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Date: 22-4-2019
1520
Date: 22-4-2019
1242
Date: 13-8-2018
2882
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Let
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(2) |
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(3) |
(OEIS A104457), where is the golden ratio, and
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(5) |
(OEIS A002390).
Define the Fibonacci hyperbolic sine by
(6) |
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(8) |
The function satisfies
(9) |
and for ,
(10) |
where is a Fibonacci number. For , 2, ..., the values are therefore 1, 3, 8, 21, 55, ... (OEIS A001906).
Define the Fibonacci hyperbolic cosine by
(11) |
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(12) |
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(13) |
This function satisfies
(14) |
and for ,
(15) |
where is a Fibonacci number. For , 2, ..., the values are therefore 2, 5, 13, 34, 89, ... (OEIS A001519).
Similarly, the Fibonacci hyperbolic tangent is defined by
(16) |
and for ,
(17) |
For , 2, ..., the values are therefore 1/2, 3/5, 8/13, 21/34, 55/89, ... (OEIS A001906 and A001519).
REFERENCES:
Sloane, N. J. A. Sequences A001519/M1439, A001906/M2741, A002390/M3318, and A104457 in "The On-Line Encyclopedia of Integer Sequences."
Stakhov, A. and Tkachenko, I. "Hyperbolic Fibonacci Trigonometry." Dokl. Akad. Nauk Ukrainy, No. 7, 9-14, 1993.
Trzaska, Z. W. "On Fibonacci Hyperbolic Trigonometry and Modified Numerical Triangles." Fib. Quart. 34, 129-138, 1996.
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5 علامات تحذيرية قد تدل على "مشكل خطير" في الكبد
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