*(2*n-5)^2+55) +17*bool(n==1) for n in (1..50)] # G. C. Greubel, Apr 29 2022

a(n) = (n-2)*(9*n^2 - 45*n + 70)/2, n>=2.
G.f.: x^3*(8+2*x+17*x^2)/(1-x)^4. - Vaclav Kotesovec, Mar 24 2010
E.g.f.: 70 + 17*x + (1/2)*(-140 + 106*x - 36*x^2 + 9*x^3)*exp(x). - G. C. Greubel, Apr 29 2022

More terms from Vincenzo Librandi, May 27 2013

A172212 Number of ways to place 3 nonattacking knights on a 3 X n board.

Original entry on oeis.org

1, 12, 36, 100, 233, 456, 796, 1280, 1935, 2788, 3866, 5196, 6805, 8720, 10968, 13576, 16571, 19980, 23830, 28148, 32961, 38296, 44180, 50640, 57703, 65396, 73746, 82780, 92525, 103008, 114256, 126296, 139155, 152860, 167438, 182916, 199321

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A172134, A061989, A047659, A172202.

CoefficientList[Series[(6 x^6 - 8 x^5 + 2 x^4 + 24 x^3 - 6 x^2 + 8 x + 1) / (x - 1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (9n^3 - 45n^2 + 122n - 144)/2, n>=4.

G.f.: x*(6*x^6-8*x^5+2*x^4+24*x^3-6*x^2+8*x+1)/(x-1)^4. - Vaclav Kotesovec, Mar 25 2010

A172138 Number of ways to place 3 nonattacking zebras on an n X n board.

Original entry on oeis.org

0, 4, 84, 452, 1772, 5596, 14888, 34640, 72712, 140716, 255036, 437968, 718980, 1136092, 1737376, 2582576, 3744848, 5312620, 7391572, 10106736, 13604716, 18056028, 23657560, 30635152, 39246296, 49782956, 62574508, 77990800

Offset: 1

Vaclav Kotesovec, Jan 26 2010

A zebra is a (fairy chess) leaper [2,3].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A047659, A172124, A172134, A172137.

[0,4,84,452,1772] cat [(n^6 -27*n^4 +120*n^3 +74*n^2 -1608*n +2976)/6: n in [6..50]]; // G. C. Greubel, Apr 19 2022

CoefficientList[Series[4x(1+14*x-13*x^2+58*x^3-29*x^4-9*x^5+x^6+ 33*x^7- 45*x^8 +23*x^9-4*x^10)/(1-x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,4,84,452,1772,5596,14888,34640,72712,140716,255036,437968},30] (* Harvey P. Dale, Mar 11 2023 *)

[0,4,84,452,1772]+[(n^6 -27*n^4 +120*n^3 +74*n^2 -1608*n +2976)/6 for n in (6..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^6 - 27*n^4 + 120*n^3 + 74*n^2 - 1608*n + 2976)/6, n >=6.

G.f.: 4*x^2*(1 + 14*x - 13*x^2 + 58*x^3 - 29*x^4 - 9*x^5 + x^6 + 33*x^7 - 45*x^8 + 23*x^9 - 4*x^10)/(1-x)^7. - Vaclav Kotesovec, Mar 25 2010

A173429 Number of ways to place 3 nonattacking nightriders on an n X n board.

Original entry on oeis.org

0, 4, 36, 276, 1152, 3920, 10568, 25348, 53848, 106292, 194732, 339416, 562652, 899796, 1388008, 2083908, 3044992, 4356344, 6102144, 8404204, 11380564, 15199100, 20019856, 26067112, 33551812, 42766092, 53981600, 67570804

Offset: 1

Vaclav Kotesovec, Feb 18 2010

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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.
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Wikipedia, Nightrider (chess)
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -2, 2, 0, 1, 0, 0, -3, 0, 2, 0, 4, -4, 0, -2, 0, 3, 0, 0, -1, 0, -2, 2, 0, 1, 0, -2, 1).

Cf. A172141, A047659, A172134.

CoefficientList[Series[-(36 x^29 + 124 x^28 + 496 x^27 + 1128 x^26 + 2632 x^25 + 4280 x^24 + 7160 x^23 + 9296 x^22 + 12936 x^21 + 14828 x^20 + 18828 x^19 + 20164 x^18 + 23820 x^17 + 23684 x^16 + 25460 x^15 + 22972 x^14 + 22412 x^13 + 18532 x^12 + 16820 x^11 + 12996 x^10 + 10912 x^9 + 7552 x^8 + 5428 x^7 + 3012 x^6 + 1652 x^5 + 604 x^4 + 204 x^3 + 28 x^2 + 4 x) / ((x + 1)^4 (x - 1)^7 (x^2 + 1) (x^2 + x + 1) (x^8 + x^6 + x^4 + x^2 + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)
LinearRecurrence[{2,0,-1,0,-2,2,0,1,0,0,-3,0,2,0,4,-4,0,-2,0,3,0,0,-1,0,-2,2,0,1,0,-2,1},{0,4,36,276,1152,3920,10568,25348,53848,106292,194732,339416,562652,899796,1388008,2083908,3044992,4356344,6102144,8404204,11380564,15199100,20019856,26067112,33551812,42766092,53981600,67570804,83876732,103365728,126463668},30] (* Harvey P. Dale, Dec 27 2015 *)

