cp's OEIS Frontend

A270439 Alternating sum of nonsquares (A000037).

%I A270439 #26 Nov 15 2022 02:37:43
%S A270439 2,-1,4,-2,5,-3,7,-4,8,-5,9,-6,11,-7,12,-8,13,-9,14,-10,16,-11,17,-12,
%T A270439 18,-13,19,-14,20,-15,22,-16,23,-17,24,-18,25,-19,26,-20,27,-21,29,
%U A270439 -22,30,-23,31,-24,32,-25,33,-26,34,-27,35,-28,37,-29,38,-30,39,-31,40,-32,41,-33,42,-34,43,-35,44,-36,46,-37,47
%N A270439 Alternating sum of nonsquares (A000037).
%C A270439 Interleaving of nontriangular numbers (A014132) and negative integers (A001478).
%F A270439 a(n) = Sum_{k = 1..n} (-1)^(k+1)*(k + floor(1/2 + sqrt(k))).
%F A270439 a(n) = Sum_{k = 1..n} (-1)^(k+1)*A000037(k).
%F A270439 a(2^k*m) = -2^(k-1) * m, k > 0.
%F A270439 a(2k - 1) = k + floor(1/2 + sqrt(2*k)), a(2k) = -k, k > 0.
%F A270439 a(2k - 1) = A014132(k), a(2k) = A001478(k).
%e A270439 a(1) = a(2*1-1) = 1 + floor(1/2 + sqrt(2*1)) = 2;
%e A270439 a(2) = a(2*1) = -1;
%e A270439 a(3) = a(2*2-1) = 2 + floor(1/2 + sqrt(2*2)) = 4;
%e A270439 a(4) = a(2*2) = -2;
%e A270439 a(5) = a(2*3-1) = 3 + floor(1/2 + sqrt(2*3)) = 5;
%e A270439 a(6) = a(2*3) = -3, etc.
%e A270439 or
%e A270439 a(1) = 2;
%e A270439 a(2) = 2 - 3 = -1;
%e A270439 a(3) = 2 - 3 + 5 = 4;
%e A270439 a(4) = 2 - 3 + 5 - 6 = -2;
%e A270439 a(5) = 2 - 3 + 5 - 6 + 7 = 5;
%e A270439 a(6) = 2 - 3 + 5 - 6 + 7 - 8 = -3, etc.
%e A270439 (2, 3, 5, 6, 7, 8, ... is the nonsquares).
%t A270439 Table[Sum[(-1)^(k + 1) (k + Floor[1/2 + Sqrt[k]]), {k, n}], {n, 75}]
%o A270439 (PARI) a(n)=if(n%2, sqrtint(4*n-3)+n+2, -n)\2 \\ _Charles R Greathouse IV_, Aug 03 2016
%o A270439 (Python)
%o A270439 from math import isqrt
%o A270439 def A270439(n): return (n>>1)+1+(m:=isqrt(n+1))+int(n-m*(m+1)>=0) if n&1 else -(n>>1) # _Chai Wah Wu_, Nov 14 2022
%Y A270439 Cf. A000037, A001478, A014132, A086849.
%K A270439 easy,sign
%O A270439 1,1
%A A270439 _Ilya Gutkovskiy_, Jul 12 2016