A316774 -id:A316774 - cp's OEIS frontend

A329934 a(1)=1, a(2)=1, a(n) = (number of times a(n-1) has appeared before) + (number of times a(n-2) has appeared before).

1, 1, 4, 3, 2, 2, 4, 4, 6, 4, 5, 5, 4, 7, 6, 3, 4, 8, 7, 3, 5, 6, 6, 8, 6, 7, 8, 6, 9, 7, 5, 8, 8, 10, 6, 8, 13, 7, 6, 13, 10, 4, 9, 9, 6, 12, 10, 4, 11, 9, 5, 9, 10, 9, 10, 11, 7, 8, 13, 10, 9, 13, 11, 7, 10, 14, 8, 9, 16, 9, 10, 17, 9, 11, 14, 6, 12, 12, 6, 14, 14

Offset: 1

Joshua Oliver, Nov 24 2019

nonn

Conjecture: This sequence grows logarithmically.

			a(n)=4 where n=3 because 1 (a(n-1)) has appeared twice before, and 1 (a(n-2)) has appeared twice before as well. 2+2 = 4.

Joshua Oliver, Table of n, a(n) for n = 1..4000
Rémy Sigrist, Density plot of the first 10000000 terms

Cf. A158416, A306246, A316774.

b:= proc(n) option remember; `if`(n=0, 0, b(n-1)+x^a(n)) end:
a:= proc(n) option remember; `if`(n<3, 1, (p->
      coeff(p, x, a(n-1))+coeff(p, x, a(n-2)))(b(n-1)))
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 24 2019

A={1,1};For[n=3,n<=81,n++,A=Append[A,Sum[Count[Table[Part[A,i],{i,1,n-1}],Part[A,n-k]],{k,2}]]];A

o=vector(17); for (n=1, 81, print1 (v=if (n<3, 1, o[pp]+o[p]) ", "); o[v]++; [pp,p]=[p,v]) \\ Rémy Sigrist, Nov 27 2019

A330332 a(n) = (number of times a(n-1) has already appeared) + (number of times a(n-2) has already appeared) + (number of times a(n-3) has already appeared), starting with a(n) = n for n<3.

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 5, 6, 5, 7, 5, 9, 6, 7, 5, 9, 9, 11, 7, 7, 9, 12, 9, 11, 8, 8, 6, 7, 10, 9, 12, 9, 16, 10, 10, 7, 12, 12, 14, 9, 13, 10, 13, 8, 9, 14, 14, 15, 7, 11, 11, 15, 10, 11, 12, 15, 13, 11, 12, 15, 16, 12, 13, 13, 17, 11, 13, 14, 17, 12, 14, 15, 18, 11, 14, 15, 20, 13, 14, 15, 21, 15

Offset: 0

N. J. A. Sloane, Dec 14 2019

Generalizes A316774, which looks at the frequencies of the two previous terms. Here we look at three previous terms.

If we look at just one previous term, we get 0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, ..., which is A133622 prefixed by 0, 1, or A152271 with its initial 1 changed to 0.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Density plot of the first 2^22 terms

Cf. A316774, A133622, A152271.

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= `if`(n<3, n, b(a(n-1))+b(a(n-2))+b(a(n-3)));
      b(t):= b(t)+1; t
    end:
[seq(a(n), n=0..200)]; # Following Alois P. Heinz's program for A316774

b[_] = 0;
a[n_] := a[n] = Module[{t}, t = If[n<3, n, b[a[n-1]] + b[a[n-2]] + b[a[n-3]]]; b[t]++; t];
a /@ Range[0, 200] (* Jean-François Alcover, Nov 09 2020, after Maple *)

A330448 a(n) = A330447(n) - A316905(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 8, 0, 4, 0, 0, 3, 0, 0, 0, 0, 0, 0, 23, 0, 45, 0, 0, 0, 74, 8, 0, 56, 47, 0, 0, 105, 67, 85, 64, 0, 130, 45, 111, 0, 21, 136, 74, 98, 0, 97, 0, 0, 35, 0, 0, 0, 37, 0, 0, 0, 52, 0, 104, 52, 67, 0, 0, 70, 0, 0, 0, 0, 0, 152, 0, 0, 0, 0, 0

Offset: 0

Alois P. Heinz, Dec 15 2019

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

Cf. A316774, A316905, A330447.

