A194074 -id:A194074 - cp's OEIS frontend

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

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

A194075 Natural interspersion of A194073; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 7, 5, 3, 13, 8, 6, 10, 19, 14, 9, 16, 11, 28, 20, 15, 22, 17, 12, 37, 29, 21, 31, 23, 18, 25, 49, 38, 30, 40, 32, 24, 34, 26, 61, 50, 39, 52, 41, 33, 43, 35, 27, 76, 62, 51, 64, 53, 42, 55, 44, 36, 46, 91, 77, 63, 79, 65, 54, 67, 56, 45, 58, 47, 109, 92, 78

Offset: 1

Clark Kimberling, Aug 14 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, A194075 is a permutation of the positive integers; its inverse is A194076.

			Northwest corner:
1...4...7...13...19
2...5...8...14...20
3...6...9...15...21
10..16..22..31...40
11..17..23..32...41

Cf. A194029, A194074, A194073, A194076.

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 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula