A330440 -id:A330440 - 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 }.

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 *)

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 *)

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A316905 a(n) is the index of the first 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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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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