a(n) = 1/6*n^6-5/6*n^5+4031/1440*n^4-621/100*n^3+3313/288*n^2-2623/150*n+82321/43200 + (1/4*n^3-25/32*n^2+77/50*n-43/64)*(-1)^n - (1+(-1)^n)/8*cos(Pi*n/2) + 8/27*(-1)^n*cos(Pi*n/3) + (-4*(-1)^n+(sqrt(5)+3+(1-sqrt(5)/5)*(-1)^n)*n)*4/25*cos(Pi*n/5) + (sqrt(58*sqrt(5)+130)-sqrt(50-22*sqrt(5))*(-1)^n/5)*16/25*sin(Pi*n/5) + (-4+(sqrt(5)/5+1+(3-sqrt(5))*(-1)^n)*n)*4/25*cos(2*Pi*n/5) + (sqrt(22*sqrt(5)+50)/5-sqrt(130-58*sqrt(5))*(-1)^n)*16/25*sin(2*Pi*n/5).
Recurrence: a(n) = 2*a(n-1)-a(n-3)-2*a(n-5)+2*a(n-6)+a(n-8)-3*a(n-11)+2*a(n-13)+4*a(n-15)-4*a(n-16)-2*a(n-18)+3*a(n-20)-a(n-23)-2*a(n-25)+2*a(n-26)+a(n-28)-2*a(n-30)+a(n-31), n>=32.
G.f.: -(36*x^30+124*x^29+496*x^28+1128*x^27+2632*x^26+4280*x^25+7160*x^24+9296*x^23+12936*x^22+14828*x^21+18828*x^20+20164*x^19+23820*x^18+23684*x^17+25460*x^16+22972*x^15+22412*x^14+18532*x^13+16820*x^12+12996*x^11+10912*x^10+7552*x^9+5428*x^8+3012*x^7+1652*x^6+604*x^5+204*x^4+28*x^3+4*x^2)/((x+1)^4*(x-1)^7*(x^2+1)*(x^2+x+1)*(x^8+x^6+x^4+x^2+1)^2). - Vaclav Kotesovec, Mar 22 2010

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

A172201 Number of ways to place 3 nonattacking amazons (superqueens) on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 48, 424, 1976, 6616, 17852, 41544, 86660, 166288, 298616, 508200, 827168, 1296744, 1968676, 2907016, 4189772, 5910944, 8182400, 11136168, 14926536, 19732600, 25760588, 33246664, 42459476, 53703216, 67320392, 83695144

Offset: 1

Vaclav Kotesovec, Jan 29 2010

An amazon (superqueen) moves like a queen and a knight.

Panos Louridas, idee & form 93/2007, pp. 2936-2938.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (5,-8,0,14,-14,0,8,-5,1).

Cf. A047659, A051223, A051224, A061989, A172200.

R:=PowerSeriesRing(Integers(), 40); [0,0,0,0] cat Coefficients(R!( 4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7 ))); // G. C. Greubel, Apr 29 2022

CoefficientList[Series[4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7), {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)

[(1/24)*(4*n^6 -40*n^5 +62*n^4 +628*n^3 -2904*n^2 +4074*n -1341 +3*(-1)^n*(2*n-1)) -4*(5*bool(n==1) +2*bool(n==2) -bool(n==3)) for n in (1..40)] # G. C. Greubel, Apr 29 2022

Explicit formula (Panos Louridas, 2007): a(n) = (2*n^6 - 20*n^5 + 31*n^4 + 314*n^3 - 1452*n^2 + 2040*n - 672)/12 if n is even (n >= 4) and a(n) = (2*n^6 - 20*n^5 + 31*n^4 + 314*n^3 - 1452*n^2 + 2034*n - 669)/12 if n is odd (n >= 5).
G.f.: 4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7). - Vaclav Kotesovec, Mar 24 2010
a(n) = (1/24)*(4*n^6 - 40*n^5 + 62*n^4 + 628*n^3 - 2904*n^2 + 4074*n - 1341 + 3*(-1)^n*(2*n-1)) - 20*[n=1] - 8*[n=2] + 4*[n=3]. - G. C. Greubel, Apr 29 2022

A190395 Number of ways to place 3 nonattacking grasshoppers on a chessboard of size n x n.

