A185868 (Odd,odd)-polka dot array in the natural number array A000027, by antidiagonals.

1, 4, 6, 11, 13, 15, 22, 24, 26, 28, 37, 39, 41, 43, 45, 56, 58, 60, 62, 64, 66, 79, 81, 83, 85, 87, 89, 91, 106, 108, 110, 112, 114, 116, 118, 120, 137, 139, 141, 143, 145, 147, 149, 151, 153, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 254, 256, 258, 260, 262, 264, 266, 268, 270, 272, 274, 276, 301, 303, 305, 307, 309, 311, 313, 315, 317, 319, 321, 323, 325, 352, 354, 356, 358, 360, 362, 364, 366, 368, 370, 372, 374, 376, 378

Offset: 1

Author

Clark Kimberling, Feb 05 2011

Keywords

• nonn
• tabl

Comments

This is one of four polka dot arrays in the natural number array A000027:

(odd,odd): A185868

(odd,even): A185869

(even,odd): A185870

(even,even): A185871

row 1: A084849

col 1: A000384

col 2: A091823

diag (1,13,...): A102083

diag (4,24,...): A085250

antidiagonal sums: A059722

Examples

			The natural number array A000027 has northwest corner
  1...2...4...7...11
  3...5...8...12..17
  6...9...13..18..24
  10..14..19..25..32
  15..20..26..33..41
The numbers in (odd,odd) positions comprise A185868:
  1....4....11...22...37
  6....13...24...39...58
  15...26...41...60...83
  28...43...62...85...112
		

Links

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

Crossrefs

Cf. A000027 (as an array), A185872, A185869, A185870, A185871.

Programs

Mathematica

f[n_,k_]:=2n-1+(n+k-2)(2n+2k-3);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

Python

from math import isqrt, comb
def A185868(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-comb(a,2)
    y = a-x+1
    return y*((y+(c:=x<<1)<<1)-7)+x*(c-5)+5 # Chai Wah Wu, Jun 18 2025

Formula

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