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

Author

Clark Kimberling, Feb 02 2011

Keywords

• nonn
• tabl

Comments

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

See the Mathematica code and A144112.

Examples

			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
		

Links

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

Crossrefs

Cf. A144112, A185739, A185740.

Programs

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

Formula

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