A364449 -id:A364449 - cp's OEIS frontend

A364448 Lexicographically earliest sequence where n is banned for n^2 terms after each appearance.

1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 3, 1, 7, 1, 2, 1, 8, 1, 4, 1, 2, 1, 3, 1, 9, 1, 2, 1, 10, 1, 5, 1, 2, 1, 3, 1, 4, 1, 2, 1, 11, 1, 12, 1, 2, 1, 3, 1, 6, 1, 2, 1, 13, 1, 4, 1, 2, 1, 3, 1, 5, 1, 2, 1, 7, 1, 14, 1, 2, 1, 3, 1, 4, 1, 2, 1, 15, 1, 16, 1, 2, 1, 3, 1, 8, 1, 2, 1, 5, 1, 4

Offset: 1

Rok Cestnik, Jul 25 2023

After 12427560 terms the sequence is periodic with period 2985984. The largest periodic term is 158.
The largest term is 159 which appears only 3 times: a(4942686), a(7756992) and a(9818928).
If banning for n terms the sequence just repeats [1,2,1,3] (A364447). If banning for n^3 terms the sequence continues growing forever (A364449).
The first 41 terms are shared with A028920.

			a(n)   ban 1  2  3  4  5  6  7  ...
 1         |  |  |  |  |  |  |
 2         x  |  |  |  |  |  |
 1         |  x  |  |  |  |  |
 3         x  x  |  |  |  |  |
 1         |  x  x  |  |  |  |
 4         x  x  x  |  |  |  |
 1         |  |  x  x  |  |  |
 2         x  |  x  x  |  |  |
 1         |  x  x  x  |  |  |
 5         x  x  x  x  |  |  |
 1         |  x  x  x  x  |  |
 6         x  x  x  x  x  |  |
 1         |  |  x  x  x  x  |
 2         x  |  |  x  x  x  |
 1         |  x  |  x  x  x  |
 3         x  x  |  x  x  x  |
 1         |  x  x  x  x  x  |
 7         x  x  x  x  x  x  |
 .
 .
 .

Rok Cestnik, Table of n, a(n) for n = 1..10000

Cf. A364447, A364449, A028920.

#include
int main(void){
    int N = 1000;
    int *a = (int*)malloc((N+1)*sizeof(int));
    int *ban = (int*)malloc(160*sizeof(int));
    for(int n = 0; n < N; ++n){
        for(int can = 1; can < 160; ++can){
            if(ban[can] == 0){
                a[n] = can;
                ban[can] = can*can+1;
                break;
            }
        }
        for(int can = 1; can < 160; ++can) if(ban[can]) --ban[can];
    }
    free(a); free(ban);
    return 0;
}

a = []
ban = [0 for n in range(160)]
for i in range(1000):
    can = ban.index(0,1)
    ban = [max(b-1,0) for b in ban]
    a.append(can)
    ban[can] = can**2

a(n) = a(n-2985984) for n >= 15413545.

A364447 Repeat [1,2,1,3].

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1

Offset: 1

Rok Cestnik, Jul 25 2023

Continued fraction of sqrt(5) - 3/2 = 0.7360679... (without integer part); and (4*sqrt(5) + 6)/11 = 1.3585701... (with integer part).

Lexicographically earliest sequence in which n is banned for n terms after each appearance (see A364448 for n^2 and A364449 for n^3).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A131743 (repeat [0,1,0,2]).
Cf. A000034, A010882, A131534.
Cf. A364448, A364449.

PadRight[{}, 100, {1, 2, 1, 3}] (* Paolo Xausa, Jan 23 2025 *)

def A364447(n): return (3,1,2,1)[n&3] # Chai Wah Wu, Jul 29 2023

a(n) = A131743(n-1) + 1.

A364603 Lexicographically earliest sequence where after the m-th appearance of term z, it is banned from re-appearing in the next m*z terms.

