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%I #18 Jul 01 2024 15:27:22
%S 0,0,1,9,15,1,41,0,55,72,9,156,36,204,262,144,135,289,209,316,111,117,
%T 406,309,527,261,342,860,804,36,954,1200,624,605,1257,969,1400,741,
%U 849,1856,1639,0,1721,2076,855,701,1770,1101,1719,397,426,1980,1416,2449,1142
%N a(n) is the difference between T = A000217(n^2) and the greatest square not exceeding T.
%H Michael De Vlieger, <a href="/A373330/b373330.txt">Table of n, a(n) for n = 0..9998</a>
%H Hugo Pfoertner, <a href="/A373330/a373330.pdf">Logarithmic plot of a(n) vs n</a>, n <= 10^5, lower envelope of terms > 0 shown in red, zoom for details.
%F a(n) = A000217(n^2) - A373329(n)^2.
%F a(A002315(n)) = 0.
%t Array[PolygonalNumber[#^2] - Floor[Sqrt[(#^4 + #^2)/2]]^2 &, 55, 0] (* _Michael De Vlieger_, Jun 02 2024 *)
%o (PARI) a(n) = my(T=(n^4+n^2)/2); T-sqrtint(T)^2
%o (Python)
%o from sympy import integer_nthroot
%o def A373330(n): return (T:=(n**4 + n**2) // 2)-(integer_nthroot(T,2)[0])**2
%o # _Karl-Heinz Hofmann_, Jul 01 2024
%Y Cf. A000217, A000290, A002315, A037270, A061288, A373329, A373333.
%Y A373331 and A373332 are the coordinates of the observed lower envelope of this sequence.
%K nonn,look
%O 0,4
%A _Hugo Pfoertner_, Jun 02 2024
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