login
A275124
Multiples of 5 where Pisano periods of Fibonacci numbers A001175 and Lucas numbers A106291 agree.
3
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
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
Sequence in context: A323070 A118151 A277273 * A282768 A217429 A154031
KEYWORD
nonn
AUTHOR
Dan Dart, Jul 18 2016
STATUS
approved