A194104 -id:A194104 - cp's OEIS frontend

A194105 Inverse permutation of A194104; every positive integer occurs exactly once.

1, 3, 2, 5, 6, 10, 4, 8, 9, 14, 15, 7, 12, 13, 19, 20, 21, 28, 11, 17, 18, 25, 26, 27, 35, 36, 16, 23, 24, 32, 33, 34, 43, 44, 45, 22, 30, 31, 40, 41, 42, 52, 53, 54, 55, 66, 29, 38, 39, 49, 50, 51, 62, 63, 64, 65, 77, 78, 37, 47, 48, 59, 60, 61, 73, 74, 75, 76, 89

Offset: 1

Clark Kimberling, Aug 15 2011

nonn

Cf. A194104.

Mathematica
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(See A194104.)
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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.

Original entry on oeis.org

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

A194103 Natural fractal sequence of A194102.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4

Offset: 1

Clark Kimberling, Aug 15 2011

nonn

See A194029 for definitions of natural fractal sequence and natural interspersion.

Cf. A194029, A194102, A194104, A001951.

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

A194105 Inverse permutation of A194104; every positive integer occurs exactly once.

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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.

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A194103 Natural fractal sequence of A194102.

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