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

1,1

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)

Links

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

Formula

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

Example

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

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

Prog

(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

Crossrefs

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

Sequence in context: A386519 A286242 A360137 * A037007 A357999 A066025

Adjacent sequences: A185868 A185869 A185870 * A185872 A185873 A185874

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 05 2011

Status

approved