A316774 -id:A316774 - cp's OEIS frontend

A316905 a(n) is the index of the first occurrence of n in A316774.

0, 1, 2, 5, 4, 8, 11, 22, 14, 32, 28, 42, 48, 45, 68, 71, 77, 89, 108, 115, 92, 140, 95, 149, 216, 268, 194, 260, 310, 254, 263, 340, 362, 257, 295, 277, 298, 476, 346, 431, 365, 560, 539, 424, 486, 462, 576, 479, 579, 692, 657, 707, 754, 794, 757, 797, 928

Offset: 0

Alois P. Heinz, Jul 18 2018

			a(4) = 4 because A316774(j) = 4 for j in {4,7,12,13,36,49,55} with minimal element 4.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A316774, A316973, A316984, A330440 (a sorted version of this), A330447, A330448.

b:= proc() 0 end:
g:= proc(n) option remember; local t;
      t:= `if`(n<2, n, b(g(n-1))+b(g(n-2)));
      b(t):= b(t)+1; t
    end:
a:= proc() local t, a; t, a:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= g(t);
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..100);

b[_] = 0;
g[n_] := g[n] = Module[{t}, t = If[n < 2, n, b[g[n - 1]] + b[g[n - 2]]];       b[t] = b[t] + 1; t];
a[n_] := Module[{t = -1, a}, a[_] = -1; Module[{h}, While[a[n] == -1, t = t + 1; h = g[t]; If[a[h] == -1, a[h] = t]]; a[n]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 28 2023, after Alois P. Heinz *)

a(n) = min { j >= 0 : A316774(j) = n }.

A316973 a(n) is the number of occurrences of n in A316774.

Original entry on oeis.org

1, 1, 3, 4, 7, 7, 15, 12, 18, 16, 25, 23, 14, 22, 26, 33, 23, 38, 32, 35, 45, 49, 41, 44, 58, 49, 59, 39, 49, 69, 54, 57, 57, 75, 79, 62, 75, 76, 74, 81, 75, 78, 74, 90, 86, 86, 88, 78, 78, 105, 106, 109, 129, 91, 92, 111, 108, 101, 126, 130, 116, 129, 133, 140

Offset: 0

Alois P. Heinz, Jul 17 2018

These values may be only conjectural, in which case the b-file should be changed to an a-file and marked as conjectured. - N. J. A. Sloane, Dec 14 2019

			a(4) = 7 because A316774(j) = 4 for j in s = {4,7,12,13,36,49,55} and |s| = 7.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Altug Alkan, A plot of a(n)-2*n (conjectural)

Cf. A316774, A316905, A316984.

a(n) = |{ j : A316774(j) = n }|.

A316984 a(n) is the index of the last occurrence of n in A316774.

Original entry on oeis.org

0, 1, 6, 16, 55, 488, 314, 367, 279, 562, 519, 799, 500, 694, 792, 876, 1322, 1598, 5826, 1814, 6102, 5345, 2042, 5105, 2949, 3288, 5260, 4971, 6353, 5172, 8596, 3193, 6528, 8065, 8194, 6538, 6881, 6966, 7963, 5993, 8114, 9090, 8457, 8948, 12703, 9994, 6568

Offset: 0

Alois P. Heinz, Jul 18 2018

[It may be that these values are only conjectural. If so, the b-file should be changed to an a-file and marked as conjectured. - N. J. A. Sloane, Dec 14 2019]

			a(4) = 55 because A316774(j) = 4 for j in {4,7,12,13,36,49,55} with maximal element 55.

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A316774, A316905, A316973.

a(n) = max { j >= 0 : A316774(j) = n }.

A330439 Number of times g(n) appears in [g(0),g(1),...,g(n)], where g = A316774.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 3, 1, 3, 4, 1, 2, 4, 2, 3, 4, 2, 5, 1, 6, 2, 3, 3, 7, 1, 4, 4, 5, 1, 8, 2, 2, 5, 3, 6, 3, 4, 7, 1, 8, 5, 1, 9, 3, 1, 6, 4, 4, 9, 2, 2, 7, 6, 3, 7, 5, 2, 5, 6, 3, 8, 3, 4, 10, 1, 5, 10, 1, 6, 7, 4, 7, 8, 1, 9, 6, 2, 11, 5, 2, 8, 7, 3, 8, 9, 1, 9, 10, 1, 10, 11, 1, 4, 5, 10, 4, 2, 11, 6

Offset: 0

Alois P. Heinz, Dec 14 2019

Alois P. Heinz, Table of n, a(n) for n = 0..65536

Cf. A316774, A330440.

b:= proc() 0 end:
g:= proc(n) option remember; local t;
      t:= `if`(n<2, n, b(g(n-1))+b(g(n-2)));
      b(t):= b(t)+1; t
    end:
a:= proc(n) option remember; b(g(n)) end:
seq(a(n), n=0..200);

b[_] = 0;
g[n_] := g[n] = Module[{t},
     t = If[n<2, n, b[g[n-1]]+b[g[n-2]]];
     b[t]++; t];
a[n_] := a[n] = b[g[n]];
a /@ Range[0, 200] (* Jean-François Alcover, Mar 30 2021, after Alois P. Heinz *)

