Fibonacci Numbers
These are sometimes called the Fibonacci 2-step numbers or 2-bonacci numbers. The sequence starts with two predetermined terms and each term afterwards is the sum of the preceding two terms.
\(F_0 = 0\)
\(F_1 = 1\)
\(F_n = F_{n−1} + F_{n−2}\), for integer \(n \ge 2\).
See OEIS A000045.
{
a = [0, 1];
for (x = 3, 30,
\(\qquad\)y = a[x – 1] + a[x – 2];
\(\qquad\)a = concat(a, y);
);
print(a);
}
Fibonacci Numbers 0–30
Base-2 (Binary)
0, 1, 1, 10, 11, 101, 1000, 1101, 10101, 100010, 110111, 1011001, 10010000, 11101001, 101111001, 1001100010, 1111011011, 11000111101, 101000011000, 1000001010101, 1101001101101, 10101011000010, 100010100101111, 110111111110001, 1011010100100000, 10010010100010001, 11101101000110001, 101111111101000010, 1001101100101110011, 1111101100010110101, 11001011001000101000
Base-3
0, 1, 1, 2, 10, 12, 22, 111, 210, 1021, 2001, 10022, 12100, 22122, 111222, 211121, 1100120, 2012011, 10112201, 12201212, 100021120, 120000102, 220021222, 1110022101, 2100121100, 10210220201, 20011112001, 100222102202, 121010221210, 222010101112, 1120021100022
Base-4
0, 1, 1, 2, 3, 11, 20, 31, 111, 202, 313, 1121, 2100, 3221, 11321, 21202, 33123, 120331, 220120, 1001111, 1221231, 2223002, 10110233, 12333301, 23110200, 102110101, 131220301, 233331002, 1031211303, 1331202311, 3023020220
Base-5
0, 1, 1, 2, 3, 10, 13, 23, 41, 114, 210, 324, 1034, 1413, 3002, 4420, 12422, 22342, 40314, 113211, 204030, 322241, 1031321, 1404112, 2440433, 4400100, 12341033, 22241133, 40132221, 112423404, 203111130
Base-6
0, 1, 1, 2, 3, 5, 12, 21, 33, 54, 131, 225, 400, 1025, 1425, 2454, 4323, 11221, 15544, 31205, 51153, 122402, 213555, 340401, 554400, 1335201, 2334001, 4113202, 10451203, 15004405, 25500012
Base-7
0, 1, 1, 2, 3, 5, 11, 16, 30, 46, 106, 155, 264, 452, 1046, 1531, 2610, 4441, 10351, 15122, 25503, 43625, 102431, 146356, 252120, 431506, 1013626, 1445435, 2462364, 4241132, 10033526
Base-8 (Octal)
0, 1, 1, 2, 3, 5, 10, 15, 25, 42, 67, 131, 220, 351, 571, 1142, 1733, 3075, 5030, 10125, 15155, 25302, 42457, 67761, 132440, 222421, 355061, 577502, 1154563, 1754265, 3131050
Base-9
0, 1, 1, 2, 3, 5, 8, 14, 23, 37, 61, 108, 170, 278, 458, 747, 1316, 2164, 3481, 5655, 10246, 16012, 26258, 43271, 70540, 123821, 204461, 328382, 533853, 863345, 1507308
Base-10 (Decimal)
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040
Base-16 (Hexadecimal)
0, 1, 1, 2, 3, 5, 8, D, 15, 22, 37, 59, 90, E9, 179, 262, 3DB, 63D, A18, 1055, 1A6D, 2AC2, 452F, 6FF1, B520, 12511, 1DA31, 2FF42, 4D973, 7D8B5, CB228
Fibonacci Numbers of Higher Orders
Tribonacci Numbers
Sometimes called the Fibonacci 3-step numbers or 3-bonacci numbers, the tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms.
\(F_0 = F_1 = 0\)
\(F_2 = 1\)
\(F_n = F_{n−1} + F_{n−2} + F_{n−3}\), for integer \(n \ge 3\).
