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%I #29 Sep 08 2022 08:45:28
%S -1,0,1,0,-1,0,-1,-2,-1,0,-1,0,1,0,-1,0,1,0,1,2,1,0,1,0,-1,0,1,0,-1,0,
%T -1,-2,-1,0,-1,0,1,0,-1,0,-1,-2,-1,0,-1,-2,-1,-2,-3,-2,-1,-2,-1,0,-1,
%U -2,-1,0,-1,0,1,0,-1,0,-1,-2,-1,0,-1,0,1,0,-1,0,1,0,1,2,1,0,1,0,-1,0,1,0,-1,0,-1,-2,-1,0,-1,0,1,0,-1,0,1,0
%N Partial sums of (-1)^floor(n*sqrt(2)).
%C Conjecture: A001652(n) is the index of the first occurrence of n in sequence A123737, A001108(n) is the index of the first occurrence of -n in sequence A123737. - _Vaclav Kotesovec_, Jun 02 2015
%H T. D. Noe, <a href="/A123737/b123737.txt">Table of n, a(n) for n = 1..10000</a>
%H Kevin O'Bryant, Bruce Reznick and Monika Serbinowska, <a href="http://www.math.uiuc.edu/~reznick/ors.pdf">Almost alternating sums</a>, Amer. Math. Monthly, Vol. 113 (October 2006), pp. 673-688.
%F O'Bryant, Reznick, & Serbinowska show that |a(n)| <= k log n + 1, with k = 1/(2 log (1 + sqrt(2))), and further -a(n) > k log n + 0.78 infinitely often. - _Charles R Greathouse IV_, Feb 07 2013
%p ListTools:-PartialSums([seq((-1)^floor(n*sqrt(2)),n=1..100)]); # _Robert Israel_, Jun 02 2015
%t Rest[FoldList[Plus,0,(-1)^Floor[Sqrt[2]*Range[120]]]]
%t Accumulate[(-1)^Floor[Range[120]Sqrt[2]]] (* _Harvey P. Dale_, Jan 16 2012 *)
%t (* The positions of the first occurrences of n and -n in this sequence: *) stab = Rest[FoldList[Plus,0,(-1)^Floor[Sqrt[2]*Range[1000000]]]]; Print[Table[FirstPosition[stab,n][[1]],{n,1,8}]]; Print[Table[FirstPosition[stab,-n][[1]],{n,1,8}]]; (* _Vaclav Kotesovec_, Jun 02 2015 *)
%o (PARI) a(n)=sum(i=1,n,(-1)^sqrtint(2*i^2)) \\ _Charles R Greathouse IV_, Feb 07 2013
%o (Magma) [&+[(-1)^Floor(j*Sqrt(2)): j in [1..n]]: n in [1..130]]; // _G. C. Greubel_, Sep 05 2019
%o (Sage) [sum((-1)^floor(j*sqrt(2)) for j in (1..n)) for n in (1..130)] # _G. C. Greubel_, Sep 05 2019
%Y Cf. A123724 (sum for 2^(1/3)), A123738 (sum for Pi), A123739 (sum for e).
%Y Cf. A001652, A001951, A228639.
%K easy,sign
%O 1,8
%A _T. D. Noe_, Oct 11 2006
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