A104188 -id:A104188 - cp's OEIS frontend

A187850 T(n,k) is the number of n-step king-knight's tours (piece capable of both kinds of moves) on a k X k board summed over all starting positions.

1, 4, 0, 9, 12, 0, 16, 56, 24, 0, 25, 132, 304, 24, 0, 36, 240, 1056, 1400, 0, 0, 49, 380, 2312, 7620, 5328, 0, 0, 64, 552, 4048, 20952, 49776, 16032, 0, 0, 81, 756, 6264, 41652, 177040, 292776, 35328, 0, 0, 100, 992, 8960, 69456, 408048, 1400168, 1533064, 49536, 0, 0

Offset: 1

R. H. Hardin, Mar 14 2011

			Table starts:
.1..4.....9.......16........25........36........49........64.......81.....100
.0.12....56......132.......240.......380.......552.......756......992....1260
.0.24...304.....1056......2312......4048......6264......8960....12136...15792
.0.24..1400.....7620.....20952.....41652.....69456....104268...146088..194916
.0..0..5328....49776....177040....408048....744696...1183632..1723120.2362864
.0..0.16032...292776...1400168...3807828...7700944..13082348.19910456
.0..0.35328..1533064..10353632..33908456..76860784.140714528
.0..0.49536..7067600..71450504.288493336.741624088
.0..0.32256.28260592.458862208
.0..0.....0.96217616
Some n=4 solutions for 4 X 4:
..1..2..0..0....0..1..0..0....1..0..0..0....0..0..0..0....0..0..0..4
..0..0..3..0....2..0..0..0....0..2..0..0....0..0..0..0....0..1..0..3
..0..0..0..0....0..3..0..0....0..3..0..0....0..2..0..0....0..0..2..0
..0..0..0..4....0..0..0..4....0..4..0..0....0..1..3..4....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..99

Row 2 is A104188(n-1).

Empirical: T(1,k) = k^2.
Empirical: T(2,k) = 16*k^2 - 36*k + 20.
Empirical: T(3,k) = 240*k^2 - 904*k + 832 for k>3.
Empirical: T(4,k) = 3504*k^2 - 17748*k + 21996 for k>5.
Empirical: T(5,k) = 50128*k^2 - 312688*k + 476944 for k>7.
Empirical: T(6,k) = 706880*k^2 - 5180252*k + 9274644 for k>9.
Empirical: T(7,k) = 9862808*k^2 - 82444808*k + 168212080 for k>11.
Empirical: T(8,k) = 136526552*k^2 - 1275583564*k + 2906368876 for k>13.

A187027 T(n,k) is the number of n-step one or two collinear space at a time queen's tours on a k X k board summed over all starting positions.

Original entry on oeis.org

1, 4, 0, 9, 12, 0, 16, 56, 24, 0, 25, 132, 296, 24, 0, 36, 240, 1008, 1344, 0, 0, 49, 380, 2232, 7056, 5120, 0, 0, 64, 552, 3936, 19568, 45152, 15760, 0, 0, 81, 756, 6120, 39348, 161256, 263000, 36816, 0, 0, 100, 992, 8784, 66360, 376248, 1251720, 1384152, 57904, 0, 0

Offset: 1

R. H. Hardin, Mar 02 2011

			Table starts:
.1..4.....9.......16........25........36.......49........64.......81.....100
.0.12....56......132.......240.......380......552.......756......992....1260
.0.24...296.....1008......2232......3936.....6120......8784....11928...15552
.0.24..1344.....7056.....19568.....39348....66360....100380...141408..189444
.0..0..5120....45152....161256....376248...696992...1121176..1647008.2273384
.0..0.15760...263000...1251720...3443028..7080688..12213336.18821144
.0..0.36816..1384152...9151912..30203792.69641344.129718288
.0..0.57904..6516592..62903536.254189928
.0..0.45856.27116200.405255984
.0..0.....0.98268864
Some n=4 solutions for 4 X 4:
..0..0..0..0....0..0..0..0....0..0..0..0....0..4..0..0....0..0..1..0
..0..0..0..0....0..2..0..0....4..0..1..0....0..0..3..0....0..0..2..0
..3..0..4..0....0..4..0..0....0..0..2..0....1..0..0..0....0..0..3..0
..0..2..0..1....0..3..0..1....0..0..3..0....2..0..0..0....0..4..0..0

R. H. Hardin, Table of n, a(n) for n = 1..98

Row 2 is A104188(n-1).

