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A253421 Number of (n+2)X(5+2) nonnegative integer arrays with all values the knight distance from the upper left minus as much as 1, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero. 1

%I #9 Sep 27 2015 17:43:33

%S 1581,10411,215037,1432500,9787192,31393746,187516434,402193875,

%T 1885771797,2402676252,10791372467,13668072759,29495644141,

%U 31712757323,83214252197,92712352801,154900321588,160193732925,330016199502

%N Number of (n+2)X(5+2) nonnegative integer arrays with all values the knight distance from the upper left minus as much as 1, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero.

%C Column 5 of A253424.

%H R. H. Hardin, <a href="/A253421/b253421.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = a(n-1) +4*a(n-4) -4*a(n-5) -6*a(n-8) +6*a(n-9) +4*a(n-12) -4*a(n-13) -a(n-16) +a(n-17) for n>35.

%F Empirical for n mod 4 = 0: a(n) = (36118528/3)*n^4 - (1157303200/3)*n^3 + (520897460059/96)*n^2 - (927400346113/24)*n + 113007591457 for n>18.

%F Empirical for n mod 4 = 1: a(n) = (36118528/3)*n^4 - (1103125408/3)*n^3 + (483238622299/96)*n^2 - (1699831438517/48)*n + (3301652661093/32) for n>18.

%F Empirical for n mod 4 = 2: a(n) = (36118528/3)*n^4 - (1247599520/3)*n^3 + (600461819995/96)*n^2 - (567913877425/12)*n + (1175858264163/8) for n>18.

%F Empirical for n mod 4 = 3: a(n) = (36118528/3)*n^4 - 337609696*n^3 + (402428173147/96)*n^2 - (426295607213/16)*n + (2227476245245/32) for n>18.

%e Some solutions for n=2

%e ..0..2..2..3..2..2..3....0..2..1..3..2..3..3....0..2..2..2..2..2..3

%e ..2..3..1..2..2..3..2....2..3..1..2..2..3..2....2..3..1..2..3..3..2

%e ..1..1..3..2..2..2..3....2..1..3..2..2..2..3....1..1..3..2..1..3..3

%e ..4..2..2..1..2..3..3....4..1..3..2..2..3..3....4..1..2..2..3..3..2

%e Knight distance matrix for n=2

%e ..0..3..2..3..2..3..4

%e ..3..4..1..2..3..4..3

%e ..2..1..4..3..2..3..4

%e ..5..2..3..2..3..4..3

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 31 2014

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)