Original entry on oeis.org

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

Offset: 1

Rok Cestnik, Jul 29 2023

			a(n)   ban 1  2  3  4  5  6  7 ...
 1         |  |  |  |  |  |  |
 2         x  |  |  |  |  |  |
 1         |  x  |  |  |  |  |
 3         x  x  |  |  |  |  |
 2         x  |  x  |  |  |  |
 1         |  x  x  |  |  |  |
 4         x  x  x  |  |  |  |
 3         x  x  |  x  |  |  |
 5         x  x  x  x  |  |  |
 1         |  |  x  x  x  |  |
 2         x  |  x  x  x  |  |
 4         x  x  x  |  x  |  |
 6         x  x  x  x  x  |  |
 7         x  x  x  x  x  x  |
 1         |  x  |  x  |  x  x
 3         x  x  |  x  |  x  x
 5         x  x  x  x  |  x  x
 2         x  |  x  x  x  x  x
 8         x  x  x  x  x  x  x
 6         x  x  x  x  x  |  x
 1         |  x  x  |  x  x  x
 .
 .
 .

Rok Cestnik, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.
Michael De Vlieger, Plot [b(k) > n] at (x,y) = (n,k), for n = 1..2^12, where k may appear at index b(k) and brackets are Iverson. Hence if b(k) > n, k is banned and shown in black else k is permitted and shown in white. This image is similar to the Example but rotated 90 degrees counterclockwise.

Cf. A364604, A364448, A364449.

c[] := 0; q[] := 1; r = k = 1; nn = 120; Do[Set[{a[n], c[k]}, {k, n + k*q[k] + 1}]; q[k]++; If[k > r, r = k]; k = MinimalBy[Range[r], c, 1][[1]]; If[c[k] > n + 1, k = r + 1], {n, nn}]; Array[a, nn] (* Michael De Vlieger, Aug 13 2023 *)

A364603(N) = {my(a=vector(N),z=1); for(s=1,N, if(a[s],next); my(m=0); for(i=s,N, if(!a[i], a[i]=z; i+=(z*(m++)))); z++); a}

def A364603(N):
    a = [0]*N; z=s=0
    while(s

A364604 Lexicographically earliest sequence, where after every appearance, a term is banned from re-appearing for twice as long as last time; first appearance bans it for 1 term.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 4, 3, 5, 1, 2, 4, 6, 5, 7, 6, 3, 7, 8, 9, 4, 6, 5, 7, 8, 9, 1, 2, 7, 8, 6, 9, 10, 3, 8, 10, 9, 4, 7, 5, 10, 11, 12, 8, 11, 9, 10, 6, 11, 12, 13, 14, 12, 11, 13, 7, 10, 12, 13, 1, 2, 8, 9, 11, 13, 14, 3, 12, 14, 15, 4, 15, 5, 10, 13, 14, 15, 16

Offset: 1

Rok Cestnik, Jul 29 2023

nonn

			a(n)   ban 1  2  3  4  5  6 ...
 1         |  |  |  |  |  |
 2         x  |  |  |  |  |
 1         |  x  |  |  |  |
 2         x  |  |  |  |  |
 3         x  x  |  |  |  |
 1         |  x  x  |  |  |
 2         x  |  |  |  |  |
 3         x  x  |  |  |  |
 4         x  x  x  |  |  |
 5         x  x  x  x  |  |
 1         |  x  |  |  x  |
 2         x  |  |  |  |  |
 3         x  x  |  |  |  |
 4         x  x  x  |  |  |
 5         x  x  x  x  |  |
 6         x  x  x  x  x  |
 4         x  x  x  |  x  x
 3         x  x  |  x  |  |
 5         x  x  x  x  |  |
 1         |  x  x  x  x  |
 .
 .
 .

Cf. A364603, A364448, A364449.

A364604(N) = {my(a=vector(N),z=1); for(s=1,N, if(a[s],next); my(m=-1); for(i=s,N, if(!a[i], a[i]=z; i+=(1<<(m++)))); z++); a}

def A364604(N):
    a = [0]*N; z=s=0
    while(s

cp's OEIS Frontend

A364448 Lexicographically earliest sequence where n is banned for n^2 terms after each appearance.

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A364447 Repeat [1,2,1,3].

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A364603 Lexicographically earliest sequence where after the m-th appearance of term z, it is banned from re-appearing in the next m*z terms.

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A364604 Lexicographically earliest sequence, where after every appearance, a term is banned from re-appearing for twice as long as last time; first appearance bans it for 1 term.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Python