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Search: a129660 -id:a129660
Displaying 1-10 of 15 results found. page 1 2
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A129404 Decimal expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3. +10
24
8, 8, 4, 0, 2, 3, 8, 1, 1, 7, 5, 0, 0, 7, 9, 8, 5, 6, 7, 4, 3, 0, 5, 7, 9, 1, 6, 8, 7, 1, 0, 1, 1, 8, 0, 7, 7, 4, 7, 9, 4, 6, 1, 8, 6, 1, 1, 7, 6, 5, 8, 9, 3, 4, 7, 8, 2, 5, 8, 7, 4, 1, 4, 7, 4, 9, 1, 1, 5, 6, 6, 7, 0, 3, 3, 3, 2, 3, 1, 8, 7, 0, 1, 6, 3, 5, 9, 6, 3, 6, 4, 6, 8, 9, 5, 5, 3, 6, 0, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Contributed to OEIS on April 15, 2007 -- the 300th anniversary of the birth of Leonhard Euler.
REFERENCES
Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292.
LINKS
Table of n, a(n) for n=0..99.
FORMULA
chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
Series: L(3, chi3) = Sum_{k>=1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
Equals 1 + Sum_{k>=1} ( 1/(3*k+1)^3 - 1/(3*k-1)^3 ). - Sean A. Irvine, Aug 17 2021
Equals Product_{p prime} (1 - Kronecker(-3, p)/p^3)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p^3)^(-1). - Amiram Eldar, Nov 06 2023
EXAMPLE
L(3, chi3) = 0.8840238117500798567430579168710118077...
MATHEMATICA
nmax = 1000; First[ RealDigits[4 Pi^3/(81 Sqrt[3]) - (1/2) * 10^(-nmax), 10, nmax] ]
CROSSREFS
Cf. A129405, A129406, A129407, A129408, A129409, A129410, A129411.
Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665.
Cf. A344778, A344727, A344688.
KEYWORD
nonn,cons,easy
AUTHOR
Stuart Clary, Apr 15 2007
EXTENSIONS
Offset corrected by R. J. Mathar, Feb 05 2009
STATUS
approved
A129405 Expansion of L(3, chi3) in base 2, where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3. +10
15
1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Contributed to OEIS on Apr 15 2007 -- the 300th anniversary of the birth of Leonhard Euler.
REFERENCES
Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
LINKS
Table of n, a(n) for n=0..103.
FORMULA
chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
Series: L(3, chi3) = sum_{k >= 1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
EXAMPLE
L(3, chi3) = A129404 = (0.111000100100111101100010011100000101101...)_2
MATHEMATICA
nmax = 1000; First[ RealDigits[4 Pi^3/(81 Sqrt[3]) - (1/2) * 2^(-nmax), 2, nmax] ]
CROSSREFS
Cf. A129404, A129406, A129407, A129408, A129409, A129410, A129411.
Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665
KEYWORD
nonn,base,cons,easy
AUTHOR
Stuart Clary, Apr 15 2007
EXTENSIONS
Offset corrected by R. J. Mathar, Feb 05 2009
STATUS
approved
A129406 Expansion of L(3, chi3) in base 3, where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3. +10
15
2, 1, 2, 2, 1, 2, 1, 1, 0, 0, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2, 1, 0, 2, 2, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 1, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, 2, 1, 1, 2, 2, 2, 0, 0, 2, 2, 1, 0, 2, 0, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 0, 2, 2, 0, 2, 1, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Contributed to OEIS on Apr 15 2007 -- the 300th anniversary of the birth of Leonhard Euler.
REFERENCES
Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
LINKS
Table of n, a(n) for n=0..99.
FORMULA
chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
Series: L(3, chi3) = sum_{k >= 1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
EXAMPLE
L(3, chi3) = 0.8840238117500798567430579168710118077... = (0.2122121100201111101022022020002102211...)_3
MATHEMATICA
nmax = 1000; First[ RealDigits[4 Pi^3/(81 Sqrt[3]) - (1/2) * 3^(-nmax), 3, nmax] ]
CROSSREFS
Cf. A129404, A129405, A129407, A129408, A129409, A129410, A129411.
Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665
KEYWORD
nonn,base,cons,easy
AUTHOR
Stuart Clary, Apr 15 2007
EXTENSIONS
Offset corrected by R. J. Mathar, Feb 05 2009
STATUS
approved
A129407 Balanced ternary expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3. +10
15
1, 0, -1, 0, 0, -1, -1, 1, 1, 0, 1, -1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, -1, 1, 0, -1, 1, -1, 0, 0, 1, -1, 1, 1, 0, -1, 1, 1, 0, 0, 1, 1, 1, 1, -1, 1, 1, 1, 0, 1, 0, -1, 0, 0, 0, 1, -1, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 1, 1, 0, -1, -1, 0, 0, -1, 0, 1, 0, -1, 1, 1, -1, 1, -1, 0, 0, 0, -1, -1, 1, 1, 1, 1, 1, 0, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Contributed to OEIS on Apr 15 2007 --- the 300th anniversary of the birth of Leonhard Euler.
