A185738 Rectangular array T(n,k) = 2^n + k - 2, by antidiagonals.

1, 2, 3, 3, 4, 7, 4, 5, 8, 15, 5, 6, 9, 16, 31, 6, 7, 10, 17, 32, 63, 7, 8, 11, 18, 33, 64, 127, 8, 9, 12, 19, 34, 65, 128, 255, 9, 10, 13, 20, 35, 66, 129, 256, 511, 10, 11, 14, 21, 36, 67, 130, 257, 512, 1023, 11, 12, 15, 22, 37, 68, 131, 258, 513, 1024, 2047, 12, 13, 16, 23, 38, 69, 132, 259, 514, 1025, 2048, 4095, 13, 14, 17, 24, 39, 70, 133, 260, 515, 1026, 2049, 4096, 8191, 14, 15, 18, 25, 40, 71, 134, 261, 516, 1027, 2050, 4097, 8192, 16383

Offset

1,2

Comments

This array fits in a chain: ...->(weight array)->A185738->(accumulation array->...

See the Mathematica code and A144112.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

T(n,k) = 2^n + k - 2, n>=1, k>=1.

Example

Northwest corner:
1....2....3....4....5
3....4....5....6....7
7....8....9....10...11
15...16...17...18...19
31...32...33...34...35

Mathematica

(* This program prints the array T=A185738, the accumulation array A185739 of T, and the weight array A185740 of T. *)
f[n_, 0]:=0; f[0, k_]:=0;
f[n_, k_]:=2^n+k-2;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]  (* Array A185738 *)
Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten
s[n_, k_]:=Sum[f[i, j], {i, 1, n}, {j, 1, k}]; (* accumulation array of {f(n, k)} *)
FullSimplify[s[n, k]]  (* formula for accumulation array *)
TableForm[Table[s[n, k], {n, 1, 10}, {k, 1, 15}]]  (* Array A185739 *)
Table[s[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten
w[m_, n_]:=f[m, n]+f[m-1, n-1]-f[m, n-1]-f[m-1, n]/; Or[m>0, n>0];
TableForm[Table[w[n, k], {n, 1, 10}, {k, 1, 15}]] (* Array A185740 *)
Table[w[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten

Crossrefs

Cf. A144112, A185739, A185740.

Sequence in context: A078467 A154217 A361644 * A239361 A266362 A241956

Adjacent sequences: A185735 A185736 A185737 * A185739 A185740 A185741

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 02 2011

Status

approved