|
%I #10 Sep 27 2015 17:43:51
%S 5716,41139,1399041,20832926,187516434,1042904812,10608304158,
%T 31966946561,220223373747,451565510308,2982127961746,5158972747725,
%U 25580058296829,30372010984513,146090574735814,173965238271521
%N Number of (n+2)X(7+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 7 of A253424.
%H R. H. Hardin, <a href="/A253423/b253423.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>55.
%F Empirical for n mod 4 = 0: a(n) = (718127759360/3)*n^4 - (39490511684608/3)*n^3 + (14255146023889861/48)*n^2 - (38862279334406495/12)*n + 14257489007470705 for n>38.
%F Empirical for n mod 4 = 1: a(n) = (718127759360/3)*n^4 - 12729635040256*n^3 + (13453895148949957/48)*n^2 - (24046237081399511/8)*n + (209379461771811511/16) for n>38.
%F Empirical for n mod 4 = 2: a(n) = (718127759360/3)*n^4 - (41061416158208/3)*n^3 + (15381923770175941/48)*n^2 - (43432473054074767/12)*n + (65925438070533675/4) for n>38.
%F Empirical for n mod 4 = 3: a(n) = (718127759360/3)*n^4 - (36618000647168/3)*n^3 + (12344285144411077/48)*n^2 - (63323055557477789/24)*n + (175889901064136459/16) for n>38.
%e Some solutions for n=2
%e ..0..3..2..3..2..3..4..4..3....0..3..2..3..2..2..3..4..3
%e ..2..3..1..2..3..4..2..4..4....3..3..1..2..3..3..3..3..4
%e ..2..1..3..3..2..3..4..4..3....2..1..3..2..2..3..3..4..4
%e ..4..2..3..2..3..3..3..3..4....4..2..3..2..2..3..3..3..4
%e Knight distance matrix for n=2
%e ..0..3..2..3..2..3..4..5..4
%e ..3..4..1..2..3..4..3..4..5
%e ..2..1..4..3..2..3..4..5..4
%e ..5..2..3..2..3..4..3..4..5
%K nonn
%O 1,1
%A _R. H. Hardin_, Dec 31 2014
| 