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%I #9 Sep 27 2015 12:20:45
%S 27015,306822,5136108,65795210,900035309,11880244815,143441370184,
%T 1729766633007,22265080961901,168952079085709,2291315315054161,
%U 13548765068035154,126091710238160991,392553127438900097
%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 3, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero.
%C Column 5 of A253342.
%H R. H. Hardin, <a href="/A253339/b253339.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = a(n-1) +12*a(n-4) -12*a(n-5) -66*a(n-8) +66*a(n-9) +220*a(n-12) -220*a(n-13) -495*a(n-16) +495*a(n-17) +792*a(n-20) -792*a(n-21) -924*a(n-24) +924*a(n-25) +792*a(n-28) -792*a(n-29) -495*a(n-32) +495*a(n-33) +220*a(n-36) -220*a(n-37) -66*a(n-40) +66*a(n-41) +12*a(n-44) -12*a(n-45) -a(n-48) +a(n-49) for n>91.
%F Empirical for n mod 4 = 0: a(n) = (1253718133571584/467775)*n^12 - (1513575466139648/2835)*n^11 + (2293916482476703744/42525)*n^10 - (10240538993757282304/2835)*n^9 + (2498319895108335604444/14175)*n^8 - (56372942016156158682847/8640)*n^7 + (518664943050032916670372864177/2786918400)*n^6 - (189950558749707182726511656527/46448640)*n^5 + (11893630815834705657880230536429/174182400)*n^4 - (2440524928938759640404827034881/2903040)*n^3 + (47997287704865098816381223181763/6652800)*n^2 - (3475619264569625236509990347/90)*n + 97227899315922344041473464 for n>42.
%F Empirical for n mod 4 = 1: a(n) = (1253718133571584/467775)*n^12 - (81366073437323264/155925)*n^11 + (2204607325781622784/42525)*n^10 - (9721122824862588928/2835)*n^9 + (2350080923280126442204/14175)*n^8 - (1843494536839979967367421/302400)*n^7 + (482245712450116369126333216177/2786918400)*n^6 - (352060495755776403334188807107/92897280)*n^5 + (176018690343312950929169938308659/2786918400)*n^4 - (180483651448192705985047774869181/232243200)*n^3 + (22730977993898379305602673223789221/3406233600)*n^2 - (4062932882110474918962264294976801/113541120)*n + (5921177910506699205563221003321/65536) for n>42.
%F Empirical for n mod 4 = 2: a(n) = (1253718133571584/467775)*n^12 - (127971771809792/231)*n^11 + (2460988065933426688/42525)*n^10 - (3775085148094078976/945)*n^9 + (2842392669764506704604/14175)*n^8 - (153767402849239130686003/20160)*n^7 + (622591666829741083907718967729/2786918400)*n^6 - (13003640128490986320870174787/2580480)*n^5 + (60178034678938929572606181280721/696729600)*n^4 - (1056451534992381069260369948851/967680)*n^3 + (2048259502276179188648286046083857/212889600)*n^2 - (10428214142894520033522078422561/197120)*n + (140156750324367273059426454255/1024) for n>42.
%F Empirical for n mod 4 = 3: a(n) = (1253718133571584/467775)*n^12 - (78231778103394304/155925)*n^11 + (2036184077462142976/42525)*n^10 - (1725141750208323584/567)*n^9 + (286232913699354347332/2025)*n^8 - (503275686098940984003967/100800)*n^7 + (379150800535770732951179416753/2786918400)*n^6 - (265450410346187732124530812801/92897280)*n^5 + (127119169081152486853990503111899/2786918400)*n^4 - (124671656538414998327787479596231/232243200)*n^3 + (2142222256636383869016639843851267/486604800)*n^2 - (2555607822575278265396684930499931/113541120)*n + (3545181274023444705684213351905/65536) for n>42.
%e Some solutions for n=1
%e ..0..2..1..1..1..1..1....0..0..0..1..0..1..1....0..1..1..2..2..2..2
%e ..2..1..1..1..1..1..1....0..1..0..0..0..2..0....2..3..1..1..2..2..2
%e ..1..0..2..1..1..1..2....0..0..1..1..0..0..1....1..0..2..2..2..2..2
%e Knight distance matrix for n=1
%e ..0..3..2..3..2..3..4
%e ..3..4..1..2..3..4..3
%e ..2..1..4..3..2..3..4
%K nonn
%O 1,1
%A _R. H. Hardin_, Dec 30 2014
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