OFFSET
1,3
COMMENTS
A fractal sequence : If one deletes the first occurrence of 1, 2, 3, ... the original sequence is reproduced.
Subsequent runs of consecutive terms which are these first occurrences are separated by a term whose square root yields the length of the preceding run (when rounded down).
Motivated by Project Euler problem 535, see LINKS.
LINKS
Martin Møller Skarbiniks Pedersen, Table of n, a(n) for n = 1..1000
Project Euler, Problem 535: Fractal Sequence
Clark Kimberling, Interspersions and Fractal Sequences Associated with Fractions c^j/d^k, Journal of Integer Sequences, Issue 5, Volume 10 (2007), Article 07.5.1
FORMULA
The sequence contains marked numbers and non-marked numbers.
The marked numbers are consecutive starting with a(1)=1.
Immediately preceding each non-marked number in a(n), there are exactly floor(sqrt(a(n)) [= A000196(a(n))] adjacent marked numbers.
EXAMPLE
The runs of first occurrences of the positive integers are {1}, {2}, {3}, {4}, {5}, {6}, {7, 8}, {9}, {10, 11}, ... each separated from the next one by, respectively, 1, 1, 2, 1, 3, 2, 4, 1, 5, ... where 4 and 5 follow the groups {7, 8} and {10, 11} of length 2 = sqrt(4) = floor(sqrt(5)). - M. F. Hasler, Dec 13 2015
PROG
(C)
#include <stdio.h>
#include <math.h>
#define SIZE 1000
unsigned int numbers[SIZE];
int main() {
unsigned int pointer=0, next=1, circle_count=1, next_circle_number=2, sqrt_non_circle=1;
numbers[0]=1; printf("1");
while (next<SIZE) {
if (circle_count==sqrt_non_circle) {
numbers[next]=numbers[pointer]; circle_count=0; pointer++;
sqrt_non_circle=sqrt(numbers[pointer]);
} else {
circle_count++; numbers[next]=next_circle_number;
next_circle_number++;
}
printf(", %u", numbers[next]); next++;
}
}
(PARI) A265650(n, list=0, a=[1], cc=0, nc=1, p=0)={for(i=2, n, a=concat(a, if(0<=cc-=1, nc+=1, cc=sqrtint(a[!!p+p+=1]); a[p]))); list&&return(a); a[n]} \\ Set 2nd optional arg.to 1 to return the whole list. - M. F. Hasler, Dec 13 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Martin Møller Skarbiniks Pedersen, Dec 11 2015
EXTENSIONS
New name from M. F. Hasler, Dec 13 2015
STATUS
approved