A275124 -id:A275124 - cp's OEIS frontend

A275167 Pisano periods of A275124.

20, 60, 60, 40, 40, 60, 60, 60, 120, 140, 80, 120, 60, 120, 120, 100, 220, 60, 40, 60, 260, 120, 420, 140, 100, 240, 120, 120, 180, 120, 120, 180, 380, 120, 300, 180, 280, 240, 80, 240, 660, 60, 120, 60, 500, 240, 780, 120, 540, 120, 420, 420, 80, 360, 300, 200, 240, 620, 140, 120, 220, 60, 240, 180, 440, 120, 120, 120, 180, 1140, 520, 120

Offset: 1

Dan Dart, Jul 18 2016

nonn

			A275124(1) = 55, and PisanoPeriod(55) = 20, etc.

Cf. A001175, A275124.

let bases = [],
    Fs = [],
    Ls = [],
    agrees = [],
    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;
    }
    Fs[base] = fibs.length;
    Ls[base] = lucas.length;
    if (fibs.length != lucas.length)
        bases.push(base);
    //console.log('F', fibs.join(','), 'L:', lucas.join(','));
}
for (let i = 1; i <= baselimit/5; i++) {
    if (!bases.includes(i * 5)) {
        agrees.push(Fs[i * 5]);
    }
}
console.log(agrees.join(','));

a(n) = A001175(A275124(n)).

A344258 Integers m for which F (mod m) does not have rotational symmetry although F (mod k) does have rotational symmetry for all factors k of m.

Original entry on oeis.org

4, 11, 15, 19, 21, 29, 31, 35, 39, 51, 59, 69, 71, 79, 91, 101, 111, 115, 119, 131, 139, 141, 151, 159, 179, 181, 183, 191, 199, 205, 211, 215, 219, 229, 235, 239, 251, 259, 267, 271, 287, 291, 299, 301, 309, 311, 327, 329, 331, 335, 339, 349, 359, 371, 379, 381

Offset: 1

Michel Marcus, May 13 2021

nonn

Flanagan et al. define F (mod m) as the set of points [x_i, y_i] (mod m) where x_i = Fibonacci(i) and y_i = Fibonacci(i+1).

Patrick Flanagan, Marc S. Renault, and Josh Updike, Symmetries of Fibonacci Points, Mod m, Fibonacci Quart. 53 (2015), no. 1, 34-41. See p. 5.

Cf. A000045, A001175, A275124, A344281.

\\ where pisano(n) is A001175
hasrot(m) = {if (m==2, return (1)); my(j=pisano(m)/2); my(vf = [fibonacci(j), fibonacci(j+1)]); Mod(vf, m) == [0, -1];}
isok(m) = {if ((m>1) && !hasrot(m), fordiv(m, d, if ((d>1) && (d

A344281 Integers m for which F (mod m) has rotational symmetry.

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 10, 13, 14, 17, 18, 23, 25, 26, 27, 34, 37, 41, 43, 46, 47, 49, 50, 53, 54, 61, 65, 67, 73, 74, 81, 82, 83, 85, 86, 89, 94, 97, 98, 103, 106, 107, 109, 113, 122, 123, 125, 127, 129, 130, 134, 137, 146, 149, 157, 161, 162, 163, 166, 167, 169, 170

Offset: 1

Michel Marcus, May 14 2021

nonn

Flanagan et al. define F (mod m) as the set of points [x_i, y_i] (mod m) where x_i = Fibonacci(i) and y_i = Fibonacci(i+1).

Patrick Flanagan, Marc S. Renault, and Josh Updike, Symmetries of Fibonacci Points, Mod m, Fibonacci Quart. 53 (2015), no. 1, 34-41. See p. 5.

Cf. A000045, A001175, A275124, A344258.

\\ where pisano(n) is A001175
hasrot(m) = {if (m==1, return (0)); if (m==2, return (1)); my(j = pisano(m)/2); my(vf = [fibonacci(j), fibonacci(j+1)]); Mod(vf, m) == [0, -1];}

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A275167 Pisano periods of A275124.

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A344258 Integers m for which F (mod m) does not have rotational symmetry although F (mod k) does have rotational symmetry for all factors k of m.

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PARI

A344281 Integers m for which F (mod m) has rotational symmetry.

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PARI