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 A253338 Number of (n+2)X(4+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. 1

%I

%S 7494,59519,722826,7184212,65795210,621408160,6203113283,46327921378,

%T 431150608201,2513803675142,21694110698916,91728436841790,

%U 701214300325482,2062603211477260,13146712894951064,29023341800426220

%N Number of (n+2)X(4+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 4 of A253342.

%H R. H. Hardin, <a href="/A253338/b253338.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = a(n-1) +12*a(n-2) -12*a(n-3) -66*a(n-4) +66*a(n-5) +220*a(n-6) -220*a(n-7) -495*a(n-8) +495*a(n-9) +792*a(n-10) -792*a(n-11) -924*a(n-12) +924*a(n-13) +792*a(n-14) -792*a(n-15) -495*a(n-16) +495*a(n-17) +220*a(n-18) -220*a(n-19) -66*a(n-20) +66*a(n-21) +12*a(n-22) -12*a(n-23) -a(n-24) +a(n-25) for n>55.

%F Empirical for n mod 2 = 0: a(n) = (8992587776/93555)*n^12 - (1754527694848/155925)*n^11 + (4454499745792/6075)*n^10 - (97850932330496/2835)*n^9 + (503920339091456/405)*n^8 - (163857161734094848/4725)*n^7 + (31746767654773668352/42525)*n^6 - (34889484014881237376/2835)*n^5 + (184886920914132728164/1215)*n^4 - (19246032261612188624813/14175)*n^3 + (485106832651191238006429/59400)*n^2 - (403481645478857418044233/13860)*n + 44942549657886439191 for n>30.

%F Empirical for n mod 2 = 1: a(n) = (8992587776/93555)*n^12 - (524891979776/51975)*n^11 + (734730911744/1215)*n^10 - (75896779177984/2835)*n^9 + (370458004717568/405)*n^8 - (113927599393374208/4725)*n^7 + (4158011324223812096/8505)*n^6 - (7143754455560924288/945)*n^5 + (105914562844059145828/1215)*n^4 - (10178334481445614975453/14175)*n^3 + (46495471701987542783201/11880)*n^2 - (84473612843189842494151/6930)*n + (120539555998778748191/8) for n>30.

%e Some solutions for n=2

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

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

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

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

%e Knight distance matrix for n=2

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

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

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

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

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 30 2014

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Last modified March 22 12:52 EDT 2023. Contains 361430 sequences. (Running on oeis4.)