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A355881
Table read by descending antidiagonals: T(k,n) (k >= 0, n>= 1) is number of ways to (k+2)-color a 3 X n grid ignoring the variations of two colors.
3
1, 1, 2, 1, 9, 3, 1, 41, 49, 4, 1, 187, 801, 169, 5, 1, 853, 13095, 7141, 441, 6, 1, 3891, 214083, 301741, 38897, 961, 7, 1, 17749, 3499929, 12749989, 3430789, 153921, 1849, 8, 1, 80963, 57218481, 538747549, 302602093, 24653151, 488401, 3249, 9
OFFSET
0,3
COMMENTS
With variations, the number of ways to color a 3 X 1 grid is (k+2)*(k+1)^2. The number of variations of two colors is (k+2)*(k+1). Therefore, T(k,1) = k+1. Only for k=1, the number of variations of two colors equals the number of permutations of all colors, see A020698.
T(0,n) = A000012(n) = constant 1
T(1,n) = A020698(n-1)
T(2,n) = A355882(n)
T(3,n) = A355883(n)
FORMULA
T(k,n) = k*(k^2 + k + 3) * T(k,n-1) - (k^4 + k^3 + k^2-1) * T(k,n-2)
with T(k,1) = k+1, T(k,2) = (k^2+k+1)^2.
G.f.: x*(k + 1 - (k^2 + k - 1)*x) / (1 - k*(k^2 + k + 3)*x + (k^4 + k^3 + k^2 - 1)*x^2).
EXAMPLE
Table begins:
k\n_1____2______3_________4___________5_____________6________________7
0: 1 1 1 1 1 1 1
1: 2 9 41 187 853 3891 17749
2: 3 49 801 13095 214083 3499929 57218481
3: 4 169 7141 301741 12749989 538747549 22764640981
4: 5 441 38897 3430789 302602093 26690078241 2354115497017
5: 6 961 153921 24653151 3948635061 632443246191 101296892084301
6: 7 1849 488401 129007867 34076567743 9001098120361 2377580042199049
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gerhard Kirchner, Jul 24 2022
STATUS
approved