A194103 -id:A194103 - cp's OEIS frontend

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

Clark Kimberling, Aug 15 2011

nonn

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

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

Cf. A001951, A194029, A194103, A194104.

[(&+[Floor(k*Sqrt(2)): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jun 05 2018

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

apply( A194102(n)=sum(k=1,n,sqrtint(k^2*2)), [1..99]) \\ M. F. Hasler, Jan 16 2021

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

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

A194104 Natural interspersion of A194102; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 12, 8, 9, 6, 19, 13, 14, 10, 11, 27, 20, 21, 15, 16, 17, 36, 28, 29, 22, 23, 24, 18, 47, 37, 38, 30, 31, 32, 25, 26, 59, 48, 49, 39, 40, 41, 33, 34, 35, 73, 60, 61, 50, 51, 52, 42, 43, 44, 45, 88, 74, 75, 62, 63, 64, 53, 54, 55, 56, 46, 104, 89, 90

Offset: 1

Clark Kimberling, Aug 15 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194100 is a permutation of the positive integers; its inverse is A194101.

			Northwest corner:
1...3...7...12...19
2...4...8...13...20
5...9...14..21...29
6...10..15..22...30
11..16..23..31...40

Cf. A194029, A194102, A194103, A194105.

z = 40; g = Sqrt[2];
c[k_] := Sum[Floor[j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A194102 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194103  new *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194104 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194105 *)

cp's OEIS Frontend

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

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