A194075 -id:A194075 - cp's OEIS frontend

A194076 Inverse permutation to A194075; every positive integer occurs exactly once.

1, 3, 6, 2, 5, 9, 4, 8, 13, 10, 15, 21, 7, 12, 18, 14, 20, 27, 11, 17, 24, 19, 26, 34, 28, 36, 45, 16, 23, 31, 25, 33, 42, 35, 44, 54, 22, 30, 39, 32, 41, 51, 43, 53, 64, 55, 66, 78, 29, 38, 48, 40, 50, 61, 52, 63, 75, 65, 77, 90, 37, 47, 58, 49, 60, 72, 62, 74, 87

Offset: 1

Clark Kimberling, Aug 14 2011

nonn

Cf. A194075.

Mathematica
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(See A194075.)
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A194073 a(n) = 1 + floor((3/4)*n^2).

Original entry on oeis.org

1, 4, 7, 13, 19, 28, 37, 49, 61, 76, 91, 109, 127, 148, 169, 193, 217, 244, 271, 301, 331, 364, 397, 433, 469, 508, 547, 589, 631, 676, 721, 769, 817, 868, 919, 973, 1027, 1084, 1141, 1201, 1261, 1324, 1387, 1453, 1519, 1588, 1657, 1729, 1801

Offset: 1

Clark Kimberling, Aug 14 2011

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A002620, A194074 (natural fractal sequence of A194073), A194075 (natural interspersion of A194074).

c[k_]:=1+Floor[(3/4)k^2];
Table[c[k],{k,1,90}]

a(n)=3*n^2\4+1 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 1 + floor((3/4)*n^2).
G.f.: x*(1+2*x-x^2+x^3) / ( (1+x)*(1-x)^3 ). - R. J. Mathar, Aug 25 2011
a(n) = 1 + 3*A002620(n). - R. J. Mathar, Aug 25 2011
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Wesley Ivan Hurt, Jun 26 2025

A194074 Natural fractal sequence of A194073.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Aug 14 2011

nonn

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

Cf. A194073, A194075.

z = 70;
c[k_] := 1 + Floor[(3/4) k^2];
c = Table[c[k], {k, 1, z}]  (* A194073 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194074 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 14}, {k, 1, n}]] (* A194075 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]]  (* A194076 *)

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A194076 Inverse permutation to A194075; every positive integer occurs exactly once.

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A194073 a(n) = 1 + floor((3/4)*n^2).

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A194074 Natural fractal sequence of A194073.

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