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():
h:= proc(n) option remember; `if`(n<0, 0,
      max(h(n-1), f(n)))
    end:
a:= n-> h(n)-f(n):
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]]];
h[n_] := h[n] = If[n < 0, 0, Max[h[n - 1], f[n]]];
a[n_] := h[n] - f[n];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 13 2023, after Alois P. Heinz *)

A317015 a(n) = n for n < 2, a(n) = a(freq(a(n-1),n)) + a(freq(a(n-2),n)) for n >= 2, where freq(i, j) is the number of times i appears in the first j terms.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 2, 3, 4, 4, 4, 8, 5, 2, 5, 6, 3, 3, 8, 6, 4, 5, 5, 8, 6, 4, 4, 6, 7, 5, 4, 7, 6, 5, 5, 6, 5, 6, 7, 5, 6, 8, 8, 6, 7, 8, 6, 6, 16, 9, 2, 4, 7, 7, 4, 6, 9, 7, 5, 7, 8, 7, 7, 8, 8, 8, 8, 16, 10, 3, 4, 11, 9, 3, 4, 7, 13, 9, 5, 12, 9, 4, 5, 7, 10, 7, 4, 7, 10, 7, 8, 11, 7, 5, 5, 10, 9

Offset: 0

Altug Alkan, Jul 19 2018

nonn

Inspired by A316774.

Let b(n) = n for n < 3, b(n) = b(freq(b(n-1),n)) for n >= 3, where freq(i, j) is the number of times i appears in the first j terms and b(n) has offset 0. For n >= 1, b(n) - 1 are 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, ... (cf. A093879). While b(n) has one parent spot, this entry (a(n)) has two parent spots which are freq(a(n-1),n) and freq(a(n-2),n).

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

Cf. A316774.

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= `if`(n<2, n, a(b(a(n-1)))+a(b(a(n-2))));
      b(t):= b(t)+1; t
    end:
seq(a(n), n=0..200);  # Alois P. Heinz, Jul 19 2018

Nest[Append[#, #[[Count[#, #[[-1]] ] + 1]] + #[[Count[#, #[[-2]] ] + 1 ]] ] &, {0, 1}, 95] (* Michael De Vlieger, Jul 20 2018 *)

A317127 a(0) = a(1) = a(2) = 1; for n >= 3, a(n) = freq(a(n-1),n) + freq(a(n-3),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

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

Offset: 0

Altug Alkan, Jul 21 2018

Inspired by A316774.

In this sequence, it is obvious that we have exactly three 1’s that are a(0) = a(1) = a(2) = 1. Can we determine the frequency characteristics of some other positive integers? For example, are there infinitely many 2's in this sequence?

Antti Karttunen, Table of n, a(n) for n = 0..65537
Altug Alkan, A line graph of a(n) for n <= 500

Cf. A316774.

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= `if`(n<3, 1, b(a(n-1))+b(a(n-3)));
      b(t):= b(t)+1; t
    end:
seq(a(n), n=0..100); # after Alois P. Heinz at A316774

c = <||>; f[n_] := If[KeyExistsQ[c,n],c[n],0]; a[n_] := a[n] = Block[{v}, v = If[n<3, 1, f[a[n-1]] + f[a[n-3]]]; If[f[v]>0, c[v] = c[v]+1, c[v]=1]; v]; Array[a, 93, 0] (* Giovanni Resta, Jul 24 2018 *)

up_to = 5000;
listA317127off1(up_to) = { my(v = vector(up_to),c); v[1] = v[2] = v[3] = 1; for(n=4,up_to, c=0; for(k=1,(n-1), c += ((v[k]==v[n-1])+(v[k]==v[n-3]))); v[n] = c); (v); };
listA317127off1(up_to) = { my(v = vector(up_to), m = Map(), c); v[1] = v[2] = v[3] = 1; mapput(m, 1, 3); for(n=4,up_to, c = (mapget(m, v[n-1])+mapget(m,v[n-3])); v[n] = c; mapput(m, c, if(!mapisdefined(m, c), 1, 1+mapget(m, c)))); (v); }; \\ Faster!
v317127 = listA317127off1(1+up_to);
A317127(n) = v317127[1+n]; \\ Antti Karttunen, Jul 23 2018

A330588 a(n) is the first index m such that A330439(m) = n.

Original entry on oeis.org

0, 3, 6, 13, 21, 23, 27, 33, 46, 67, 81, 104, 107, 114, 129, 166, 169, 172, 193, 261, 267, 276, 287, 311, 373, 430, 457, 478, 485, 590, 596, 656, 691, 768, 789, 796, 873, 941, 969, 1047, 1093, 1149, 1170, 1239, 1303, 1349, 1491, 1533, 1555, 1567, 1805, 1808

Offset: 1

Alois P. Heinz, Dec 18 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..16000

Row n=1 of A330587.

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() -1 end;
      proc(k) local h;
        while l(k)<0 do t:= t+1; h:= f(t);
          if l(h)<0 then l(h):= t fi
        od: l(k)
      end
    end():
seq(a(n), n=1..60);

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]]];
a[k_] := A[1, k];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 13 2023, after Alois P. Heinz *)

A330589 The n-th index m such that A330439(m) = n.