See OEIS A000073.
a = [0, 0, 1];
for (x = 4, 30,
\(\qquad\)y = a[x – 1] + a[x – 2] + a[x – 3];
\(\qquad\)a = concat(a, y);
);
print(a);
}
\(\qquad\)
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064]
Tetranacci numbers
Sometimes called the Fibonacci 4-step numbers or 4-bonacci numbers, the tetranacci numbers start with four predetermined terms and each term afterwards is the sum of the preceding four terms.
\(F_0 = F_1 = F_2 = 0\)
\(F_3 = 1\)
\(F_n = F_{n−1} + F_{n−2} + F_{n−3} + F_{n−4}\), for integer \(n \ge 4\).
See OEIS A000078.
a = [0, 0, 0, 1];
for (x = 5, 30,
\(\qquad\)y = a[x – 1] + a[x – 2] + a[x – 3] + a[x – 4];
\(\qquad\)a = concat(a, y);
);
print(a);
}
\(\qquad\)
[0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, 1055026, 2033628, 3919944, 7555935, 14564533]
Pentanacci Numbers
Sometimes called the Fibonacci 5-step numbers or 5-bonacci numbers, the pentanacci numbers start with five predetermined terms and each term afterwards is the sum of the preceding five terms.
\(F_0 = F_1 = F_2 = F_3 = 0\)
\(F_4 = 1\)
\(F_n = F_{n−1} + F_{n−2} + F_{n−3} + F_{n−4} + F_{n−5}\), for integer \(n \ge 5\).
See OEIS A001591.
a = [0, 0, 0, 0, 1];
for (x = 6, 30,
\(\qquad\)y = a[x – 1] + a[x – 2] + a[x – 3] + a[x – 4] + a[x – 5];
\(\qquad\)a = concat(a, y);
);
print(a);
}
\(\qquad\)
[0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641]
Hexanacci Numbers
Sometimes called the Fibonacci 6-step numbers or 6-bonacci numbers, the hexanacci numbers start with six predetermined terms and each term afterwards is the sum of the preceding six terms.
\(F_0 = F_1 = F_2 = F_3 = F_4 = 0\)
\(F_5 = 1\)
\(F_n = F_{n−1} + F_{n−2} + F_{n−3} + F_{n−4} + F_{n−5} + F_{n−6}\), for integer \(n \ge 6\).
See OEIS A001592.
a = [0, 0, 0, 0, 0, 1];
for (x = 7, 30,
\(\qquad\)y = a[x – 1] + a[x – 2] + a[x – 3] + a[x – 4] + a[x – 5] + a[x – 6];
\(\qquad\)a = concat(a, y);
);
print(a);
}
\(\qquad\)
[0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, 29970, 59448, 117920, 233904, 463968, 920319, 1825529, 3621088, 7182728]
Heptanacci Numbers
Sometimes called the Fibonacci 7-step numbers or 7-bonacci numbers, the heptanacci numbers start with seven predetermined terms and each term afterwards is the sum of the preceding seven terms.
\(F_0 = F_1 = F_2 = F_3 = F_4 = F_5 = 0\)
\(F_6 = 1\)
\(F_n = F_{n−1} + F_{n−2} + F_{n−3} + F_{n−4} + F_{n−5} + F_{n−6} + F_{n−7}\), for integer \(n \ge 7\).
See OEIS ______.
a = [0, 0, 0, 0, 0, 0, 1];
for (x = 8, 30,
\(\qquad\)y = a[x – 1] + a[x – 2] + a[x – 3] + a[x – 4] + a[x – 5] + a[x – 6] + a[x – 7];
\(\qquad\)a = concat(a, y);
);
print(a);
}
\(\qquad\)
[0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776, 987568, 1967200, 3918592]
Octanacci Numbers
Sometimes called the Fibonacci 8-step numbers or 8-bonacci numbers, the octanacci numbers start with eight predetermined terms and each term afterwards is the sum of the preceding eight terms.
