A083044 -id:A083044 - cp's OEIS frontend

A163491 A fractal sequence (if we delete the first occurrence of n we get the sequence itself).

1, 1, 1, 2, 1, 2, 3, 1, 2, 4, 3, 1, 5, 2, 4, 6, 3, 1, 7, 5, 2, 8, 4, 6, 9, 3, 1, 10, 7, 5, 11, 2, 8, 12, 4, 6, 13, 9, 3, 14, 1, 10, 15, 7, 5, 16, 11, 2, 17, 8, 12, 18, 4, 6, 19, 13, 9, 20, 3, 14, 21, 1, 10, 22, 15, 7, 23, 5, 16, 24, 11, 2, 25, 17, 8, 26, 12

Offset: 1

Benoit Cloitre, Jul 29 2009

nonn

Write the positive integers with two spaces between each integer: 1,,,2,,,3,,,4,,,5,,,6,..., and fill undefined places with the sequence itself. A003602 is obtained by starting from 1,,2,,3,,4,,5,_,6,....
From Peter Munn, Aug 02 2020: (Start)
a(n) - 1 is the row of A083044 in which n occurs.
The m-th occurrence of m is at position A083045(m-1).
(End)

			  1,_,_,2,_,_,3,_,_,4,... -->
  1,1,_,2,_,_,3,_,_,4,... -->
  1,1,1,2,_,_,3,_,_,4,... -->
  1,1,1,2,1,_,3,_,_,4,... -->
  1,1,1,2,1,2,3,_,_,4,... -->
  1,1,1,2,1,2,3,_,_,4,... -->
  1,1,1,2,1,2,3,1,_,4,... -->
  1,1,1,2,1,2,3,1,2,4,... -->
  ...

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A003602, A007494, A061419, A083044, A083045.

Ordinal transform of A087088.

a[n_] := a[n] = If[Mod[n, 3] == 1, (n+2)/3, a[Floor[2n/3]]];
Array[a, 100] (* Jean-François Alcover, Jan 10 2022 *)

a(n) = n+=2; my(q,r); while([q,r]=divrem(n,3); r, n-=q); q; \\ Kevin Ryde, Jan 16 2021

def a(n): return (n+2)//3 if n%3==1 else a(n*2//3)
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Jan 16 2021

a(3n-2) = n.
A083044(a(n) - 1, A087088(n) - 1) = n. - Peter Munn, Aug 02 2020
From Rémy Sigrist, Jan 15 2021: (Start)
a(n+ceiling(n/2)) = a(n).
a(n) = 1 iff n belongs to A061419.
(End)
a(n) = (n+2)/3 if n == 1 (mod 3), otherwise a(n) = a(floor(n*2/3)). - Michael S. Branicky and Kevin Ryde, Jan 16 2021

Terms after a(70) corrected by Jon E. Schoenfield, Nov 26 2015

A083048 Main diagonal of table A083047.

Original entry on oeis.org

1, 5, 17, 36, 80, 164, 300, 575, 1020, 1884, 3426, 5921, 10568, 18697, 31850, 55716, 94332, 163579, 282388, 474625, 814328, 1363979, 2328358, 3963781, 6609951, 11209355, 18969158, 31524783, 53186480, 88235842, 148471479, 249459365

Offset: 0

Paul D. Hanna, Apr 18 2003

nonn

Cf. A083044, A083047, A083050, A083049.

a(n) = T(n, n), where T(n, 0) = floor(n*x/(x-1)) + 1, T(n, k+1) = ceiling(x*T(n, k)), for n >= 0, k >= 0, with x = (sqrt(5)+1)/2.

A083049 Antidiagonal sums of table A083047.

Original entry on oeis.org

1, 5, 15, 34, 68, 127, 225, 387, 652, 1084, 1787, 2927, 4775, 7769, 12616, 20462, 33160, 53709, 86962, 140769, 227834, 368711, 596658, 965488, 1562270, 2527887, 4090292, 6618319, 10708756, 17327225, 28036136, 45363522, 73399824, 118763517

Offset: 0

Paul D. Hanna, Apr 18 2003

nonn

Cf. A083044, A083047, A083050, A083048.

a(n) = Sum_{k=0..n} T(k, n-k), where T(n, 0) = floor(n*x/(x-1))+1, T(n, k+1) = ceiling(x*T(n, k)), for n>=0, k>=0, with x = (sqrt(5)+1)/2.

A083052 Main diagonal of table A083050.

Original entry on oeis.org

1, 6, 15, 33, 60, 107, 176, 283, 463, 732, 1156, 1773, 2702, 4170, 6333, 9611, 14365, 21842, 32281, 48041, 71690, 106136, 158196, 232380, 340846, 504318, 740949, 1089909, 1590748, 2348540, 3411262, 4978297, 7278002, 10600284, 15494846

Offset: 0

Paul D. Hanna, Apr 18 2003

nonn

Cf. A083044, A083047, A083050, A083053.

a(n) = T(n, n), where T(n, 0) = floor(n*x/(x-1)) + 1, T(n, k+1) = ceiling(x*T(n, k)), for n >= 0, k >= 0, with x=sqrt(2).

