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

1,1

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.

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-3), k>=1, n>=1.

Example

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

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

Prog

(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

Crossrefs

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

Sequence in context: A128699 A104816 A181022 * A341938 A341939 A244353

Adjacent sequences: A185867 A185868 A185869 * A185871 A185872 A185873

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 05 2011

Status

approved