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A185868 (Odd,odd)-polka dot array in the natural number array A000027, by antidiagonals.

%I A185868 #18 Jun 18 2025 11:54:17
%S A185868 1,4,6,11,13,15,22,24,26,28,37,39,41,43,45,56,58,60,62,64,66,79,81,83,
%T A185868 85,87,89,91,106,108,110,112,114,116,118,120,137,139,141,143,145,147,
%U A185868 149,151,153,172,174,176,178,180,182,184,186,188,190,211,213,215,217,219,221,223,225,227,229,231,254,256,258,260,262,264,266,268,270,272,274,276,301,303,305,307,309,311,313,315,317,319,321,323,325,352,354,356,358,360,362,364,366,368,370,372,374,376,378
%N A185868 (Odd,odd)-polka dot array in the natural number array A000027, by antidiagonals.
%C A185868 This is one of four polka dot arrays in the natural number array A000027:
%C A185868 (odd,odd): A185868
%C A185868 (odd,even): A185869
%C A185868 (even,odd): A185870
%C A185868 (even,even): A185871
%C A185868 row 1: A084849
%C A185868 col 1: A000384
%C A185868 col 2: A091823
%C A185868 diag (1,13,...): A102083
%C A185868 diag (4,24,...): A085250
%C A185868 antidiagonal sums: A059722
%H A185868 G. C. Greubel, <a href="/A185868/b185868.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185868 T(n,k) = 2*n-1+(n+k-2)*(2*n+2*k-3).
%e A185868 The natural number array A000027 has northwest corner
%e A185868   1...2...4...7...11
%e A185868   3...5...8...12..17
%e A185868   6...9...13..18..24
%e A185868   10..14..19..25..32
%e A185868   15..20..26..33..41
%e A185868 The numbers in (odd,odd) positions comprise A185868:
%e A185868   1....4....11...22...37
%e A185868   6....13...24...39...58
%e A185868   15...26...41...60...83
%e A185868   28...43...62...85...112
%t A185868 f[n_,k_]:=2n-1+(n+k-2)(2n+2k-3);
%t A185868 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
%t A185868 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%o A185868 (Python)
%o A185868 from math import isqrt, comb
%o A185868 def A185868(n):
%o A185868     a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
%o A185868     x = n-comb(a,2)
%o A185868     y = a-x+1
%o A185868     return y*((y+(c:=x<<1)<<1)-7)+x*(c-5)+5 # _Chai Wah Wu_, Jun 18 2025
%Y A185868 Cf. A000027 (as an array), A185872, A185869, A185870, A185871.
%K A185868 nonn,tabl
%O A185868 1,2
%A A185868 _Clark Kimberling_, Feb 05 2011