A054777 a(n) = 4n(4n-1)(4n-2)(4*n-3).

Original entry on oeis.org

0, 24, 1680, 11880, 43680, 116280, 255024, 491400, 863040, 1413720, 2193360, 3258024, 4669920, 6497400, 8814960, 11703240, 15249024, 19545240, 24690960, 30791400, 37957920, 46308024, 55965360, 67059720, 79727040, 94109400, 110355024, 128618280, 149059680, 171845880

Offset: 0

Henry Bottomley, May 19 2000

L. B. W. Jolley, Summation of Series, Dover, 1961.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 268.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A052762, A060541, A104188.

[4*n*(4*n-1)*(4*n-2)*(4*n-3): n in [0..30]]; // Vincenzo Librandi, Oct 04 2011

a[n_] := 4*n*(4*n-1)*(4*n-2)*(4*n-3); Array[a, 40, 0] (* Amiram Eldar, Mar 08 2022 *)

a(n) = A052762(4n) = 24*A060541(n).
Sum_{n>=1} 1/a(n) = log(2)/4 - Pi/24 = 0.0423871012404116... [Jolley eq. 242] - Benoit Cloitre, Apr 05 2002
G.f. -24*x*(1 + 65*x + 155*x^2 + 35*x^3) / (x-1)^5. - R. J. Mathar, Oct 03 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = log(sqrt(2)-1)/(6*sqrt(2)) - log(2)/24 + (1/(6*sqrt(2)) - 1/16)*Pi. - Amiram Eldar, Mar 08 2022

A257942 a(n) = (n+1)*(n+2)/A014695(n+1), where A014695 is repeat (1, 2, 2, 1).

Original entry on oeis.org

1, 3, 12, 20, 15, 21, 56, 72, 45, 55, 132, 156, 91, 105, 240, 272, 153, 171, 380, 420, 231, 253, 552, 600, 325, 351, 756, 812, 435, 465, 992, 1056, 561, 595, 1260, 1332, 703, 741, 1560, 1640, 861, 903, 1892, 1980, 1035, 1081, 2256, 2352, 1225, 1275, 2652

Offset: 0

Paul Curtz, Jul 14 2015

Consider, for n >= 0, a sequence s(n). A useful transform is wi(n) = s(0), s(2), s(3), ..., i.e., s(n) without s(1).
For s(n) = 1/(n+1), wi(n)= 1, 1/3, 1/4, 1/5, ..., whose inverse binomial transform is f(n) = 1, -2/3, 7/12, -11/20, 8/15, -11/21, 29/56, -37/72, 23/45, -28/55, 67/132, -79/156, 46/91, -53/105, 121/240, -137/272, ...
The denominator of f(n) is a(n), for n >= 0.
If the numerator of f(n) is b(n), then it can be seen that b(n+1) = -(-1)^n* A226089(n).
Alternating a(n) - b(n) with a(n) + b(n) yields 0, 1, 5, 9, ... = A160050(n+1).
a(4n+1) is linked to the Rydberg-Ritz spectra of hydrogen.
h(n) = 0, 0, 1, 1, 4, 4, 3, 3, 8, 8, 5, 5, ... = duplicated A022998(n).
A022998(n) is linked to the Balmer series (see A246943(n)).
With an initial 0 and offset=0, a(-n) = a(n). Then (a(n+10) - a(n-10))/10 = 1, 6, 10, 7, 9, 22, 26, ... = g(n). a(n) mod 9 is of period 20.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A002378(n+1), A014695(n+1)=A130658(n+2), A014634, A033567(n+1), A104188(n+1), 4*A007742(n+1), A160050 (in A226089), A022998, A109613, A000217(n+1), A246943.

