A340156
Square array read by upward antidiagonals: T(n, k) is the number of n-ary strings of length k containing 00.
Original entry on oeis.org
1, 1, 3, 1, 5, 8, 1, 7, 21, 19, 1, 9, 40, 79, 43, 1, 11, 65, 205, 281, 94, 1, 13, 96, 421, 991, 963, 201, 1, 15, 133, 751, 2569, 4612, 3217, 423, 1, 17, 176, 1219, 5531, 15085, 20905, 10547, 880, 1, 19, 225, 1849, 10513, 39186, 86241, 92935, 34089, 1815
Offset: 2
For n = 3 and k = 4, there are 21 strings: {0000, 0001, 0002, 0010, 0011, 0012, 0020, 0021, 0022, 0100, 0200, 1000, 1001, 1002, 1100, 1200, 2000, 2001, 2002, 2100, 2200}.
Square table T(n,k):
k=2: k=3: k=4: k=5: k=6: k=7:
n=2: 1 3 8 19 43 94
n=3: 1 5 21 79 281 963
n=4: 1 7 40 205 991 4612
n=5: 1 9 65 421 2569 15085
n=6: 1 11 96 751 5531 39186
n=7: 1 13 133 1219 10513 87199
n=8: 1 15 176 1849 18271 173608
n=9: 1 17 225 2665 29681 317817
-
m[r_] := Normal[With[{p = 1/n}, SparseArray[{Band[{1, 2}] -> p, {i_, 1} /; i <= r -> 1 - p, {r + 1, r + 1} -> 1}]]];
T[n_, k_, r_] := MatrixPower[m[r], k][[1, r + 1]]*n^k;
Reverse[Table[T[n, k - n + 2, 2], {k, 2, 11}, {n, 2, k}], 2] // Flatten (* Robert P. P. McKone, Jan 26 2021 *)
A341050
Cube array read by upward antidiagonals ignoring zero and empty terms: T(n, k, r) is the number of n-ary strings of length k, containing r consecutive 0's.
Original entry on oeis.org
1, 1, 1, 3, 1, 1, 3, 1, 5, 8, 1, 1, 3, 1, 5, 8, 1, 7, 21, 19, 1, 1, 3, 1, 5, 8, 1, 7, 21, 20, 1, 9, 40, 81, 43, 1, 1, 3, 1, 5, 8, 1, 7, 21, 20, 1, 9, 40, 81, 47, 1, 11, 65, 208, 295, 94, 1, 1, 3, 1, 5, 8, 1, 7, 21, 20, 1, 9, 40, 81, 48, 1, 11, 65, 208, 297, 107, 1, 13, 96, 425, 1024, 1037, 201
Offset: 2
For n = 5, k = 6 and r = 4, there are 65 strings: {000000, 000001, 000002, 000003, 000004, 000010, 000011, 000012, 000013, 000014, 000020, 000021, 000022, 000023, 000024, 000030, 000031, 000032, 000033, 000034, 000040, 000041, 000042, 000043, 000044, 010000, 020000, 030000, 040000, 100000, 100001, 100002, 100003, 100004, 110000, 120000, 130000, 140000, 200000, 200001, 200002, 200003, 200004, 210000, 220000, 230000, 240000, 300000, 300001, 300002, 300003, 300004, 310000, 320000, 330000, 340000, 400000, 400001, 400002, 400003, 400004, 410000, 420000, 430000, 440000}
The first seven slices of the tetrahedron (or pyramid) are:
-----------------Slice 1-----------------
1
-----------------Slice 2-----------------
1
1 3
-----------------Slice 3-----------------
1
1 3
1 5 8
-----------------Slice 4-----------------
1
1 3
1 5 8
1 7 21 19
-----------------Slice 5-----------------
1
1 3
1 5 8
1 7 21 20
1 9 40 81 43
-----------------Slice 6-----------------
1
1 3
1 5 8
1 7 21 20
1 9 40 81 47
1 11 65 208 295 94
-----------------Slice 7-----------------
1
1 3
1 5 8
1 7 21 20
1 9 40 81 48
1 11 65 208 297 107
1 13 96 425 1024 1037 201
Cf.
A005408,
A003215,
A005917,
A022521,
A022522,
A022523,
A022524,
A022525,
A022526,
A022527,
A022528,
A022529,
A022530,
A022531,
A022532,
A022533,
A022534,
A022535,
A022536,
A022537,
A022538,
A022539,
A022540 (k=x, r=1, where x is the x-th Nexus Number).
Cf.
A000567 [(k=4, r=2),(k=5, r=3),(k=6, r=4),...,(k=x, r=x-2)].
Cf.
A103532 [(k=6, r=3),(k=7, r=4),(k=8, r=5),...,(k=x, r=x-3)].
-
m[r_, n_] := Normal[With[{p = 1/n}, SparseArray[{Band[{1, 2}] -> p, {i_, 1} /; i <= r -> 1 - p, {r + 1, r + 1} -> 1}]]]; T[n_, k_, r_] := MatrixPower[m[r, n], k][[1, r + 1]]*n^k; DeleteCases[Transpose[PadLeft[Reverse[Table[T[n, k, r], {k, 2, 8}, {r, 2, k}, {n, 2, r}], 2]], 2 <-> 3], 0, 3] // Flatten
Showing 1-2 of 2 results.