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Decimal expansion of the sum of reciprocals of the products of 3 consecutive Fibonacci numbers.
+10
3
7, 1, 0, 8, 5, 5, 3, 5, 1, 4, 2, 9, 3, 2, 8, 4, 1, 6, 8, 8, 7, 6, 9, 4, 4, 9, 0, 3, 8, 4, 2, 7, 0, 8, 3, 3, 0, 4, 5, 1, 1, 8, 0, 4, 8, 4, 1, 0, 3, 0, 8, 6, 3, 9, 9, 7, 4, 9, 7, 3, 5, 1, 4, 9, 3, 6, 9, 6, 4, 2, 3, 8, 2, 6, 1, 1, 3, 5, 4, 4, 8, 4, 1, 7, 5, 8, 8, 4, 1, 6, 8, 1, 7, 1, 4, 8, 5, 8, 5, 7, 6, 8, 5, 4, 9
FORMULA
Equals 1 - A158933 (Melham, 2003). (End)
EXAMPLE
0.71085535142932841688769449038427083304511804841030863997497351493696423826...
MATHEMATICA
RealDigits[ Sum[ N[ 1/Product[ Fibonacci@j, {j, k, k + 2}], 128], {k, 177}], 10, 111][[1]]
PROG
(PARI) suminf(n=1, 1/(fibonacci(n)*fibonacci(n+1)*fibonacci(n+2))) \\ Michel Marcus, Feb 19 2019
Decimal expansion of Sum_{n>=1} 1/F(n)^n, where F= A000045 (Fibonacci numbers).
+10
2
2, 1, 3, 7, 6, 6, 9, 5, 0, 9, 6, 7, 2, 6, 9, 8, 4, 3, 3, 3, 1, 7, 1, 4, 9, 8, 1, 6, 9, 0, 3, 2, 6, 1, 9, 4, 1, 9, 0, 3, 9, 6, 6, 6, 3, 1, 7, 4, 4, 2, 0, 9, 7, 5, 8, 4, 7, 2, 1, 2, 1, 4, 7, 1, 0, 5, 2, 3, 8, 7, 1, 0, 1, 1, 6, 3, 4, 5, 5, 0, 5, 2, 5, 3, 9, 6, 5, 8, 8, 6, 2, 6, 3, 0, 5, 3, 3, 3, 6, 6, 0, 8, 6, 8, 0
EXAMPLE
2.13766950967269843331714981... = 1/1^1 + 1/1^2+ 1/2^3+ 1/3^4 +1/5^5 +1/8^6 +...
MAPLE
with(combinat, fibonacci):Digits:=120:s:=sum( evalf(1/ fibonacci(n)^n), n=1..200):print(s):
MATHEMATICA
digits = 105; NSum[1/Fibonacci[n]^n, {n, 1, Infinity}, NSumTerms -> digits, WorkingPrecision -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 21 2014 *)
PROG
(PARI) suminf(n=1, 1/fibonacci(n)^n); \\ Michel Marcus, Feb 21 2014
Decimal expansion of Sum_{n = 1 .. infinity} (-1)^(n+1)/F(n)^n where F= A000045 is the Fibonacci sequence.
+10
1
1, 1, 2, 9, 7, 0, 5, 2, 2, 2, 0, 0, 5, 9, 7, 7, 4, 2, 2, 3, 8, 0, 4, 0, 6, 7, 7, 9, 0, 4, 2, 8, 7, 9, 4, 3, 4, 0, 8, 6, 1, 9, 1, 4, 5, 0, 2, 3, 1, 6, 4, 4, 8, 6, 2, 1, 1, 2, 1, 0, 5, 0, 7, 6, 7, 7, 7, 0, 1, 9, 5, 3, 8, 3, 2, 7, 3, 0, 7, 9, 8, 9, 2, 9, 2, 6, 3, 4, 6, 4, 8, 2, 2, 8, 9, 4, 3, 8, 9, 6, 9, 3, 7, 8, 8
EXAMPLE
0.1129705222005977422380406779... = 1/1^1 - 1/1^2 + 1/2^3 - 1/3^4 + 1/5^5 - ...
MAPLE
with(combinat, fibonacci):Digits:=120:s:=sum( evalf(((-1)^(n+1))/ fibonacci(n)^n), n=1..200):print(s):
MATHEMATICA
RealDigits[N[Sum[((-1)^(n+1))/Fibonacci[n]^n, {n, 1, 105}], 105]][[1]]
Decimal expansion of Sum_{k>=1} (-1)^(k+1)/L(k) where L(k) is the k-th Lucas number ( A000032).
+10
1
8, 3, 0, 5, 0, 2, 8, 2, 1, 5, 8, 6, 8, 7, 6, 6, 8, 2, 3, 1, 6, 9, 3, 6, 4, 8, 6, 2, 5, 1, 0, 5, 9, 5, 1, 9, 1, 7, 7, 3, 0, 4, 6, 2, 1, 4, 3, 0, 4, 0, 8, 2, 8, 0, 1, 4, 6, 0, 2, 6, 4, 1, 3, 9, 0, 7, 9, 1, 0, 4, 9, 8, 4, 8, 6, 0, 4, 3, 0, 0, 6, 7, 4, 9, 3, 3, 0
COMMENTS
André-Jeannin (1989) proved that this constant is irrational, and Tachiya (2004) proved that it does not belong to the quadratic number field Q(sqrt(5)).
FORMULA
Equals Sum_{k>=1} (-1)^(k+1) * Fibonacci(k)/Fibonacci(2*k).
Equals Sum_{k>=1} (-1)^(k+1)/(phi^k + (1-phi)^k), where phi is the golden ratio ( A001622). Equals Sum_{k>=0} 1/(phi^(2*k+1) + (-1)^k).
EXAMPLE
0.83050282158687668231693648625105951917730462143040...
MATHEMATICA
RealDigits[Sum[(-1)^(n+1)/LucasL[n], {n, 1, 1000}], 10, 120][[1]]
Decimal expansion of Sum_{k>=1} (-1)^(k+1)*k/Fibonacci(2*k).
+10
1
5, 8, 0, 0, 0, 4, 7, 3, 9, 5, 0, 7, 7, 7, 0, 6, 8, 0, 0, 6, 7, 4, 7, 0, 9, 8, 1, 8, 9, 5, 5, 2, 2, 8, 0, 2, 6, 9, 8, 5, 0, 1, 2, 6, 0, 9, 6, 4, 6, 1, 6, 3, 9, 0, 1, 5, 7, 7, 5, 6, 1, 0, 0, 1, 7, 7, 6, 7, 3, 7, 5, 7, 5, 2, 1, 9, 9, 7, 8, 4, 8, 9, 4, 9, 2, 1, 0, 4, 4, 7, 8, 6, 6, 9, 4, 0, 2, 2, 3, 7, 1, 4, 1, 1, 5
FORMULA
Equals Sum_{k>=1} (-1)^(k+1)*k/ A001906(k). Equals (1/sqrt(5)) * Sum_{k>=1} 1/Fibonacci(2*k-1)^2 (Jennings, 1994).
EXAMPLE
0.58000473950777068006747098189552280269850126096461...
MATHEMATICA
RealDigits[Sum[(-1)^(k+1)*k/Fibonacci[2*k], {k, 1, 300}], 10, 100][[1]]
PROG
(PARI) sumalt(k=1, (-1)^(k+1)*k/fibonacci(2*k)) \\ Michel Marcus, Sep 10 2022
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