Original entry on oeis.org

0, 4, 76, 516, 2172, 6860, 17904, 40796, 83976, 159732, 285220, 483604, 785316, 1229436, 1865192, 2753580, 3969104, 5601636, 7758396, 10566052, 14172940, 18751404, 24500256, 31647356, 40452312, 51209300, 64250004, 79946676

Offset: 1

Vaclav Kotesovec, May 10 2011

The Grasshopper moves on the same lines as a queen, but must jump over a hurdle to land on the square immediately beyond.

Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p.829

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)

Cf. A047659.

CoefficientList[Series[-4 x (3 x^5 - 7 x^4 + 4 x^3 + 17 x^2 + 12 x + 1) / (x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

Explicit formula (C. Poisson, 1990): a(n) = 1/6*(n-1)(n^5 +n^4 -2*n^3 -22*n^2 +76*n -72).

G.f.: -4*x^2*(3*x^5 -7*x^4 +4*x^3 +17*x^2 +12*x +1)/(x-1)^7.

A202654 Number of ways to place 3 nonattacking semi-queens on an n X n board.

Original entry on oeis.org

0, 0, 3, 52, 370, 1620, 5285, 14168, 33012, 69240, 133815, 242220, 415558, 681772, 1076985, 1646960, 2448680, 3552048, 5041707, 7018980, 9603930, 12937540, 17184013, 22533192, 29203100, 37442600, 47534175, 59796828, 74589102, 92312220, 113413345, 138388960

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A099152, A047659, A103220, A202655, A202656, A202657.

Rest@ CoefficientList[Series[-x^3*(17 x^3 + 69 x^2 + 31 x + 3)/(x - 1)^7, {x, 0, 32}], x] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = 1/6*(n-2)*(n-1)*n*(n^3-5*n^2+8*n-3).

G.f.: -x^3*(17*x^3 + 69*x^2 + 31*x + 3)/(x-1)^7.

A172207 Number of ways to place 3 nonattacking bishops on a 3 X n board.

Original entry on oeis.org

1, 6, 26, 86, 211, 426, 758, 1234, 1881, 2726, 3796, 5118, 6719, 8626, 10866, 13466, 16453, 19854, 23696, 28006, 32811, 38138, 44014, 50466, 57521, 65206, 73548, 82574, 92311, 102786, 114026, 126058, 138909, 152606, 167176, 182646, 199043, 216394, 234726, 254066

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A172124, A061989, A047659, A061996.

CoefficientList[Series[(2 x^6 + 14 x^3 + 8 x^2 + 2 x + 1) / (x - 1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (9n^3 - 45n^2 + 106n - 108)/2, n>=4.

G.f.: x*(2*x^6+14*x^3+8*x^2+2*x+1)/(x-1)^4. - Vaclav Kotesovec, Mar 25 2010

A178717 Degree of denominator of GF for number of ways to place k nonattacking queens on an n X n board.

Original entry on oeis.org

3, 5, 9, 17, 37, 81, 197, 477, 1197, 3077, 7989, 20649, 53885, 140601, 366917, 959685, 2511477, 6571681, 17202449, 45027677, 117871345, 308581637, 807852685, 2114904397, 5536838045, 14495554593, 37949503089, 99352690141, 260108204933

Offset: 1

Vaclav Kotesovec, Jun 07 2010

nonn

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

Cf. A036464, A047659, A061994, A108792, A176186, A000010, A000045.

Table[2*k + 1 + Sum[Sum[2*j*EulerPhi[i], {i, Fibonacci[k - j] + 1, Fibonacci[k - j + 1]}], {j, 1, k - 1}], {k, 1, 20}]

Explicit formula (Vaclav Kotesovec, May 31 2010), for k>1 : d(k) = 2*k+1+Sum[Sum[2*j*EulerPhi[i],{i,Fibonacci[k-j]+1,Fibonacci[k-j+1]}],{j,1,k-1}].

Asymptotic formula: d(k) ~ 6/(5*Pi^2)*((1+Sqrt[5])/2)^(2*k+1) or d(k) ~ 3*(1+Sqrt[5])/Pi^2*Fibonacci[k]^2.

cp's OEIS Frontend

A172202 Number of ways to place 3 nonattacking kings on a 3 X n board.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A172212 Number of ways to place 3 nonattacking knights on a 3 X n board.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A172138 Number of ways to place 3 nonattacking zebras on an n X n board.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A173429 Number of ways to place 3 nonattacking nightriders on an n X n board.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A172201 Number of ways to place 3 nonattacking amazons (superqueens) on an n X n board.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A190395 Number of ways to place 3 nonattacking grasshoppers on a chessboard of size n x n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A202654 Number of ways to place 3 nonattacking semi-queens on an n X n board.

Views