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Empirical for column k:

k=1: a(n) = 3*a(n-1) -2*a(n-2)

k=2: a(n) = 2*a(n-1) +3*a(n-2) -4*a(n-3)

k=3: a(n) = 3*a(n-1) +2*a(n-2) -4*a(n-3)

k=4: a(n) = a(n-1) +13*a(n-2) +3*a(n-3) -16*a(n-4)

k=5: a(n) = 4*a(n-1) +15*a(n-2) -38*a(n-3) -52*a(n-4) +72*a(n-5)

k=6: (order 9 recurrence)

k=7: (order 10 recurrence)

Empirical: a(n) = a(n-1) + 4*a(n-2).

Conjectures from Colin Barker, Feb 27 2018: (Start)

G.f.: 4*x*(3 + 5*x) / (1 - x - 4*x^2).

a(n) = (2^(-1-n)*((1-sqrt(17))^n*(-19+5*sqrt(17)) + (1+sqrt(17))^n*(19+5*sqrt(17)))) / sqrt(17).

(End)

Empirical: a(n) = 13*a(n-2) + 16*a(n-3).

Empirical g.f.: 16*x*(3 + 13*x + 13*x^2) / (1 - 13*x^2 - 16*x^3). - Colin Barker, Apr 04 2018

Empirical: a(n) = 3*a(n-1) + 18*a(n-2) - 20*a(n-3) - 72*a(n-4).

Empirical g.f.: 16*x*(6 + 15*x - 39*x^2 - 94*x^3) / ((1 + 2*x)*(1 - 5*x - 8*x^2 + 36*x^3)). - Colin Barker, Apr 04 2018

Empirical: a(n) = 3*a(n-1) + 48*a(n-2) - 78*a(n-3) - 720*a(n-4) + 612*a(n-5) + 4232*a(n-6) - 1392*a(n-7) - 8192*a(n-8).

Empirical g.f.: 16*x*(12 + 49*x - 287*x^2 - 1123*x^3 + 2127*x^4 + 8058*x^5 - 4668*x^6 - 17344*x^7) / (1 - 3*x - 48*x^2 + 78*x^3 + 720*x^4 - 612*x^5 - 4232*x^6 + 1392*x^7 + 8192*x^8). - Colin Barker, Apr 04 2018

Empirical: a(n)=7*a(n-1)+57*a(n-2)-332*a(n-3)-1122*a(n-4)+5412*a(n-5)+9248*a(n-6)-35984*a(n-7)-26624*a(n-8)+82304*a(n-9)

Empirical: a(n)=7*a(n-1)+162*a(n-2)-935*a(n-3)-10301*a(n-4)+50843*a(n-5)+343104*a(n-6)-1473463*a(n-7)-6611678*a(n-8)+24960936*a(n-9)+75737800*a(n-10)-253751744*a(n-11)-503746592*a(n-12)+1513486656*a(n-13)+1775133568*a(n-14)-4833383936*a(n-15)-2525235200*a(n-16)+6285426688*a(n-17)