A185740 -id:A185740 - cp's OEIS frontend

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

Clark Kimberling, Feb 02 2011

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

See the Mathematica code and A144112.

			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

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

Cf. A144112, A185739, A185740.

(* 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

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

A185739 Accumulation array of A185738, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 6, 10, 11, 10, 18, 25, 26, 15, 28, 42, 56, 57, 21, 40, 62, 90, 119, 120, 28, 54, 85, 128, 186, 246, 247, 36, 70, 111, 170, 258, 378, 501, 502, 45, 88, 140, 216, 335, 516, 762, 1012, 1013, 55, 108, 172, 266, 417, 660, 1030, 1530, 2035, 2036, 66, 130, 207, 320, 504, 810, 1305, 2056, 3066, 4082, 4083, 78, 154, 245, 378, 596, 966, 1587, 2590, 4106, 6138, 8177, 8178, 91

Offset: 1

Clark Kimberling, Feb 02 2011

This arrays is a member of a chain; see A185738.

			Northwest corner:
1....3....6....10....15
4....10...18...28....40
11...25...42...62....85
26...56...90...128...170

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

Rows 1 to 4: A000217, A028562, A140675, 2*A098847
Columns 1 to 3: A000295, A000247, A068293.
Cf. A185738, A185740.

(* See A185738 *)
f[n_, k_] := (k/2)*(4*(2^n - 1) + (k - 3)*n);
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 10}]]  (* Array A185739 *)
Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 11 2017 *)

T(n,k) = k*(4*(2^n-1)+(k-3)*n), k>=1, n>=1.

cp's OEIS Frontend

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

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