A185869 (Odd,even)-polka dot array in the natural number array A000027; read by antidiagonals.

2, 7, 9, 16, 18, 20, 29, 31, 33, 35, 46, 48, 50, 52, 54, 67, 69, 71, 73, 75, 77, 92, 94, 96, 98, 100, 102, 104, 121, 123, 125, 127, 129, 131, 133, 135, 154, 156, 158, 160, 162, 164, 166, 168, 170, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252, 277, 279, 281, 283, 285, 287, 289, 291, 293, 295, 297, 299, 326, 328, 330, 332, 334, 336, 338, 340, 342, 344, 346, 348, 350, 379, 381, 383, 385, 387, 389, 391, 393, 395, 397, 399, 401, 403, 405

Offset: 1

Author

Clark Kimberling, Feb 05 2011

Keywords

• nonn
• tabl

Comments

This is the second of four polka dot arrays; see A185868.

row 1: A130883;

row 2: A100037;

row 3: A100038;

row 4: A100039;

col 1: A014107;

col 2: A033537;

col 3: A100040;

col 4: A100041;

diag (2,18,...): A077591;

diag (7,31,...): A157914;

diag (16,48,...): A035008;

diag (29,69,...): A108928;

antidiagonal sums: A033431;

antidiagonal sums: 2*(1^3, 2^3, 3^3, 4^3,...) = 2*A000578.

A060432(n) + n is odd if and only if n is in this sequence. - Peter Kagey, Feb 03 2016

Examples

			Northwest corner:
  2....7....16...29...46
  9....18...31...48...69
  20...33...50...71...96
  35...52...73...98...127
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000027 (as an array), A060432, A185868, A185870, A185871.

Programs

Haskell

a185869 n = a185869_list !! (n - 1)
a185869_list = scanl (+) 2 $ a' 1
  where  a' n = 2 * n + 3 : replicate n 2 ++ a' (n + 1)
-- Peter Kagey, Sep 02 2016

Mathematica

f[n_,k_]:=2n-1+(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 isqrt, comb
def A185869(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)+2 # Chai Wah Wu, Jun 18 2025

Formula

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