A189150 -id:A189150 - cp's OEIS frontend

A189145 Number of n X 2 array permutations with each element making zero or one knight moves.

1, 1, 4, 16, 36, 81, 225, 625, 1600, 4096, 10816, 28561, 74529, 194481, 509796, 1336336, 3496900, 9150625, 23961025, 62742241, 164249856, 429981696, 1125736704, 2947295521, 7716041281, 20200652641, 52886200900, 138458410000

Offset: 1

R. H. Hardin, Apr 17 2011

nonn

Column 2 of A189150.

a(n+2) is number of ways to place k non-attacking knights on a 2 x n board, sum over all k>=0.

			All solutions for 3X2
..0..1....0..4....5..1....5..4
..2..3....2..3....2..3....2..3
..4..5....1..5....4..0....1..0

R. H. Hardin, Table of n, a(n) for n = 1..200

Table[FullSimplify[LucasL[2n+4]/25 + (3*Fibonacci[n+1] + Fibonacci[n]) * (2*Cos[(Pi*n)/2] + Sin[(Pi*n)/2])*2/25 + 7*(-1)^n/50 + 1/10], {n,1,20}] (* Vaclav Kotesovec, Nov 07 2011 *)

Empirical: a(n) = 3*a(n-1) -3*a(n-2) +6*a(n-3) -6*a(n-5) +3*a(n-6) -3*a(n-7) +a(n-8).
Empirical: G.f. -x*(1-2*x+4*x^2+x^3+3*x^5+x^7-6*x^4-3*x^6) / ( (x-1)*(1+x)*(x^2-3*x+1)*(x^4+3*x^2+1) ). - R. J. Mathar, Oct 15 2011
Explicit formula: ((3+sqrt(5))/2)^(n+2)/25 + ((3-sqrt(5))/2)^(n+2)/25 + (((sqrt(5)+1)/2)^(n+2) + ((sqrt(5)-1)/2)^(n+2))*4*cos((Pi*n)/2)/25 + (((sqrt(5)+1)/2)^(n+2) - ((sqrt(5)-1)/2)^(n+2))*2*sin((Pi*n)/2)/25 + 1/10 + 7/50*(-1)^n. - Vaclav Kotesovec, Nov 07 2011

A189146 Number of n X 3 array permutations with each element making zero or one knight moves.

Original entry on oeis.org

1, 4, 49, 569, 4372, 42689, 412189, 3988132, 38271921, 375573977, 3665309372, 35872284105, 350949375581, 3439343559628, 33682318930233, 330021363385529, 3233215326749252, 31680809629578289, 310402921706993341

Offset: 1

R. H. Hardin, Apr 17 2011

nonn

Column 3 of A189150.

			Some solutions for 4 X 3:
..0..8..3....0..1..2....0..1..7....0..8..7....5..1..2....0..1..7....7..6..3
..2..9..6....8.11.10....3..4..6....3..9.10....8..9..0....8..4..5....2..4.10
..5..7..1....6..7..3....5..2..9...11..2..1....6..7..3...11..2..3....1..0..9
..4.10.11....9..5..4....8.10.11....4..5..6....4.10.11....9.10..6....8..5.11

