In mathematics, the tribonacci ratio is a geometrical proportion, given by the unique real solution of the equation x3 = x2 + x + 1. Its decimal expansion begins with 1.839286755214161... (sequence A058265 in the OEIS). The moniker tribonacci was introduced by high-school student Mark Feinberg in an article published in the Fibonacci Quarterly of october 1963.
Definition
Three quantities a > b > c > 0 are in the tribonacci ratio if
a + b + c a = a b = b c = τ {\displaystyle {\frac {a+b+c}{a}}={\frac {a}{b}}={\frac {b}{c}}=\tau }
This ratio is commonly denoted τ . {\displaystyle \tau .} Substituting b = τ c {\displaystyle b=\tau c} and a = τ b = τ 2 c {\displaystyle a=\tau b=\tau ^{2}c} in the first fraction gives
τ = c ( τ 2 + τ + 1 ) τ 2 c . {\displaystyle \tau ={\frac {c(\tau ^{2}+\tau +1)}{\tau ^{2}c}}.} It follows that the tribonacci ratio is the unique real solution of the cubic equation τ 3 = τ 2 + τ + 1 {\displaystyle \tau ^{3}=\tau ^{2}+\tau +1} . Closed-form expressions for τ {\displaystyle \tau } are found by solving the depressed cubic y 3 − 4 3 y − 38 27 {\displaystyle y^{3}-{\tfrac {4}{3}}y-{\tfrac {38}{27}}} , which has real zero τ − 1 3 {\displaystyle \tau -{\tfrac {1}{3}}} .
τ = 1 3 ( 1 + 19 + 3 33 3 + 19 − 3 33 3 ) = 1 3 ( 1 + 4 cosh ( 1 3 arcosh ( 19 8 ) ) ) . {\displaystyle {\begin{aligned}\tau &={\frac {1}{3}}\left(1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}\right)\\&={\frac {1}{3}}\left(1+4\cosh \left({\frac {1}{3}}\operatorname {arcosh} \left({\frac {19}{8}}\right)\right)\right).\end{aligned}}}
The iteration x ← 1 2 + x 3 {\displaystyle x\gets {\sqrt[{3}]{{\tfrac {1}{2}}+x}}} with fixed point 1 τ − 1 {\displaystyle {\frac {1}{\tau -1}}} results in the continued radical
τ − 1 = 1 / 1 2 + 1 2 + 1 2 + ⋯ 3 3 3 {\displaystyle \tau -1=1/{\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+\cdots }}}}}}}
Since the iteration derives from 2 x 3 = 2 x + 1 {\displaystyle 2x^{3}=2x+1} , alternative expressions for τ {\displaystyle \tau } are
w 1 , 2 = ( 1 ± 1 3 11 3 ) / 4 τ = 1 + ( w 1 3 + w 2 3 ) − 1 = 1 + 3 2 sech ( 1 3 arcosh ( 3 3 4 ) ) . {\displaystyle {\begin{aligned}w_{1,2}&=\left(1\pm {\frac {1}{3}}{\sqrt {\frac {11}{3}}}\right)/4\\\tau &=1+({\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}})^{-1}\\&=1+{\frac {\sqrt {3}}{2}}\operatorname {sech} \left({\frac {1}{3}}\operatorname {arcosh} \left({\frac {3{\sqrt {3}}}{4}}\right)\right).\end{aligned}}}
1 τ − 1 {\displaystyle {\frac {1}{\tau -1}}} is the superstable fixed point of the Newton iteration x ← ( 2 x 3 + 1 2 ) / ( 3 x 2 − 1 ) {\displaystyle x\gets (2x^{3}+{\tfrac {1}{2}})/(3x^{2}-1)} .
