A252816 - OEIS

%I #8 Dec 06 2018 10:10:19

%S 7,40,187,684,2078,5490,13015,28299,57338,109549,199168,347035,582831,

%T 947837,1498290,2309416,3480225,5139158,7450681,10622926,14916484,

%U 20654460,28233905,38138745,50954332,67383747,88265990,114596197

%N Number of n X 4 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

%H R. H. Hardin, <a href="/A252816/b252816.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (1/8064)*n^8 + (1/288)*n^7 + (103/2880)*n^6 + (17/90)*n^5 + (703/1152)*n^4 + (451/288)*n^3 + (2869/3360)*n^2 + (209/120)*n + 2.

%F Conjectures from _Colin Barker_, Dec 06 2018: (Start)

%F G.f.: x*(7 - 23*x + 79*x^2 - 147*x^3 + 176*x^4 - 138*x^5 + 67*x^6 - 18*x^7 + 2*x^8) / (1 - x)^9.

%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>9.

%F (End)

%e Some solutions for n=4:

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

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

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

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

%Y Column 4 of A252820.

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 22 2014