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n) = 9*a(n-1) +48*a(n-2) -371*a(n-3) +173*a(n-4) -4636*a(n-5) -21700*a(n-6) +321034*a(n-7) -261016*a(n-8) -939316*a(n-9) +8804712*a(n-10) -89239632*a(n-11) +89709648*a(n-12) +830637056*a(n-13) -2499914752*a(n-14) +6390654336*a(n-15) -987233536*a(n-16) -98495866880*a(n-17) +227291122176*a(n-18) -22499119616*a(n-19) -574642074624*a(n-20) +4090433287168*a(n-21) -8710999644160*a(n-22) -9262792159232*a(n-23) +33961426997248*a(n-24) -59705619185664*a(n-25) +164268932415488*a(n-26) +206996041138176*a(n-27) -959952477028352*a(n-28) +148261211865088*a(n-29) +225201818959872*a(n-30) -528765464084480*a(n-31) +2794884482203648*a(n-32) +3264721555816448*a(n-33) +9397359024799744*a(n-34) -32363504358916096*a(n-35) -14560459025809408*a(n-36) +41205664512475136*a(n-37) -65070379105779712*a(n-38) +117763114509795328*a(n-39) -26803976066301952*a(n-40) +132124530591137792*a(n-41) +236469835482005504*a(n-42) -1308758689225637888*a(n-43) +382298681149227008*a(n-44) +1688786500906385408*a(n-45) -1092767223451222016*a(n-46) -1774519408253730816*a(n-47) +3781671287689052160*a(n-48) -211699968811991040*a(n-49) -6414401302863806464*a(n-50) +3737846953229156352*a(n-51) +2510967898491584512*a(n-52) +740560663725735936*a(n-53) -4415779434636771328*a(n-54) +2508504992445366272*a(n-55) +54043195528445952*a(n-56) -1152921504606846976*a(n-57) +576460752303423488*a(n-58)
Contribution from Vaclav Kotesovec, Sep 01 2012: (Start)
Empirical: G.f.: -(1 - 8*x - 53*x^2 + 336*x^3 + 134*x^4 + 2846*x^5 + 19852*x^6 - 260036*x^7 + 6880*x^8 + 1292304*x^9 - 4702832*x^10 + 63471872*x^11 - 41560704*x^12 - 709453568*x^13 + 1351929600*x^14 - 3128104192*x^15 - 659457024*x^16 + 70691011072*x^17 - 106136367104*x^18 - 132922614784*x^19 + 318303350784*x^20 - 2065904152576*x^21 + 3225670889472*x^22 + 10372375166976*x^23 - 11884683280384*x^24 + 821650128896*x^25 - 70221817905152*x^26 - 122303538593792*x^27 + 415569654579200*x^28 + 388617697755136*x^29 - 457567875104768*x^30 + 122123978801152*x^31 - 134229549645824*x^32 - 7331344765943808*x^33 - 2581369716736000*x^34 + 28061409178812416*x^35 + 3074337254932480*x^36 - 27158215842070528*x^37 + 16916901758238720*x^38 - 8637373579526144*x^39 - 42490723492167680*x^40 - 200005314030862336*x^41 + 176918825432776704*x^42 + 705725146060554240*x^43 - 362309834634166272*x^44 - 1007318127542796288*x^45 + 222508717868843008*x^46 + 1439135376433217536*x^47 - 1225049467388952576*x^48 - 273971909362712576*x^49 + 2010997971109281792*x^50 - 2011772027295236096*x^51 + 145522562959409152*x^52 + 534802455750246400*x^53 + 22517998136852480*x^54 - 288230376151711744*x^55 + 144115188075855872*x^56)/( - 1 + 9*x + 48*x^2 - 371*x^3 + 173*x^4 - 4636*x^5 - 21700*x^6 + 321034*x^7 - 261016*x^8 - 939316*x^9 + 8804712*x^10 - 89239632*x^11 + 89709648*x^12 + 830637056*x^13 - 2499914752*x^14 + 6390654336*x^15 - 987233536*x^16 - 98495866880*x^17 + 227291122176*x^18 - 22499119616*x^19 - 574642074624*x^20 + 4090433287168*x^21 - 8710999644160*x^22 - 9262792159232*x^23 + 33961426997248*x^24 - 59705619185664*x^25 + 164268932415488*x^26 + 206996041138176*x^27 - 959952477028352*x^28 + 148261211865088*x^29 + 225201818959872*x^30 - 528765464084480*x^31 + 2794884482203648*x^32 + 3264721555816448*x^33 + 9397359024799744*x^34 - 32363504358916096*x^35 - 14560459025809408*x^36 + 41205664512475136*x^37 - 65070379105779712*x^38 + 117763114509795328*x^39 - 26803976066301952*x^40 + 132124530591137792*x^41 + 236469835482005504*x^42 - 1308758689225637888*x^43 + 382298681149227008*x^44 + 1688786500906385408*x^45 - 1092767223451222016*x^46 - 1774519408253730816*x^47 + 3781671287689052160*x^48 - 211699968811991040*x^49 - 6414401302863806464*x^50 + 3737846953229156352*x^51 + 2510967898491584512*x^52 + 740560663725735936*x^53 - 4415779434636771328*x^54 + 2508504992445366272*x^55 + 54043195528445952*x^56 - 1152921504606846976*x^57 + 576460752303423488*x^58)
Asymptotic: 0.045707910845127735589456 * 9.7983760587433722777622517835675^n
(End)

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A189145 Number of n X 2 array permutations with each element making zero or one knight moves.

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A189146 Number of n X 3 array permutations with each element making zero or one knight moves.

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A189147 Number of n X 4 array permutations with each element making zero or one knight moves.

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A189148 Number of nX5 array permutations with each element making zero or one knight moves.

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A189149 Number of nX6 array permutations with each element making zero or one knight moves.

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