A377030 Period 19: repeat [0, 3, 6, 9, 7, 4, 1, 2, 5, 8, 8, 5, 2, 1, 4, 7, 9, 6, 3].

0, 3, 6, 9, 7, 4, 1, 2, 5, 8, 8, 5, 2, 1, 4, 7, 9, 6, 3, 0, 3, 6, 9, 7, 4, 1, 2, 5, 8, 8, 5, 2, 1, 4, 7, 9, 6, 3, 0, 3, 6, 9, 7, 4, 1, 2, 5, 8, 8, 5, 2, 1, 4, 7, 9, 6, 3, 0, 3, 6, 9, 7, 4, 1, 2, 5, 8, 8, 5, 2, 1, 4, 7, 9, 6, 3, 0, 3, 6, 9, 7, 4, 1, 2, 5, 8, 8, 5, 2, 1, 4, 7, 9, 6, 3, 0

Offset: 0

Author

Neil Vaughan, Oct 13 2024

Keywords

• nonn
• easy

Comments

Difference between the multiples of 3 (A008585) and the closest multiple of 19 (A008601). For any two numbers (in this case p=3 and q=19), a sequence is produced consisting of a palindrome cycle. If p and q are coprime, then the cycle length is equal to max(p,q). The biggest number in the sequence will be at most half of max(p,q). Here is a plot of the first cycle of this sequence:

9 X X

8 X X

7 X X

6 X X

5 X X

4 X X

3 X X

2 X X

1 X X

0 X X

Links

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

Crossrefs

Cf. A008585, A008601.

Programs

C

int p = 3;
int q = 19;
for (int t = 0;t <= 99;t++) {
	int closest = 999;
	for (int i = 0;i <= 99;i++) {
		int dist=abs(i * q - t * p);
		if (dist < closest) {
			closest = dist;
		}
	}
	printf("%i, ", closest);
}

Mathematica

LinearRecurrence[{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1},{0, 3, 6, 9, 7, 4, 1, 2, 5, 8, 8, 5, 2, 1, 4, 7, 9, 6, 3},96] (* James C. McMahon, Oct 31 2024 *)

PARI

a(n) = my(n3 = 3*n); min(n3 - 19*floor(n3/19), ceil(19*ceil(n3/19) - n3)) \\ David A. Corneth, Oct 14 2024

Python

def A377030(n): return (0, 3, 6, 9, 7, 4, 1, 2, 5, 8, 8, 5, 2, 1, 4, 7, 9, 6, 3)[n%19] # Chai Wah Wu, Oct 31 2024

Formula

a(n) = a(n-19). - David A. Corneth, Oct 14 2024

a(n) = min(3*n-19*floor(3*n/19), 19*ceil(3*n/19)-3*n).