Properties
The tribonacci ratio can be written in terms of itself as fractions
τ = τ 2 + 1 τ 2 − 1 τ 2 = τ + 1 τ − 1 τ 3 = τ 4 + 1 2 . {\displaystyle {\begin{aligned}\tau &={\frac {\tau ^{2}+1}{\tau ^{2}-1}}\\\tau ^{2}&={\frac {\tau +1}{\tau -1}}\\\tau ^{3}&={\frac {\tau ^{4}+1}{2}}.\end{aligned}}}
Similarly as the infinite geometric series
τ 2 + 1 2 = ∑ n = 0 ∞ τ − n τ + 1 2 = ∑ n = 0 ∞ τ − 2 n 1 τ − 1 = ∑ n = 0 ∞ τ − 3 n . {\displaystyle {\begin{aligned}{\frac {\tau ^{2}+1}{2}}&=\sum _{n=0}^{\infty }\tau ^{-n}\\{\frac {\tau +1}{2}}&=\sum _{n=0}^{\infty }\tau ^{-2n}\\{\frac {1}{\tau -1}}&=\sum _{n=0}^{\infty }\tau ^{-3n}.\end{aligned}}}
For every integer n {\displaystyle n} one has
τ n = τ n − 1 + τ n − 2 + τ n − 3 = 2 τ n − 2 + 2 τ n − 3 + τ n − 4 = 3 τ n − 2 + τ n − 4 + τ n − 6 {\displaystyle {\begin{aligned}\tau ^{n}&=\tau ^{n-1}+\tau ^{n-2}+\tau ^{n-3}\\&=2\tau ^{n-2}+2\tau ^{n-3}+\tau ^{n-4}\\&=3\tau ^{n-2}+\tau ^{n-4}+\tau ^{n-6}\end{aligned}}}
from this an infinite number of further relations can be found. A notable example is τ + τ − 3 = 2 {\displaystyle \tau +\tau ^{-3}=2} . Continued fraction pattern of a few low powers
τ − 1 = [ 0 ; 1 , 1 , 5 , 4 , 2 , 305 , 1 , 8 , 2 , . . . ] ≈ 0.5437 ( 31 57 ) τ 0 = [ 1 ] τ 1 = [ 1 ; 1 , 5 , 4 , 2 , 305 , 1 , 8 , 2 , 1 , . . . ] ≈ 1.8393 ( 103 56 ) τ 2 = [ 3 ; 2 , 1 , 1 , 1 , 1 , 2 , 1 , 152 , 2 , . . . ] ≈ 3.3830 ( 159 47 ) τ 3 = [ 6 ; 4 , 2 , 305 , 1 , 8 , 2 , 1 , 4 , 6 , . . . ] ≈ 6.2223 ( 56 9 ) τ 4 = [ 11 ; 2 , 4 , 152 , 1 , 17 , 1 , 2 , 2 , . . . ] ≈ 11.4445 ( 103 9 ) {\displaystyle {\begin{aligned}\tau ^{-1}&=[0;1,1,5,4,2,305,1,8,2,...]\approx 0.5437\;({\tfrac {31}{57}})\\\tau ^{0}&=[1]\\\tau ^{1}&=[1;1,5,4,2,305,1,8,2,1,...]\approx 1.8393\;({\tfrac {103}{56}})\\\tau ^{2}&=[3;2,1,1,1,1,2,1,152,2,...]\approx 3.3830\;({\tfrac {159}{47}})\\\tau ^{3}&=[6;4,2,305,1,8,2,1,4,6,...]\approx 6.2223\;({\tfrac {56}{9}})\\\tau ^{4}&=[11;2,4,152,1,17,1,2,2,...]\approx 11.4445\;({\tfrac {103}{9}})\end{aligned}}}
The tribonacci ratio is the fourth smallest cubic Pisot number. By definition of these numbers, the absolute value 1 / τ {\displaystyle 1/{\sqrt {\tau }}} of the algebraic conjugates is smaller than 1, thus powers of τ {\displaystyle \tau } generate almost integers. For example: τ 18 = 58034.99919... {\displaystyle \tau ^{18}=58034.99919...} ≈ 58035 − 1 / 1234 {\displaystyle \approx 58035-1/1234} . After 18 rotation steps the phases of the inward spiraling conjugate pair – initially close to ± 9 π / 13 {\displaystyle \pm 9\pi /13} – nearly align with the imaginary axis.
The first implied mention of the tribonacci constant was in the eleventh century, when the Persian poet and polymath Omar Khayyam found the solution 10 τ + 1 τ {\displaystyle 10{\tfrac {\tau +1}{\tau }}} of the cubic x 3 + 200 x = 20 x 2 + 2000 {\displaystyle x^{3}+200x=20x^{2}+2000} by considering the intersection of a circle and a rectangular hyperbola.
W 11 ( x ) = x 3 − 2 x 2 + 2 x − 2 , {\displaystyle W_{11}(x)=x^{3}-2x^{2}+2x-2,} with real zero ω = τ + 1 τ = τ ( τ − 1 ) , {\displaystyle \omega ={\tfrac {\tau +1}{\tau }}=\tau (\tau -1),} is the Weber class polynomial associated with discriminant Δ = − 11 {\displaystyle \Delta =-11} . Properties of the related Klein j-invariant result in near-identity ω ≈ ( e π − Δ + 24 ) 1 / 24 . {\displaystyle \omega \approx (e^{\pi {\sqrt {-\Delta }}}+24)^{1/24}.}
Argument θ = arccos ( 1 2 τ ) {\displaystyle \theta =\arccos({\tfrac {1}{2}}\tau )\,} satisfies 4 sin ( 3 θ ) − tan ( θ ) = 11 {\displaystyle \,4\sin(3\theta )-\tan(\theta )={\sqrt {11}}} , a result which is related through distance parameter z = τ ( 1 − τ ) ⋅ 2 cos ( 2 π 11 ) {\displaystyle \,z=\tau (1-\tau )\cdot 2\cos({\tfrac {2\pi }{11}})} to the 'miraculous' neusis construction of the hendecagon, found by Benjamin and Snyder. The reciprocal 1 τ {\displaystyle {\tfrac {1}{\tau }}} of the tribonacci ratio solves the equation 2 arctan ( x ) = arccos ( x ) {\displaystyle \,2\arctan(x)=\arccos(x)} . The angle is close to 1 radian. Its complement arccos ( τ − 1 ) = arcsin ( 1 τ ) {\displaystyle \,\arccos(\tau -1)=\arcsin({\tfrac {1}{\tau }})\,} figures in the geometric construction of the tribonacci constant found by biologist Xerardo Neira. The tribonacci ratio is particularly important in the study of the snub cube.