\(F_0 = F_1 = F_2 = F_3 = F_4 = F_5 = F_6 = 0\)
\(F_7 = 1\)
\(F_n = F_{n−1} + F_{n−2} + F_{n−3} + F_{n−4} + F_{n−5} + F_{n−6} + F_{n−7} + F_{n−8}\), for integer \(n \ge 8\).
See OEIS ______.
a = [0, 0, 0, 0, 0, 0, 0, 1];
for (x = 9, 30,
\(\qquad\)y = a[x – 1] + a[x – 2] + a[x – 3] + a[x – 4] + a[x – 5] + a[x – 6] + a[x – 7]
\(\qquad\)\(\qquad\)+ a[x – 8];
\(\qquad\)a = concat(a, y);
);
print(a);
}
\(\qquad\)
[0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192, 64256, 128257, 256005, 510994, 1019960, 2035872]
Enneanacci Numbers
https://oeis.org/A104144
Sometimes called the Fibonacci 9-step numbers or 9-bonacci numbers, the enneanacci numbers start with nine predetermined terms and each term afterwards is the sum of the preceding nine terms.
\(F_0 = F_1 = F_2 = F_3 = F_4 = F_5 = F_6 = F_7 = 0\)
\(F_8 = 1\)
\(F_n = F_{n−1} + F_{n−2} + F_{n−3} + F_{n−4} + F_{n−5} + F_{n−6} + F_{n−7} + F_{n−8} + F_{n−9}\), for integer \(n \ge 9\).
See OEIS ______.
a = [0, 0, 0, 0, 0, 0, 0, 0, 1];
for (x = 10, 30,
\(\qquad\)y = a[x – 1] + a[x – 2] + a[x – 3] + a[x – 4] + a[x – 5] + a[x – 6] + a[x – 7]
\(\qquad\)\(\qquad\)+ a[x – 8] + a[x – 9];
\(\qquad\)a = concat(a, y);
);
print(a);
}
\(\qquad\)
[0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269]
Decanacci Numbers
Sometimes called the Fibonacci 10-step numbers or 10-bonacci numbers, the decanacci numbers start with 10 predetermined terms and each term afterwards is the sum of the preceding 10 terms.
\(F_0 = F_1 = F_2 = F_3 = F_4 = F_5 = F_6 = F_7 = F_8 = 0\)
\(F_9 = 1\)
\(F_n = F_{n−1} + F_{n−2} + F_{n−3} + F_{n−4} + F_{n−5} + F_{n−6} + F_{n−7} + F_{n−8} + F_{n−9} + F_{n−10}\), for integer \(n \ge 10\).
See OEIS A122265.
a = [0, 0, 0, 0, 0, 0, 0, 0, 0, 1];
for (x = 11, 30,
\(\qquad\)y = a[x – 1] + a[x – 2] + a[x – 3] + a[x – 4] + a[x – 5] + a[x – 6] + a[x – 7]
\(\qquad\)\(\qquad\)+ a[x – 8] + a[x – 9] + a[x – 10];
\(\qquad\)a = concat(a, y);
);
print(a);
}
\(\qquad\)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172, 16336, 32656, 65280, 130496, 260864, 521472]
Hendecanacci Numbers
Sometimes called the Fibonacci 11-step numbers or 11-bonacci numbers, the hendecanacci numbers start with 11 predetermined terms and each term afterwards is the sum of the preceding 11 terms.
\(F_0 = F_1 = F_2 = F_3 = F_4 = F_5 = F_6 = F_7 = F_8 = F_9 = 0\)
\(F_{10} = 1\)
\(F_n = F_{n−1} + F_{n−2} + F_{n−3} + F_{n−4} + F_{n−5} + F_{n−6} + F_{n−7} + F_{n−8} + F_{n−9} + F_{n−10} + F_{n−11}\), for integer \(n \ge 11\).
