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Search: a263401 -id:a263401
Displaying 1-10 of 12 results found. page 1 2
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A002390 Decimal expansion of natural logarithm of golden ratio.
(Formerly M3318 N1334) 		+10
72
4, 8, 1, 2, 1, 1, 8, 2, 5, 0, 5, 9, 6, 0, 3, 4, 4, 7, 4, 9, 7, 7, 5, 8, 9, 1, 3, 4, 2, 4, 3, 6, 8, 4, 2, 3, 1, 3, 5, 1, 8, 4, 3, 3, 4, 3, 8, 5, 6, 6, 0, 5, 1, 9, 6, 6, 1, 0, 1, 8, 1, 6, 8, 8, 4, 0, 1, 6, 3, 8, 6, 7, 6, 0, 8, 2, 2, 1, 7, 7, 4, 4, 1, 2, 0, 0, 9, 4, 2, 9, 1, 2, 2, 7, 2, 3, 4, 7, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
The Baxa article proves that every gamma >= this constant is the Lévy constant of a transcendental number. - Michel Marcus, Apr 09 2016
The entropy of the golden mean shift. See Capobianco link. - Michel Marcus, Jan 19 2019
Also the limiting value of the area of the function y = 1/x bounded by the abscissa of consecutive F(n) points (where F(n)=A000045(n) are the Fibonacci numbers and n > 0). - Burak Muslu, May 09 2021
REFERENCES
George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 236.
W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 31-38.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
Alexander Adamchuk's comment, Sep 01 2006 Mathematics in Russian
Christoph Baxa, Lévy constants of transcendental numbers, Proc. Amer. Math. Soc. 137 (2009), 2243-2249.
Christopher Brown, The natural logarithm of the golden section, Fibonacci Quarterly 55:5 (2017), pp. 42-44.
Silvio Capobianco, Introduction to Symbolic Dynamics. Part 4: Entropy; The entropy of the golden mean shift, Institute of Cybernetics at TUT; May 12 2010. Slides 15-17.
Simon Plouffe, Plouffe's Inverter, ln(phi) to 10000 digits
Simon Plouffe, ln(0.5+0.5*SQRT(5)) to 2000 digits
Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions
Index entries for transcendental numbers
FORMULA
Also equals arcsinh(1/2).
Equals sqrt(5)* A086466 /2. - Seiichi Kirikami, Aug 20 2011
Equals sqrt(5)*(5* A086465 -1)/4. - Jean-François Alcover, Apr 29 2013
Also equals (125*C - 55) / (24*sqrt(5)), where C = Sum_{k>=1} (-1)^(k+1)*1/Cat(k), where Cat(k) = (2k)!/k!/(k+1)! = A000108(k) - k-th Catalan number. See Sep 01 2006 comment at ref. Mathematics in Russian. - Alexander Adamchuk, Dec 27 2013
Equals sqrt(5)/4 * Sum_{n>=0} (-1)^n/((2n+1)*C(2*n,n)) = sqrt(5) *A344041 /4. - Alexander Adamchuk, Dec 27 2013
Equals sqrt((Pi^2/6 - W)/3), where W = Sum_{n>=0} (-1)^n/((2n+1)^2*C(2*n,n)) = A145436, attributed by Alexander Adamchuk to Ramanujan. See Sep 01 2006 comment at ref. Mathematics in Russian. - Alexander Adamchuk, Dec 27 2013
Equals lim_{j->infinity} Sum_{k=F(j)..F(j+1)-1} (1/k), where F = A000045, the Fibonacci sequence. Convergence is slow. For example: Sum_{k=21..33} (1/k) = 0.4910585.... - Richard R. Forberg, Aug 15 2014
Equals Sum_{k>=1} cos(Pi*k/5)/k. - Amiram Eldar, Aug 12 2020
Equals real solution to exp(x)+exp(2*x) = exp(3*x). - Alois P. Heinz, Jul 14 2022
Equals arccoth(sqrt(5)). - Amiram Eldar, Feb 09 2024
Sum_{n >= 1} 1/(n*P(n, sqrt(5))*P(n-1, sqrt(5))), where P(n, x) denotes the n-th Legendre polynomial. The first ten terms of the series gives the approximation log((1 + sqrt(5))/2) = 0.481211825059(39..), correct to 12 decimal places. - Peter Bala, Mar 16 2024
EXAMPLE
0.481211825059603447497758913424368423135184334385660519661...
MAPLE
arcsinh(1/2);
MATHEMATICA
RealDigits[N[Log[GoldenRatio], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
PROG
(PARI) asinh(1/2) \\ Charles R Greathouse IV, Jan 04 2016
CROSSREFS
Cf. A000108, A001622, A013661, A086463, A086466, A263401.
