A337212 Modulo 3 Pisano period of 'n-bonacci' series.

1, 8, 13, 26, 104, 728, 364, 80, 91, 8744, 3851, 3280, 59048, 4782968, 7174453, 3438578, 16139240, 5373368, 5235412, 1678822106, 86049704, 387420488, 47071589413, 140633637386, 2952400, 728, 757, 9526168288, 7312949144072, 49566102697280, 24477226494760

Offset

1,2

Comments

The modulo 2 variant of this sequence gives 1, 3, 4, 5, 6, 7, 8, ... (the natural numbers not including 2), and likewise, when the modulus is a power of 2, it seems that the Pisano period lengths form an arithmetic progression. (Note that both of these observations are based on empirical observation only).

a(39)=797161, a(80)=6560, a(81)=6643, a(90)=5380840, a(242)=59048, a(243)=59293, a(728)=531440, a(729)= 532171, a(2186)=4782968, a(2187)=4785157, a(6560)=43046720, a(6561)=43053283, a(19682)=387420488, a(19683)=387440173. - Chai Wah Wu, Sep 15 2020

Links

Table of n, a(n) for n=1..31.

Formula

Conjecture: a(3^k-1)=a(3^k)-3^k-2=3^(2k)-1, a(3^k)=3^k(3^k+1)+1 for k>0. - Chai Wah Wu, Sep 15 2020

Example

For n = 3, the remainders modulo 3 of the tribonacci series are 0, 1, 1, 2, 1, 1, 1, 0, 2, 0, 2, 1, 0, (these repeat indefinitely), so the Pisano period of the 'tribonacci' sequence is 13.

Prog

(PARI) a(n) = {my(v=w=concat(0, vector(n-1, i, 1))); for(k=1, oo, v=concat(v[2..n], vecsum(v)%3); if(v==w, return(k))); } \\ Jinyuan Wang, Aug 20 2020
(Python)
def A337212(n):
    x, y, k, r, m = (3**n-3)//2, (3**n-3)//2, (n-1)%3, 3**(n-1), 0
    while True:
        m += 1
        a, b = divmod(x, 3)
        x, k = a+k*r, (k+k-b)%3
        if y == x:
            return m # Chai Wah Wu, Sep 14 2020

Crossrefs

Cf. A001175 (period of Fibonacci numbers mod n).

Sequence in context: A019535 A229446 A205704 * A093023 A057097 A246639

Adjacent sequences: A337209 A337210 A337211 * A337213 A337214 A337215

Keyword

nonn

Author

Adam Bascal, Aug 19 2020

Extensions

a(20)-a(22) from Jinyuan Wang, Aug 20 2020

a(23) from Chai Wah Wu, Sep 14 2020

a(24)-a(28) from Chai Wah Wu, Sep 15 2020

a(29)-a(31) from Chai Wah Wu, Sep 21 2020

Status

approved