A257942:=n->(n+1)*(n+2)/(3/2+(-1)^((2*n+7+(-1)^n)/4)/2): seq(A257942(n), n=0..100); # Wesley Ivan Hurt, Jul 18 2015

CoefficientList[Series[-(x^6 + 9 x^4 - 8 x^3 + 9 x^2 + 1)/((x - 1)^3 (x^2 + 1)^3), {x, 0, 50}], x] (* Michael De Vlieger, Jul 14 2015 *)
(* Using inverse binomial transform *) s[0]=1; s[n_] := 1/(n+2); f[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*s[k], {k, 0, n}]; Table[f[n]//Denominator, {n, 0, 50}] (* Jean-François Alcover, Jul 14 2015 *)
LinearRecurrence[{3, -6, 10, -12, 12, -10, 6, -3, 1}, {1, 3, 12, 20, 15, 21, 56, 72, 45}, 55] (* Vincenzo Librandi, Jul 15 2015 *)

Vec(-(x^6+9*x^4-8*x^3+9*x^2+1)/((x-1)^3*(x^2+1)^3) + O(x^100)) \\ Colin Barker, Jul 14 2015

a(n)=(n+1)*(n+2)/if(n%4<2,2,1) \\ Charles R Greathouse IV, Jul 14 2015

a(4n)    = (2*n+1)*(4*n+1).
a(4n+1)  = (2*n+1)*(4*n+3).
a(4n+2)  = (4*n+3)*(4*n+4).
a(4n+3)  = (4*n+4)*(4*n+5).
a(n) = A064038(n+2) * (period 4: repeat 1, 1, 4, 4).
From Colin Barker, Jul 14 2015: (Start)
a(n) = (-1/8+i/8)*(((-3-i*3)+i*(-i)^n+i^n)*(2+3*n+n^2)) where i=sqrt(-1).
G.f.: -(x^6+9*x^4-8*x^3+9*x^2+1) / ((x-1)^3*(x^2+1)^3). (End)
a(n) = h(n+2) * A109613(n+1).
a(n) = (n+1)*(n+2) * period 4:repeat (1, 1, 2, 2) /2.
From Wesley Ivan Hurt, Jul 18 2015: (Start)
a(n) = (n+1)*(n+2)/(3/2+(-1)^((2*n+7+(-1)^n)/4)/2).
a(n) = 3*a(n-1)-6*a(n-2)+10*a(n-3)-12*a(n-4)+12*a(n-5)-10*a(n-6)+6*a(n-7)-3*a(n-8)+a(n-9), n>9. (End)
Sum_{n>=0} 1/a(n) = Pi/4 + 1. - Amiram Eldar, Aug 14 2022

cp's OEIS Frontend

A187850 T(n,k) is the number of n-step king-knight's tours (piece capable of both kinds of moves) on a k X k board summed over all starting positions.

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A187027 T(n,k) is the number of n-step one or two collinear space at a time queen's tours on a k X k board summed over all starting positions.

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A054777 a(n) = 4n(4n-1)(4n-2)(4*n-3).

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Magma

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Formula

A257942 a(n) = (n+1)*(n+2)/A014695(n+1), where A014695 is repeat (1, 2, 2, 1).

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cp's OEIS Frontend

A187850 T(n,k) is the number of n-step king-knight's tours (piece capable of both kinds of moves) on a k X k board summed over all starting positions.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A187027 T(n,k) is the number of n-step one or two collinear space at a time queen's tours on a k X k board summed over all starting positions.

Views

Author

Keywords

Examples

Links

Crossrefs

A054777 a(n) = 4*n*(4*n-1)*(4*n-2)*(4*n-3).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A257942 a(n) = (n+1)*(n+2)/A014695(n+1), where A014695 is repeat (1, 2, 2, 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A054777 a(n) = 4n(4n-1)(4n-2)(4*n-3).