A185870 (Even,odd)-polka dot array in the natural number array A000027, by antidiagonals.

3, 8, 10, 17, 19, 21, 30, 32, 34, 36, 47, 49, 51, 53, 55, 68, 70, 72, 74, 76, 78, 93, 95, 97, 99, 101, 103, 105, 122, 124, 126, 128, 130, 132, 134, 136, 155, 157, 159, 161, 163, 165, 167, 169, 171, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 233, 235, 237, 239, 241, 243, 245, 247, 249, 251, 253, 278, 280, 282, 284, 286, 288, 290, 292, 294, 296, 298, 300, 327, 329, 331, 333, 335, 337, 339, 341, 343, 345, 347, 349, 351, 380, 382, 384, 386, 388, 390, 392, 394, 396, 398, 400, 402, 404, 406

Offset: 1

Author

Clark Kimberling, Feb 05 2011

Keywords

• nonn
• tabl

Comments

This is the third of four polka dot arrays in the array A000027. See A185868.

row 1: A033816

col 1: A014105

col 2: -A168244

antidiagonal sums: A061317

antidiagonal sums: 3*(octahedral numbers) = 3*A005900.

Examples

			Northwest corner:
  3....8....17...30...47
  10...19...32...49...70
  21...34...51...72...97
  36...53...74...99...128
		

Links

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

Crossrefs

Cf. A000027 (as an array), A185868, A185869, A185871.

Programs

Mathematica

f[n_,k_]:=2n+(2n+2k-3)(n+k-1);
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 comb, isqrt
def A185870(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)-5)+x*(c-3)+3 # Chai Wah Wu, Jun 18 2025

Formula

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