See OEIS A168082.
a = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1];
for (x = 12, 30,
\(\qquad\)y = a[x – 1] + a[x – 2] + a[x – 3] + a[x – 4] + a[x – 5] + a[x – 6] + a[x – 7]
\(\qquad\)\(\qquad\)+ a[x – 8] + a[x – 9] + a[x – 10] + a[x – 11];
\(\qquad\)a = concat(a, y);
);
print(a);
}
\(\qquad\)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2047, 4093, 8184, 16364, 32720, 65424, 130816, 261568]
Dodecanacci Numbers
Sometimes called the Fibonacci 12-step numbers or 12-bonacci numbers, the dodecanacci numbers start with 12 predetermined terms and each term afterwards is the sum of the preceding 12 terms.
\(F_0 = F_1 = F_2 = F_3 = F_4 = F_5 = F_6 = F_7 = F_8 = F_9 = F_{10} = 0\)
\(F_{11} = 1\)
\(F_n = F_{n−1} + F_{n−2} + F_{n−3} + F_{n−4} + F_{n−5} + F_{n−6} + F_{n−7} + F_{n−8} + F_{n−9} + F_{n−10} + F_{n−11} + F_{n−12}\), for integer \(n \ge 12\).
See OEIS A168083.
a = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1];
for (x = 13, 30,
\(\qquad\)y = a[x – 1] + a[x – 2] + a[x – 3] + a[x – 4] + a[x – 5] + a[x – 6] + a[x – 7]
\(\qquad\)\(\qquad\)+ a[x – 8] + a[x – 9] + a[x – 10] + a[x – 11] + a[x – 12];
\(\qquad\)a = concat(a, y);
);
print(a);
}
\(\qquad\)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4095, 8189, 16376, 32748, 65488, 130960]
Other Fibonacci Properties
Negative Fibonacci Numbers
When the Fibonacci sequence is extended in the negative direction, the numbers alternate between positive and negative.
for (x = -10, 10, print1(fibonacci(x), “, “))
Cassini’s Formula
Cassini’s Formula states:
\(F_{n+1} \space \cdot \space F_{n-1} \space – \space (F_{n})^{2} \space = \space (-1)^{n}\)
for (x = 1, 10, print(x, ” = “, fibonacci(x+1) * fibonacci(x-1) – fibonacci(x)^2))
2 = 1
3 = -1
4 = 1
5 = -1
6 = 1
7 = -1
8 = 1
9 = -1
10 = 1
Cassini’s Formula Variant
A variant of Cassini’s Formula states:
\(F_{n-2} \space \cdot \space F_{n+1} \space – \space F_{n-1} \space \cdot \space F_{n} \space = \space (-1)^{n-1}\)
Lucas’ Theorem
Lucas’ Theorem states:
\(F_{m} \space \text{gcd} \space F_{n} \space = \space F_{(m \space \text{gcd} \space n)}\)
\(\text{gcd}\) = greatest common divisor
Simson’s Relation
Simson’s Relation states:
\(F_{n+1} \space \cdot \space F_{n-1} \space + \space (-1)^{n-1} \space = \space (F_{n})^{2}\)
References
The Fibonacci Association — Official website. Hosted by the Department of Mathematics & Statistics at Dalhousie University in Halifax, Nova Scotia, Canada. Accessed August 24, 2019.
Inglis, Alan N. “A249169 Fibonacci 16-step numbers, a(n) = a(n-1) + a(n-2) + … + a(n-16)”. The On-Line Encyclopedia of Integer Sequences (OEIS). October 22, 2014. Accessed July 21, 2019.
Sarcone, G. “Fibonacci Number Properties“. Archimedes’ Lab. 2003. Accessed August 16, 2019.
Wikipedia contributors. “Fibonacci number”. Wikipedia. Accessed July 21, 2019.
Wikipedia contributors. “Generalizations of Fibonacci numbers”. Wikipedia. Accessed July 21, 2019.