Original entry on oeis.org

0, 7, 12, 29, 59, 72, 85, 142, 190, 205, 238, 270, 305, 318, 402, 492, 619, 652, 795, 845, 950, 996, 1121, 1163, 1228, 1393, 1548, 1662, 1756, 1920, 1937, 2106, 2202, 2351, 2448, 2555, 2594, 2707, 2788, 3254, 3420, 3466, 3663, 3974, 4136, 4282, 4363, 4621, 4732

Offset: 1

Alois P. Heinz, Dec 18 2019

nonn

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

Main diagonal of A330587.

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 a, b, t; a, b, t:= proc() end, proc() 0 end, -1;
      proc(k) local h;
        while b(k)

a(n) = A330587(n,n).

A337063 a(n) = 1 for n < 2; a(n) = freq(a(n-1),n) * freq(a(n-2),n) for n >= 2, where freq(i,j) is the number of times i appears in [a(0),a(1),...,a(j-1)].

Original entry on oeis.org

1, 1, 4, 2, 1, 3, 3, 4, 4, 9, 3, 3, 16, 4, 4, 25, 5, 1, 4, 24, 6, 1, 5, 10, 2, 2, 9, 6, 4, 14, 7, 1, 6, 18, 3, 5, 15, 3, 6, 24, 8, 2, 4, 32, 8, 2, 10, 10, 9, 9, 16, 8, 6, 15, 10, 8, 16, 12, 3, 7, 14, 4, 18, 18, 9, 15, 15, 16, 16, 25, 10, 10, 36, 6, 6, 49, 7, 3

Offset: 0

Alex Lauber, Aug 13 2020

nonn

Does this sequence contain every number?
Does each number appear only a finite number of times?
Starting with a(0)=0 and a(1)=1 gives the same sequence offset by one place.

			a(2) = occurrences of a(1)=1 in [a(0), a(1)]=[1, 1] * occurrences of a(0)=1 in [a(0), a(1)]=[1, 1] = 2*2 = 4.
a(3) = occurrences of a(2)=4 in [a(0), a(1), a(2)]=[1, 1, 4] * occurrences of a(1)=1 in [a(0), a(1), a(2)]=[1, 1, 4] = 1*2 = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..25000
Rémy Sigrist, PARI program for A337063

Cf. A337064 (index of first occurrence of n).

Cf. A316774 (which adds the two previous terms), A316973.

PARI
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See Links section.
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More terms from Rémy Sigrist, Sep 18 2020

A337064 a(n) is the index of the first occurrence of n in A337063, or -1 if n never appears.

Original entry on oeis.org

0, 3, 5, 2, 16, 20, 30, 40, 9, 23, 90, 57, 110, 29, 36, 12, 220, 33, 342, 230, 163, 179, 494, 19, 15, 109, 128, 88, 694, 82, 744, 43, 125, 219, 169, 72, 1060, 373, 253, 85, 1205, 113, 1346, 151, 207, 564, 1726, 131, 75, 332

Offset: 1

Alex Lauber, Aug 13 2020

nonn

419 is the lowest number that does not appear in the first 100000 terms of A337063.

			A337063 starts {1, 1, 4, 2, ...} so a(2) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A337064

Cf. A337063, A316774 (adds the two previous terms), A316973 (similar index for the addition sequence).

PARI
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See Links section.
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cp's OEIS Frontend

A329934 a(1)=1, a(2)=1, a(n) = (number of times a(n-1) has appeared before) + (number of times a(n-2) has appeared before).

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A330332 a(n) = (number of times a(n-1) has already appeared) + (number of times a(n-2) has already appeared) + (number of times a(n-3) has already appeared), starting with a(n) = n for n<3.

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A330448 a(n) = A330447(n) - A316905(n).

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A317015 a(n) = n for n < 2, a(n) = a(freq(a(n-1),n)) + a(freq(a(n-2),n)) for n >= 2, where freq(i, j) is the number of times i appears in the first j terms.

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A317127 a(0) = a(1) = a(2) = 1; for n >= 3, a(n) = freq(a(n-1),n) + freq(a(n-3),n) where freq(i, j) is the number of times i appears in the terms a(0) .. a(j-1).

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A330588 a(n) is the first index m such that A330439(m) = n.

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A330589 The n-th index m such that A330439(m) = n.

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A337063 a(n) = 1 for n < 2; a(n) = freq(a(n-1),n) * freq(a(n-2),n) for n >= 2, where freq(i,j) is the number of times i appears in [a(0),a(1),...,a(j-1)].

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