A083053 Antidiagonal sums of table A083050.

Original entry on oeis.org

1, 6, 16, 35, 66, 114, 185, 290, 443, 661, 976, 1425, 2064, 2972, 4259, 6083, 8667, 12327, 17506, 24834, 35203, 49869, 70615, 99959, 141462, 200159, 283173, 400577, 566616, 801435, 1133522, 1603168, 2267350, 3206653, 4535033, 6413648, 9070416

Offset: 0

Paul D. Hanna, Apr 18 2003

nonn

Cf. A083044, A083047, A083050, A083052.

a(n) = Sum_{k=0..n} T(k,n-k), where T(n,0) = floor(n*x/(x-1)) + 1, T(n,k+1) = ceiling(x*T(n,k)), for n>=0, k>=0, with x = sqrt(2).

A191656 Dispersion of (2,4,5,7,8,10,...), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 11, 13, 16, 14, 12, 17, 20, 25, 22, 19, 15, 26, 31, 38, 34, 29, 23, 18, 40, 47, 58, 52, 44, 35, 28, 21, 61, 71, 88, 79, 67, 53, 43, 32, 24, 92, 107, 133, 119, 101, 80, 65, 49, 37, 27, 139, 161, 200, 179, 152, 121, 98, 74, 56

Offset: 1

Clark Kimberling, Jun 10 2011

Row 1:  A006999.
For a background discussion of dispersions, see A191426.
...
Each of the sequences (3n, n>0), (3n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191449=dispersion of A008583 (0 mod 3)
A191451=dispersion of A016777 (1 mod 3)
A191450=dispersion of A016789 (2 mod 3)
A191656=dispersion of A001651 (1 or 2 mod 3)
A083044=dispersion of A007494 (0 or 2 mod 3)
A191655=dispersion of A032766 (0 or 1 mod 3)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191449 has 1st col A001651, all else A008583
A191451 has 1st col A007494, all else A016777
A191450 has 1st col A032766, all else A016789
A191656 has 1st col A008583, all else A001651
A083044 has 1st col A016777, all else A083044
A191655 has 1st col A016789, all else A032766
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.

			Northwest corner:
1...2....4....7....11
3...5....8....13...20
6...10...16...25...38
9...14...22...34...52
12..19...29...44...67

Cf. A001651, A008583, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 2; b = 4; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 3*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A001651: (2+5k,4+5k, k>=0) *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i,1,10}, {j,1,10}]]          (* A191656 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]   (* A191656 sequence *)

A083090 Main diagonal of table A083087.

Original entry on oeis.org

1, 5, 25, 90, 247, 766, 2258, 5860, 16526, 45639, 124043, 313327, 837221, 2216256, 5545541, 14524810, 37810204, 97907017, 242993558, 625252309, 1602714963, 3962518559, 10109705767, 25718726896, 63402237401, 160711911464

Offset: 0

Paul D. Hanna, Apr 22 2003

nonn

Cf. A083087, A083044, A083047, A083050.

a(n) = T(n, n), where T(n, 0) = floor(n*x/(x-1)) + 1, T(n, k+1) = ceiling(x*T(n, k)), for n >= 0, k >= 0, where x = 1+sqrt(2).

A083091 Antidiagonal sums of table A083087.

Original entry on oeis.org

1, 5, 17, 49, 128, 321, 790, 1924, 4664, 11282, 27262, 65843, 158988, 383863, 926761, 2237435, 5401685, 13040863, 31483472, 76007871, 183499282, 443006506, 1069512368, 2582031320, 6233575089, 15049181582, 36331938341, 87713058356

Offset: 0

Paul D. Hanna, Apr 22 2003

nonn

Cf. A083087, A083044, A083047, A083050.

a(n) = Sum_{k=0..n} T(k, n-k), where T(n, 0) = floor(n*x/(x-1)) + 1, T(n, k+1) = ceiling(x*T(n, k)), for n>=0, k>=0, with x = 1 + sqrt(2).

cp's OEIS Frontend

A163491 A fractal sequence (if we delete the first occurrence of n we get the sequence itself).

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A083048 Main diagonal of table A083047.

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A083049 Antidiagonal sums of table A083047.

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A083052 Main diagonal of table A083050.

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A083053 Antidiagonal sums of table A083050.

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A191656 Dispersion of (2,4,5,7,8,10,...), by antidiagonals.

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A083090 Main diagonal of table A083087.

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A083091 Antidiagonal sums of table A083087.

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