A061993 Number of ways to place 7 nonattacking queens on a 7 X n board.

0, 0, 0, 0, 0, 0, 0, 40, 312, 2038, 9632, 37248, 120104, 335010, 835056, 1897702, 3998456, 7907094, 14818300, 26512942, 45562852, 75580634, 121520020, 190031678, 289879092, 432420154, 632159540, 907376502, 1280833348

Offset: 0

Author

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes, part of V. Kotesovec, Between chessboard and computer, 1996, pp. 204 - 206.
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A061989, A061990, A061991, A061992.

Programs

Mathematica

CoefficientList[Series[2*x^7*(20-4*x+331*x^2-88*x^3+1292*x^4-1356*x^5+2019*x^6 +264*x^7-2857*x^8+6472*x^9-7616*x^10+7462*x^11-7831*x^12+8326*x^13-5672*x^14 +1998*x^15-308*x^16-142*x^17+510*x^18-284*x^19-220*x^20+320*x^21-140*x^22 +24*x^23)/(1-x)^8, {x, 0, 40}], x] (* Vincenzo Librandi, May 12 2013 *)

SageMath

def p(x): return 20-4*x+331*x^2-88*x^3+1292*x^4-1356*x^5+2019*x^6 +264*x^7-2857*x^8+6472*x^9-7616*x^10+7462*x^11-7831*x^12+8326*x^13-5672*x^14 +1998*x^15-308*x^16-142*x^17+510*x^18-284*x^19-220*x^20+320*x^21-140*x^22 +24*x^23
[( 2*x^7*p(x)/(1-x)^8 ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Apr 29 2022

Formula

Explicit formula (V. Kotesovec, 1992): a(n) = n^7 - 63*n^6 + 1879*n^5 - 34411*n^4 + 417178*n^3 - 3336014*n^2 + 16209916*n - 36693996, n >= 23.

Recurrence: a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), n >= 31.

G.f.: 2*x^7*(20 - 4*x + 331*x^2 - 88*x^3 + 1292*x^4 - 1356*x^5 + 2019*x^6 + 264*x^7 - 2857*x^8 + 6472*x^9 - 7616*x^10 + 7462*x^11 - 7831*x^12 + 8326*x^13 - 5672*x^14 + 1998*x^15 - 308*x^16 - 142*x^17 + 510*x^18 - 284*x^19 - 220*x^20 + 320*x^21 - 140*x^22 + 24*x^23)/(1 - x)^8.