A330440 Numbers k such that A316774(k) != A316774(j) for all 0 <= j < k.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 11, 14, 22, 28, 32, 42, 45, 48, 68, 71, 77, 89, 92, 95, 108, 115, 140, 149, 194, 216, 254, 257, 260, 263, 268, 277, 295, 298, 310, 340, 346, 362, 365, 424, 431, 462, 476, 479, 486, 539, 560, 576, 579, 657, 692, 707, 754, 757, 794, 797, 928

Offset: 1

Alois P. Heinz, Dec 14 2019

nonn

This is a sorted version of A316905.

List of indices k such that A330439(k) = 1.

Alois P. Heinz, Table of n, a(n) for n = 1..25000

Cf. A316774, A316905, A330439.

Column k=1 of A330587.

b:= proc() 0 end:
g:= proc(n) option remember; local t;
      t:= `if`(n<2, n, b(g(n-1))+b(g(n-2)));
      b(t):= b(t)+1; t
    end:
f:= proc(n) option remember; b(g(n)) end:
a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, -1, a(n-1)) while f(k)<>1 do od; k
    end:
seq(a(n), n=1..80);

b[_] = 0;
g[n_] := g[n] = Module[{t},
     t = If[n < 2, n, b[g[n - 1]] + b[g[n - 2]]];
     b[t]++; t];
f[n_] := f[n] = b[g[n]];
a[n_] := a[n] = Module[{k}, For[k = 1 +
     If[n == 1, -1, a[n - 1]], f[k] != 1, k++]; k];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz *)

{ k >= 0 : A330439(k) = 1 }.

A330330 Record high points in A316774.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 10, 11, 13, 14, 15, 16, 17, 20, 22, 23, 26, 29, 33, 35, 36, 38, 40, 43, 45, 47, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 65, 66, 71, 72, 73, 74, 75, 77, 78, 80, 81, 83, 84, 91, 98, 102, 111, 112, 118, 120, 124, 131, 136, 139, 144, 145, 147, 149, 152, 156, 159, 160, 163

Offset: 1

N. J. A. Sloane, Dec 14 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..8000 (first 151 terms from N. J. A. Sloane)

Cf. A316774, A316905, A316984, A330331.

A330331 Positions of records in A316774.

Original entry on oeis.org

0, 1, 2, 4, 8, 11, 14, 28, 42, 45, 68, 71, 77, 89, 92, 95, 149, 194, 254, 257, 277, 298, 346, 365, 424, 462, 479, 579, 657, 707, 754, 757, 797, 928, 1107, 1110, 1147, 1150, 1295, 1298, 1453, 1456, 1462, 1590, 1593, 1673, 1676, 1819, 1822, 1982, 2095, 2395, 2593, 2665, 2874, 3355, 3724, 3873, 4131, 4359, 4510

Offset: 1

N. J. A. Sloane, Dec 14 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..8000 (first 151 terms from N. J. A. Sloane)

Cf. A316774, A316905, A316984, A330330.

A330447 a(n) is the smallest index k such that {0,1,2,...,n} is a subset of { A316774(j) : 0 <= j <= k }.

Original entry on oeis.org

0, 1, 2, 5, 5, 8, 11, 22, 22, 32, 32, 42, 48, 48, 68, 71, 77, 89, 108, 115, 115, 140, 140, 149, 216, 268, 268, 268, 310, 310, 310, 340, 362, 362, 362, 362, 362, 476, 476, 476, 476, 560, 560, 560, 560, 560, 576, 576, 579, 692, 692, 707, 754, 794, 794, 797, 928

Offset: 0

Alois P. Heinz, Dec 15 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A316774, A316905, A330448.

b:= proc() 0 end:
g:= proc(n) option remember; local t;
      t:= `if`(n<2, n, b(g(n-1))+b(g(n-2)));
      b(t):= b(t)+1; t
    end:
f:= proc() local t, a; t, a:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= g(t);
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
a:= proc(n) option remember; `if`(n<0, 0,
      max(a(n-1), f(n)))
    end:
seq(a(n), n=0..100);

b[_] = 0;
g[n_] := g[n] = Module[{t}, t = If[n < 2, n, b[g[n - 1]] + b[g[n - 2]]]; b[t] = b[t] + 1; t];
f[n_] := Module[{t, a}, t = -1; a[_] = -1; Module[{h}, While[a[n] == -1, t = t + 1; h = g[t]; If[a[h] == -1, a[h] = t]]; a[n]]];
a[n_] := a[n] = If[n < 0, 0, Max[a[n - 1], f[n]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 13 2022, after Alois P. Heinz *)

a(n) = max_{0 <= j <= n} A316905(j).
a(n) >= A316905(n).
a(n) <= a(n+1).

