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

1, 3, 7, 12, 19, 27, 36, 47, 59, 73, 88, 104, 122, 141, 162, 184, 208, 233, 259, 287, 316, 347, 379, 412, 447, 483, 521, 560, 601, 643, 686, 731, 777, 825, 874, 924, 976, 1029, 1084, 1140, 1197, 1256, 1316, 1378, 1441, 1506, 1572, 1639, 1708, 1778

Offset

1,2

Comments

The natural fractal sequence of A194102 is A194103; the natural interspersion is A194104. See A194029 for definitions.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Formula

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

Mathematica

a[n_]:=Sum[Floor[j*Sqrt[2]], {j, 1, n}]; Table[a[n], {n, 1, 90}]

Prog

(PARI) apply( A194102(n)=sum(k=1, n, sqrtint(k^2*2)), [1..99]) \\ M. F. Hasler, Jan 16 2021
(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
(Magma) [(&+[Floor(k*Sqrt(2)): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jun 05 2018

Crossrefs

Cf. A001951, A194029, A194103, A194104.

Sequence in context: A376360 A212293 A360921 * A006317 A194147 A077043

Adjacent sequences: A194099 A194100 A194101 * A194103 A194104 A194105

Keyword

nonn

Author

Clark Kimberling, Aug 15 2011

Status

approved