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

%I A185870 #16 Jun 18 2025 11:56:32
%S A185870 3,8,10,17,19,21,30,32,34,36,47,49,51,53,55,68,70,72,74,76,78,93,95,
%T A185870 97,99,101,103,105,122,124,126,128,130,132,134,136,155,157,159,161,
%U A185870 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
%N A185870 (Even,odd)-polka dot array in the natural number array A000027, by antidiagonals.
%C A185870 This is the third of four polka dot arrays in the array A000027.  See A185868.
%C A185870 row 1: A033816
%C A185870 col 1: A014105
%C A185870 col 2: -A168244
%C A185870 antidiagonal sums: A061317
%C A185870 antidiagonal sums: 3*(octahedral numbers) = 3*A005900.
%H A185870 G. C. Greubel, <a href="/A185870/b185870.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185870 T(n,k) = 2*n + (n+k-1)*(2*n+2*k-3), k>=1, n>=1.
%e A185870 Northwest corner:
%e A185870   3....8....17...30...47
%e A185870   10...19...32...49...70
%e A185870   21...34...51...72...97
%e A185870   36...53...74...99...128
%t A185870 f[n_,k_]:=2n+(2n+2k-3)(n+k-1);
%t A185870 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
%t A185870 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%o A185870 (Python)
%o A185870 from math import comb, isqrt
%o A185870 def A185870(n):
%o A185870     a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
%o A185870     x = n-comb(a,2)
%o A185870     y = a-x+1
%o A185870     return y*((y+(c:=x<<1)<<1)-5)+x*(c-3)+3 # _Chai Wah Wu_, Jun 18 2025
%Y A185870 Cf. A000027 (as an array), A185868, A185869, A185871.
%K A185870 nonn,tabl
%O A185870 1,1
%A A185870 _Clark Kimberling_, Feb 05 2011