A330587 A(n,k) is the n-th index m such that A330439(m) = k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

0, 3, 1, 6, 7, 2, 13, 10, 9, 4, 21, 16, 12, 15, 5, 23, 31, 19, 18, 17, 8, 27, 38, 36, 29, 25, 20, 11, 33, 41, 49, 44, 30, 26, 24, 14, 46, 43, 55, 56, 59, 40, 37, 34, 22, 67, 52, 64, 58, 62, 61, 50, 39, 35, 28, 81, 70, 78, 76, 73, 72, 69, 51, 47, 53, 32, 104, 94, 91, 88, 84, 75, 79, 82, 66, 57, 54, 42

Offset: 1

Alois P. Heinz, Dec 18 2019

			Square array A(n,k) begins:
   0,  3,  6, 13,  21,  23,  27,  33,  46,  67, ...
   1,  7, 10, 16,  31,  38,  41,  43,  52,  70, ...
   2,  9, 12, 19,  36,  49,  55,  64,  78,  91, ...
   4, 15, 18, 29,  44,  56,  58,  76,  88,  93, ...
   5, 17, 25, 30,  59,  62,  73,  84,  90,  98, ...
   8, 20, 26, 40,  61,  72,  75,  87, 117, 139, ...
  11, 24, 37, 50,  69,  79,  85, 121, 124, 154, ...
  14, 34, 39, 51,  82, 102, 118, 142, 155, 157, ...
  22, 35, 47, 66,  97, 110, 133, 180, 190, 202, ...
  28, 53, 57, 74, 106, 116, 164, 183, 197, 205, ...

Alois P. Heinz, Antidiagonals n = 1..365, flattened

Column k=1 gives A330440.
Row n=1 gives A330588.
Main diagonal gives A330589.
Cf. A316774, A330439.

b:= proc() 0 end:
g:= proc(n) option remember; local t;
      t:= `if`(n<2, n, b(g(n-1))+b(g(n-2)));
      b(t):= b(t)+1; t
    end:
f:= proc(n) option remember; b(g(n)) end:
A:= proc() local l, t; t, l:= -1, proc() [] end;
      proc(n,k) local h;
        while nops(l(k))

b[_] = 0;
g[n_] := g[n] = Module[{t}, t = If[n < 2, n, b[g[n - 1]] + b[g[n - 2]]]; b[t]++; t];
f[n_] := f[n] = b[g[n]];
A[n_, k_] := Module[{l, t = -1, h}, l[_] = {}; While[Length[l[k]] < n, t++; h = f[t]; AppendTo[l[h], t]]; l[k][[n]]];
Table[Table[A[n, 1 + d - n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Feb 11 2021, after Alois P. Heinz *)

A317223 a(0) = 0, a(1) = 1; for n >= 2, a(n) = freq(a(freq(a(n-1),n)),n) where freq(i, j) is the number of times i appears in the terms a(0) .. a(j-1).

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 1, 6, 3, 3, 6, 3, 6, 6, 6, 6, 4, 3, 6, 6, 2, 7, 3, 6, 7, 3, 7, 7, 7, 7, 7, 7, 2, 2, 9, 3, 2, 1, 10, 4, 10, 4, 10, 10, 10, 10, 8, 4, 10, 10, 5, 4, 8, 4, 10, 10, 2, 11, 4, 8, 11, 4, 11, 11, 11, 11, 8, 11, 11, 9, 4, 2, 4, 12, 4, 4, 9, 12, 4, 8, 12, 12, 12, 12, 8, 8, 12, 12, 14, 4, 8

Offset: 0

Altug Alkan, Jul 24 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..65536
Alois P. Heinz, Log plot of n, a(n) for n = 0..2^20

Cf. A316774, A317015, A317127.

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= `if`(n<2, n, b(a(b(a(n-1)))));
      b(t):= b(t)+1; t
    end:
seq(a(n), n=0..200);  # Alois P. Heinz, Jul 24 2018

b[_] = 0;
a[n_] := a[n] = Module[{t}, t = If[n<2, n, b[a[b[a[n-1]]]]]; b[t]++; t];
a /@ Range[0, 200] (* Jean-François Alcover, Nov 09 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A316905 a(n) is the index of the first occurrence of n in A316774.

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A316973 a(n) is the number of occurrences of n in A316774.

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A316984 a(n) is the index of the last occurrence of n in A316774.

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A330439 Number of times g(n) appears in [g(0),g(1),...,g(n)], where g = A316774.

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A330440 Numbers k such that A316774(k) != A316774(j) for all 0 <= j < k.

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A330330 Record high points in A316774.

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A330331 Positions of records in A316774.

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A330447 a(n) is the smallest index k such that {0,1,2,...,n} is a subset of { A316774(j) : 0 <= j <= k }.

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A330587 A(n,k) is the n-th index m such that A330439(m) = k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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