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A275740 Sums of the next n consecutive nonsquare integers.

%I A275740 #18 Jan 04 2024 18:07:51
%S A275740 0,2,8,21,46,83,136,210,306,426,575,758,972,1223,1519,1855,2236,2669,
%T A275740 3156,3694,4290,4956,5678,6467,7332,8269,9278,10368,11548,12804,14148,
%U A275740 15593,17126,18753,20485,22325,24262,26308,28481,30756,33148
%N A275740 Sums of the next n consecutive nonsquare integers.
%C A275740 Row sums of nonsquare integers (A000037), seen as a regular triangle:
%C A275740 .
%C A275740 2   | 2,
%C A275740 8   | 3,  5,
%C A275740 21  | 6,  7,  8,
%C A275740 46  | 10, 11, 12, 13,
%C A275740 83  | 14, 15, 17, 18, 19,
%C A275740 136 | 20, 21, 22, 23, 24, 26,
%C A275740 210 | 27, 28, 29, 30, 31, 32, 33,
%C A275740 306 | 34, 35, 37, 38, 39, 40, 41, 42,
%C A275740 ...
%C A275740 The equivalent for all integers are A006003 (starting from 1), A229183 (starting from 2) and A027480 (starting from 0).
%C A275740 There are several sequences close to nonsquares whose sum of groups of n terms starts like this sequence, notably A028761, A158276, A167759.
%H A275740 Robert Israel, <a href="/A275740/b275740.txt">Table of n, a(n) for n = 0..10000</a>
%p A275740 R:= 0: s:= 1:
%p A275740 for n from 1 to 100 do
%p A275740   if floor(sqrt(s+n)) = floor(sqrt(s)) then
%p A275740     R:= R, n*s + n*(n+1)/2; s:= s+n;
%p A275740   else
%p A275740     R:= R, n*s + n*(n+1)/2 - floor(sqrt(s+n))^2 + s+n+1; s:= s+n+1;
%p A275740   fi
%p A275740 od:
%p A275740 R; # _Robert Israel_, Oct 02 2022
%t A275740 Table[Sum[
%t A275740   i + Floor[1/2 + Sqrt[i]], {i, n (n - 1)/2 + 1, (n + 1) (n)/2}], {n,
%t A275740   0, 40}]
%t A275740 Join[{0},Module[{nn=1000,nsi,len},nsi=Select[Range[nn],!IntegerQ[Sqrt[#]]&];len=Floor[ (Sqrt[ 8*Length[nsi]+1]-1)/2];Total/@TakeList[nsi,Range[len]]]] (* _Harvey P. Dale_, Jan 04 2024 *)
%Y A275740 Cf. A000037, A006003, A229183, A027480.
%K A275740 nonn,easy
%O A275740 0,2
%A A275740 _Olivier Gérard_, Aug 07 2016
%E A275740 Definition clarified by _Harvey P. Dale_, Jan 04 2024