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A329319 Number of (directed) Hamiltonian paths in K_{n} X K_{3}. 2
6, 60, 1512, 83520, 8869680, 1621680480, 472907393280, 207307564531200, 130417226086775000, 113438068529746060800, 132325125941706622848000, 201817805274824171102208000, 393912091245344751592447334400 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Equivalently, number of full spectrum rook's walks on a (n X 3) board.

LINKS

Table of n, a(n) for n=1..13.

FORMULA

a(n) = 6*n! * Sum_{j=1..n} T(n,j) for n>0,

where

T(n,j) = Sum_{k=1..2^(j-1)-1} Sum_{p=0..2^j-1} 1/(n-j)! * Product_{i=0..2} A!*R(A,f(i,j,k)-1), for n>=j>1 with T(n,1)=1,

A = n - 2*f(i,j,k) + [i=0] + [i=h(j,k)] - Sum_{q=0..j-1} L(p,q)*[i=g(j-q,floor(k/2^q))],

h(n,k) = k mod 2 + floor((2 + k mod 2 - h(n-1,floor(k/2)))/2), for 0<=k<2^(n-1), n>1 with h(1,0) = 1,

g(n,k) = h(n,k+(-1)^k), for 0<=k<2^(n-1), n>1 with g(1,0) = 2,

f(i,n,k) = f(i,n-1,floor(k/2)) + [i=h(n,k)], for 0<=i<3, 0<=k<2^(n-1), n>1 with f(i,1,0) = [0<=i<2],

R(n,k) = C(n+k,k), and

L(n,k) = floor(n/2^k) mod 2.

(The expression for L(n,k) came from n = Sum_{k=0..floor(log_2(n))} L(n,k)*2^k.)

Also T(n,2) = (n!/(n-1))^2 = A162991(n-1) for n>1.

T(n,3) = 3*T(n,2) + ((n^2-4)/(n-1)^2)*T(n-1,2) for n>2.

To simplify computing we can get rid of R(n,k), i.e., n!*R(n,k) = (n+k)!/k!.

Also h(n,k) = 2 - A091297(k), for 0<k<2^(n-1), n>1 with h(n,0) = n mod 2 for n>0.

[h(n,k)=0] = 1 - A035263(k), for 0<k<2^(n-1), n>1 with [h(n,0)=0] = 1 - n mod 2 for n>0.

[h(n,k)=1] = 1 - A035263(floor((k+1)/2)) = A035263(k) + A035263(k+1) - 1, for 0<k<2^(n-1), n>1 with [h(n,0)=1] = n mod 2 for n>0.

[h(n,k)=2] = 1 - A035263(k+1), for 0<=k<2^(n-1), n>0.

f(2,n,k) = A329320(k), for 0<=k<2^(n-1), n>0.

f(0,n,k) = 1 + A329320(2^(n-1)-k-1), for 0<=k<2^(n-1), n>0.

f(1,n,k) = n - A329320(k) - A329320(2^(n-1)-k-1), for 0<=k<2^(n-1), n>0.

CROSSREFS

Cf. A035263, A091297, A096121, A162991, A269565, A329320.

Sequence in context: A285955 A001416 A251184 * A003267 A271681 A010574

Adjacent sequences:  A329316 A329317 A329318 * A329320 A329321 A329322

KEYWORD

nonn

AUTHOR

Mikhail Kurkov, Nov 10 2019

STATUS

approved

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Last modified June 19 12:56 EDT 2021. Contains 345129 sequences. (Running on oeis4.)