Tribonacci sequence The first muddled mention of the tribonacci sequence is in Charles Darwin's On the Origin of Species (1859), illustrating the population growth of elephants on the supposition that during their lifetime each pair of parents produces three pair of young. The number of compositions of n − 2 into parts 1, 2 and 3 is counted by the nth tribonacci number (n > 1). The tribonacci sequence is defined by the third-order recurrence relation
T n = T n − 1 + T n − 2 + T n − 3 for n > 2 , {\displaystyle T_{n}=T_{n-1}+T_{n-2}+T_{n-3}{\text{ for }}n>2,}
with initial values
T 0 = T 1 = 0 , T 2 = 1. {\displaystyle T_{0}=T_{1}=0,T_{2}=1.}
The first few terms are 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927,... (sequence A000073 in the OEIS). The limit ratio between consecutive terms is the tribonacci constant: lim n → ∞ T n + 1 / T n = τ . {\displaystyle \lim _{n\rightarrow \infty }T_{n+1}/T_{n}=\tau .}
The sequence can be extended to negative indices using
T n = T n + 3 − T n + 2 − T n + 1 , {\displaystyle T_{n}=T_{n+3}-T_{n+2}-T_{n+1},}
obtaining (0), 1,−1, 0, 2,−3, 1, 4,−8, 5, 7,−20, 18, 9,−47,... (sequence A057597 in the OEIS). The relationship between the negative and positive indexed segments of the sequence is given by
T 1 − n = T n 2 − T n + 1 T n − 1 . {\displaystyle T_{1-n}=T_{n}^{2}-T_{n+1}T_{n-1}.}
This reflection formula holds for all integers n {\displaystyle n} and can be derived from Agronomof's identity
T n + k = T k + 1 T n + 1 + ( T k + T k − 1 ) T n + T k T n − 1 . {\displaystyle T_{n+k}=T_{k+1}T_{n+1}+\left(T_{k}+T_{k-1}\right)T_{n}+T_{k}T_{n-1}.}
The sequence is related to sums of binomial coefficients by
T n + 2 = ∑ i = 0 ⌊ n / 2 ⌋ ∑ j = 0 i ( i j ) ( n − i − j i ) {\displaystyle T_{n+2}=\sum _{i=0}^{\lfloor n/2\rfloor }\sum _{j=0}^{i}{i \choose j}{n-i-j \choose i}}
Powers of the tribonacci ratio can be written with tribonacci numbers as quadratic coefficients τ n = τ 2 T n + τ ( T n − 1 + T n − 2 ) + T n − 1 , {\displaystyle \tau ^{n}=\tau ^{2}T_{n}+\tau (T_{n-1}+T_{n-2})+T_{n-1},} which is proved by mathematical induction on n . {\displaystyle n.} This relation also holds for n < 0. {\displaystyle n<0.} The order of the coefficients corresponds to the bottom row of matrix Q {\displaystyle Q} below. The generating function of the tribonacci sequence for non-negative n is given by
x 2 1 − x − x 2 − x 3 = ∑ n = 0 ∞ T n x n for x < 1 τ {\displaystyle {\frac {x^{2}}{1-x-x^{2}-x^{3}}}=\sum _{n=0}^{\infty }T_{n}x^{n}{\text{ for }}x<{\tfrac {1}{\tau }}}
Let τ {\displaystyle \tau } and complex conjugate pair β {\displaystyle \beta } and γ {\displaystyle \gamma } be the zeros of polynomial x 3 − x 2 − x − 1 {\displaystyle x^{3}-x^{2}-x-1} with discriminant − 44 {\displaystyle -44} , the tribonacci numbers are then given by the Binet formula
T n + 1 = a τ n + b β n + c γ n , {\displaystyle T_{n+1}=a\tau ^{n}+b\beta ^{n}+c\gamma ^{n},}
with real a {\displaystyle a} and conjugates b {\displaystyle b} and c {\displaystyle c} the roots of 44 y 3 − 2 y − 1 = 0. {\displaystyle 44y^{3}-2y-1=0.}
Since | b β n + c