A275124 Multiples of 5 where Pisano periods of Fibonacci numbers A001175 and Lucas numbers A106291 agree.

55, 110, 155, 165, 205, 220, 305, 310, 330, 355, 385, 410, 440, 465, 495, 505, 605, 610, 615, 620, 655, 660, 710, 715, 755, 770, 820, 880, 905, 915, 930, 935, 955, 990, 1010, 1045, 1065, 1085, 1155, 1205, 1210, 1220, 1230, 1240, 1255, 1265, 1310, 1320, 1355, 1395, 1420, 1430, 1435, 1485, 1510, 1515, 1540, 1555, 1595, 1640, 1655, 1705, 1760, 1810, 1815, 1830

Offset

1,1

Comments

Multiples of 5 where A001175 and A106291 agree. See 1st comment of A106291.

Links

Table of n, a(n) for n=1..66.

Patrick Flanagan, Marc S. Renault, and Josh Updike, Symmetries of Fibonacci Points, Mod m, Fibonacci Quart. 53 (2015), no. 1, 34-41. See p. 7. (Is this the same sequence?)

Example

55 is the first multiple of 5 where the Pisano period (Fibonacci) of n = 55 and the Pisano period (Lucas) of n = 55 agree (this is in this case 20).

Prog

(JavaScript)
let bases = [],
    basesd = [],
    baselimit = 2000;
for (let base = 2; base <= baselimit; base++) {
    let fibs = [1 % base, 1 % base],
        lucas = [2 % base, 1 % base],
        repeatingf = false,
        repeatingl = false;
    while (!repeatingf) {
        fibs.push((fibs[fibs.length - 2] + fibs[fibs.length - 1]) % base);
        if (1 == fibs[fibs.length - 2] &&
            0 == fibs[fibs.length - 1])
            repeatingf = true;
    }
    while (!repeatingl) {
        lucas.push((lucas[lucas.length - 2] + lucas[lucas.length - 1]) % base);
        if ((lucas[0] == (lucas[lucas.length - 2] + lucas[lucas.length - 1]) % base) &&
            (lucas[1] == (lucas[lucas.length - 2] + 2 *lucas[lucas.length - 1]) % base))
            repeatingl = true;
    }
    if (fibs.length != lucas.length)
        bases.push(base);
}
for (let i = 1; i <= baselimit/5; i++) {
    if (!bases.includes(i * 5)) basesd.push(i * 5);
}
console.log(basesd.join(', '));

Crossrefs

Cf. A001175, A106291.

Sequence in context: A323070 A118151 A277273 * A282768 A217429 A154031

Adjacent sequences: A275121 A275122 A275123 * A275125 A275126 A275127

Keyword

nonn

Author

Dan Dart, Jul 18 2016

Status

approved