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A309945 a(n) = floor(n - sqrt(2*n-1)).

%I A309945 #58 Aug 04 2022 15:04:59
%S A309945 0,0,0,1,2,2,3,4,4,5,6,7,8,8,9,10,11,12,12,13,14,15,16,17,18,18,19,20,
%T A309945 21,22,23,24,24,25,26,27,28,29,30,31,32,32,33,34,35,36,37,38,39,40,40,
%U A309945 41,42,43,44,45,46,47,48,49,50,50,51,52,53,54,55,56,57,58,59,60,60
%N A309945 a(n) = floor(n - sqrt(2*n-1)).
%C A309945 The subsequence consisting of numbers that appear twice is A007590.
%C A309945 Sequence as triangle:
%C A309945    0;
%C A309945    0;
%C A309945    0;  1,  2;
%C A309945    2,  3,  4;
%C A309945    4,  5,  6,  7,  8;
%C A309945    8,  9, 10, 11, 12;
%C A309945   12, 13, 14, 15, 16, 17, 18;
%C A309945   18, 19, 20, 21, 22, 23, 24;
%C A309945   ...
%C A309945 a(1) = 0; for n > 1, a(n) is the number of squares strictly between 2*n - 2 and n^2.
%H A309945 <a href="/A309945/b309945.txt">Table of n, a(n) for n = 1..999</a>
%F A309945 a(n) = n-1-floor(sqrt(2*n-2)). - _Wesley Ivan Hurt_, Dec 03 2020
%e A309945 For n = 3, 2*n - 2 = 4, n^2 = 9, no square numbers strictly between 4 and 9, a(3) = 0.
%e A309945 For n=5, 2*n - 2 = 8, n^2 = 25, two square numbers (9, 16) strictly between 8 and 25, a(5) = 2.
%t A309945 Table[Floor[n-(2*n-1)^(1/2)],{n,73}] (* _Stefano Spezia_, Aug 24 2019 *)
%o A309945 (PARI) a(n) = floor(n - sqrt(2*n-1)); \\ _Jinyuan Wang_, Aug 26 2019
%o A309945 (Python)
%o A309945 from math import isqrt
%o A309945 def A309945(n): return (m:=n-1)-isqrt(m<<1) # _Chai Wah Wu_, Aug 04 2022
%Y A309945 Cf. A007590, A080476, A016813.
%K A309945 nonn
%O A309945 1,5
%A A309945 _Zhandos Mambetaliyev_, Aug 24 2019