A185871 (Even,even)-polka dot array in the natural number array A000027, by antidiagonals.

5, 12, 14, 23, 25, 27, 38, 40, 42, 44, 57, 59, 61, 63, 65, 80, 82, 84, 86, 88, 90, 107, 109, 111, 113, 115, 117, 119, 138, 140, 142, 144, 146, 148, 150, 152, 173, 175, 177, 179, 181, 183, 185, 187, 189, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 255, 257, 259, 261, 263, 265, 267, 269, 271, 273, 275, 302, 304, 306, 308, 310, 312, 314, 316, 318, 320, 322, 324, 353, 355, 357, 359, 361, 363, 365, 367, 369, 371, 373, 375, 377, 408, 410, 412, 414, 416, 418, 420, 422, 424, 426, 428, 430, 432, 434

Offset: 1

Author

Clark Kimberling, Feb 05 2011

Keywords

• nonn
• tabl

Comments

This is the fourth of four polka dot arrays in the natural number array A000027. See A185868.

row 1: A096376

col 1: A014106

col 2: A071355

diag (5,25,...): A080856

diag (12,40,...): A033586

antidiagonal sums: A048395 (sums of consecutive squares)

Examples

			Northwest corner:
  5....12...23...38...57
  14...25...40...59...82
  27...42...61...84...111
  44...63...86...113..144
		

Links

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

Crossrefs

Cf. A000027 (as an array), A185868, A185869, A185870.

Programs

Mathematica

f[n_,k_]:=2n+(n+k-1)(2n+2k-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 A185871(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)-3)+x*(c-1)+1 # Chai Wah Wu, Jun 18 2025

Formula

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