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A194102 a(n) = Sum_{j=1..n} floor(j*sqrt(2)); n-th partial sum of Beatty sequence for sqrt(2), A001951.

%I A194102 #26 May 06 2022 07:34:47
%S A194102 1,3,7,12,19,27,36,47,59,73,88,104,122,141,162,184,208,233,259,287,
%T A194102 316,347,379,412,447,483,521,560,601,643,686,731,777,825,874,924,976,
%U A194102 1029,1084,1140,1197,1256,1316,1378,1441,1506,1572,1639,1708,1778
%N A194102 a(n) = Sum_{j=1..n} floor(j*sqrt(2)); n-th partial sum of Beatty sequence for sqrt(2), A001951.
%C A194102 The natural fractal sequence of A194102 is A194103; the natural interspersion is A194104.  See A194029 for definitions.
%H A194102 G. C. Greubel, <a href="/A194102/b194102.txt">Table of n, a(n) for n = 1..5000</a>
%F A194102 a(n) = B*(B+1)/2 - C*(C+1) - a(C) where B = floor(sqrt(2)*n) and C = floor(B/(sqrt(2)+2)). - _M. F. Hasler_, Apr 23 2022
%t A194102 a[n_]:=Sum[Floor[j*Sqrt[2]], {j, 1, n}]; Table[a[n], {n, 1, 90}]
%o A194102 (PARI) apply( A194102(n)=sum(k=1,n,sqrtint(k^2*2)), [1..99]) \\ _M. F. Hasler_, Jan 16 2021
%o A194102 (PARI) apply( {A194102(n)=if(n>1, (1+n=sqrtint(n^2*2))*n\2-A194102(n-=sqrtint(n^2\2)+1)-(1+n)*n, n)}, [1..99]) \\ _M. F. Hasler_, Apr 23 2022
%o A194102 (Magma) [(&+[Floor(k*Sqrt(2)): k in [1..n]]): n in [1..100]]; // _G. C. Greubel_, Jun 05 2018
%Y A194102 Cf. A001951, A194029, A194103, A194104.
%K A194102 nonn
%O A194102 1,2
%A A194102 _Clark Kimberling_, Aug 15 2011