REFERENCES
Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292 (for this constant); Articles 330 and 331 (for balanced ternary)
LINKS
Table of n, a(n) for n=0..105.
FORMULA
chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
Series: L(3, chi3) = sum_{k >= 1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
EXAMPLE
L(3, chi3) = 0.8840238117500798567430579168710118077... = 1 + 0/3 - 1/3^2 + 0/3^3 + 0/3^4 - 1/3^5 - 1/3^6 + 1/3^7 + 1/3^8 + ...
MATHEMATICA
nmax = 1000; prec = nmax/2 + 20 (* Normally this is sufficient precision. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; First@Transpose@NestList[{Round[3(#[[2]] - #[[1]])], 3(#[[2]] - #[[1]])}&, {Round[c], c}, nmax]
CROSSREFS
Cf. A129404, A129405, A129406, A129408, A129409, A129410, A129411.
Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665
KEYWORD
sign,easy
AUTHOR
Stuart Clary, Apr 15 2007
STATUS
approved
A129408 Continued fraction for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3. +10
15
0, 1, 7, 1, 1, 1, 1, 1, 5, 1, 1, 9, 4, 13, 4, 1, 2, 27, 1, 28, 1, 2, 2, 3, 2, 7, 1, 1, 19, 1, 8, 3, 3, 2, 1, 10, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 35, 1, 2, 91, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 16, 1, 2, 2, 1, 2, 6, 1, 1, 6, 14, 1, 5, 5, 14, 2, 8, 1, 1, 1, 1, 2, 4, 2, 10, 37, 1, 10, 2, 4, 5, 4, 5, 24, 1, 2, 7, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Contributed to OEIS on April 15, 2007 -- the 300th anniversary of the birth of Leonhard Euler.
REFERENCES
Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
LINKS
Table of n, a(n) for n=0..97.
FORMULA
chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
Series: L(3, chi3) = Sum_{k>=1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
EXAMPLE
L(3, chi3) = 0.8840238117500798567430579168710118077... = [0; 1, 7, 1, 1, 1, 1, 1, 5, 1, 1, 9, 4, 13, 4, ...].
MATHEMATICA
nmax = 1000; ContinuedFraction[4 Pi^3/(81 Sqrt[3]), nmax + 1]
CROSSREFS
Cf. A129404, A129405, A129406, A129407, A129409, A129410, A129411.
Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665.
KEYWORD
nonn,cofr,easy
AUTHOR
Stuart Clary, Apr 15 2007
STATUS
approved
A129409 Engel expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3. +10
15
2, 2, 2, 14, 94, 372, 1391, 7690, 17729, 49204, 87816, 128433, 151275, 290477, 297212, 299837, 352249, 897751, 1081032, 1646358, 2402614, 36591866, 49132456, 93538655, 141789387, 180474393, 687775235, 851204316, 1868593596, 7042652755 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Contributed to OEIS on Apr 15 2007 --- the 300th anniversary of the birth of Leonhard Euler.
REFERENCES
Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
LINKS
Table of n, a(n) for n=1..30.
FORMULA
chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
Series: L(3, chi3) = sum_{k >=1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
EXAMPLE
L(3, chi3) = 0.8840238117500798567430579168710118077... = 1/2 + 1/(2*2) + 1/(2*2*2) + 1/(2*2*2*14) + 1/(2*2*2*14*94) + ...
MATHEMATICA
nmax = 100; prec = 2000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; First@Transpose@NestList[{Ceiling[1/(#[[1]] #[[2]] - 1)], #[[1]] #[[2]] - 1}&, {Ceiling[1/c], c}, nmax - 1]
CROSSREFS
Cf. A129404, A129405, A129406, A129407, A129408, A129410, A129411.
Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665.
KEYWORD
nonn,easy
AUTHOR
Stuart Clary, Apr 15 2007
STATUS
approved
A129410 Pierce expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3. +10
15
1, 8, 13, 16, 64, 6951, 206515, 344040, 11364380, 14595803, 136951831, 417525297, 691111129, 982473113, 15154864245, 17661539909, 31435459113, 49634203300, 1454188399688, 2112564552862, 2266989878695, 5056833185437, 8740145960744 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Contributed to OEIS on April 15, 2007 -- the 300th anniversary of the birth of Leonhard Euler.
REFERENCES
Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
LINKS
Table of n, a(n) for n=1..23.
FORMULA
chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
Series: L(3, chi3) = Sum_{k>=1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
EXAMPLE
L(3, chi3) = 0.8840238117500798567430579168710118077... = 1/1 - 1/(1*8) + 1/(1*8*13) - 1/(1*8*13*16) + 1/(1*8*13*16*64) - ...
MATHEMATICA
nmax = 100; prec = 3000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; First@Transpose@NestList[{Floor[ 1/(1 - #[[1]] #[[2]]) ], 1 - #[[1]] #[[2]]}&, {Floor[1/c], c}, nmax - 1]
CROSSREFS
Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129411.
Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665.
KEYWORD
nonn,easy
AUTHOR
Stuart Clary, Apr 15 2007
STATUS
approved
A129411 Greedy Egyptian expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3. +10
15
2, 3, 20, 1449, 2879423, 31625640285294, 1162849840832612010600369938, 4013794377413687199924671384130798842309412001723286013, 32025095658857878502181254937184611855940944199483548417530154807379258429933254996925647878294253643673560013 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Contributed to OEIS on Apr 15 2007 --- the 300th anniversary of the birth of Leonhard Euler.
REFERENCES
Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
LINKS
Table of n, a(n) for n=1..9.
FORMULA
chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
Series: L(3, chi3) = sum_{k >=1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
EXAMPLE
L(3, chi3) = 0.8840238117500798567430579168710118077... = 1/2 + 1/3 + 1/20 + 1/1449 + 1/2879423 + ...
MATHEMATICA
nmax = 12; prec = 2000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; First@Transpose@NestList[{Ceiling[1/(#[[2]] - 1/#[[1]])], #[[2]] - 1/#[[1]]}&, {Ceiling[1/c], c}, nmax - 1]
CROSSREFS
Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410.
Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665.
KEYWORD
nonn,easy
AUTHOR
Stuart Clary, Apr 15 2007
STATUS
approved
A129658 Numerators of the convergents of the continued fraction for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3. +10
15
0, 1, 0, 1, 7, 8, 15, 23, 38, 61, 343, 404, 747, 7127, 29255, 387442, 1579023, 1966465, 5511953, 150789196, 156301149, 4527221368, 4683522517, 13894266402, 32472055321, 111310432365, 255092920051, 1896960872722, 2152053792773 (list; graph; refs; listen; history; text; internal format)
OFFSET
-2,5
REFERENCES
Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
LINKS
Table of n, a(n) for n=-2..26.
FORMULA
chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
Series: L(3, chi3) = Sum_{k>=1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
EXAMPLE
L(3, chi3) = 0.8840238117500798567430579168710118077... = [0; 1, 7, 1, 1, 1, 1, 1, 5, 1, 1, 9, 4, 13, 4, ...], the convergents of which are 0/1, 1/0, [0/1], 1/1, 7/8, 8/9, 15/17, 23/26, 38/43, 61/69, 343/388, 404/457, 747/845, 7127/8062, 29255/33093, 387442/438271, 1579023/1786177, ..., with brackets marking index 0. Those prior to index 0 are for initializing the recurrence.
MATHEMATICA
nmax = 100; cfrac = ContinuedFraction[4 Pi^3/(81 Sqrt[3]), nmax + 1]; Join[ {0, 1}, Numerator[ Table[ FromContinuedFraction[ Take[cfrac, j] ], {j, 1, nmax + 1} ] ] ]
CROSSREFS
Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.
Cf. A129659, A129660, A129661, A129662, A129663, A129664, A129665.
KEYWORD
nonn,frac,easy
AUTHOR
Stuart Clary, Apr 30 2007
STATUS
approved
A129659 Denominators of the convergents of the continued fraction for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3. +10
15
1, 0, 1, 1, 8, 9, 17, 26, 43, 69, 388, 457, 845, 8062, 33093, 438271, 1786177, 2224448, 6235073, 170571419, 176806492, 5121153195, 5297959687, 15717072569, 36732104825, 125913387044, 288558878913, 2145825539435, 2434384418348 (list; graph; refs; listen; history; text; internal format)
OFFSET
-2,5
REFERENCES
Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
LINKS
Table of n, a(n) for n=-2..26.
FORMULA
chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
Series: L(3, chi3) = Sum_{k=1..infinity} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
EXAMPLE
L(3, chi3) = 0.8840238117500798567430579168710118077... = [0; 1, 7, 1, 1, 1, 1, 1, 5, 1, 1, 9, 4, 13, 4, ...], the convergents of which are 0/1, 1/0, [0/1], 1/1, 7/8, 8/9, 15/17, 23/26, 38/43, 61/69, 343/388, 404/457, 747/845, 7127/8062, 29255/33093, 387442/438271, 1579023/1786177, ..., with brackets marking index 0. Those prior to index 0 are for initializing the recurrence.
MATHEMATICA
nmax = 100; cfrac = ContinuedFraction[4 Pi^3/(81 Sqrt[3]), nmax + 1]; Join[ {1, 0}, Denominator[ Table[ FromContinuedFraction[ Take[cfrac, j] ], {j, 1, nmax + 1} ] ] ]
CROSSREFS
Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.
Cf. A129658, A129660, A129661, A129662, A129663, A129664, A129665.
KEYWORD
nonn,frac,easy
AUTHOR
Stuart Clary, Apr 30 2007
STATUS
approved
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Last modified August 18 23:05 EDT 2024. Contains 375284 sequences. (Running on oeis4.)