KEYWORD
nonn,cons
AUTHOR
N. J. A. Sloane
STATUS
approved
A162891 Expansion of 1 / Product_{k>=1} (1-x^k-x^(2*k)). +10
10
1, 1, 3, 5, 11, 18, 36, 59, 109, 181, 318, 525, 902, 1481, 2492, 4087, 6788, 11090, 18274, 29776, 48772, 79332, 129411, 210172, 341958, 554728, 900872, 1460298, 2368555, 3837147, 6218652, 10070389, 16311432, 26407350, 42757335, 69208746, 112032256, 181316714 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
FORMULA
a(n) ~ p / (sqrt(5) * r^(n+1)), where r = (sqrt(5)-1)/2 and p = Product_{n>1} 1/(1 - r^n - r^(2*n)) = 4.64451592505133910330213147... . - Vaclav Kotesovec, Nov 16 2016
MAPLE
F:= n-> combinat[fibonacci](n+1):
b:= proc(n, i) option remember; `if`(n=0 or i=1, F(n),
add((t-> b(t, min(t, i-1)))(n-i*j)*F(j), j=0..n/i))
end:
a:= n-> b(n$2):
seq(a(n), n=0..39); # Alois P. Heinz, Aug 24 2019
MATHEMATICA
nmax = 50; CoefficientList[Series[1/Product[1-x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2016 *)
PROG
(PARI) al(n)=Vec(1/prod(k=1, n, 1-x^k-x^(2*k)+x*O(x^n)))
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/(&*[(1-x^k-x^(2*k)): k in [1..100]]))); // G. C. Greubel, Oct 24 2018
CROSSREFS
Cf. A000045, A000041, A001156, A003105, A263401, A276527.
KEYWORD
nonn,easy
AUTHOR
Franklin T. Adams-Watters, Jul 16 2009
STATUS
approved
A275820 Expansion of Product_{k>=1} (1 + x^(2*k) + x^(3*k)). +10
6
1, 0, 1, 1, 1, 0, 3, 1, 3, 3, 3, 2, 7, 3, 8, 7, 10, 7, 16, 8, 17, 17, 21, 17, 35, 22, 37, 36, 46, 37, 69, 46, 74, 71, 91, 81, 128, 96, 144, 139, 173, 154, 236, 185, 263, 257, 314, 286, 417, 345, 470, 462, 557, 517, 719, 617, 815, 802, 960, 904, 1211, 1068 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
FORMULA
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-2*x) + exp(-3*x)) dx = 0.60248650631158778882474716370201988195290074160793967143564...
MATHEMATICA
nmax = 100; CoefficientList[Series[Product[1+x^(2*k)+x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[3]] = 1; p[[4]] = 1; Do[Do[p[[j+1]] = p[[j+1]] + If[j < 2*k, 0, p[[j - 2*k + 1]]] + If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}]; , {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)
CROSSREFS
Cf. A000726, A263401, A264905, A266686, A275821.
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Nov 15 2016
STATUS
approved
A293138 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!). +10
6
1, 1, 3, 12, 72, 480, 3780, 35280, 372960, 4263840, 54432000, 758419200, 11436163200, 185253868800, 3214699488000, 59172265152000, 1163830187520000, 24097823253504000, 525794940582912000, 12073276215576576000, 290883846352619520000, 7318777466097377280000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..444
FORMULA
a(n) ~ c^(1/4) * exp(2*sqrt(c*n) - n) * n^(n+1/2) / (sqrt(5) * n^(3/4)), where c = -polylog(2, -1/2 - I/2) - polylog(2, -1/2 + I/2) = 0.9669456127221570300837545... Equivalently, c = -Sum_{k>=1} (-1)^k * cos(Pi*k/4) / (k^2 * 2^(k/2-1)). - Vaclav Kotesovec, Oct 01 2017
EXAMPLE
Let's consider the partitions of n where no positive integer appears more than twice. (See A000726)
For n = 5,
partition | |
--------------------------------------------------------------------
5 -> one 5 -> 1/(1!) (= 1 )
= 4 + 1 -> one 4 and one 1 -> 1/(1!*1!) (= 1 )
= 3 + 2 -> one 3 and one 2 -> 1/(1!*1!) (= 1 )
= 3 + 1 + 1 -> one 3 and two 1 -> 1/(1!*2!) (= 1/2)
= 2 + 2 + 1 -> two 2 and one 1 -> 1/(2!*1!) (= 1/2)
--------------------------------------------------------------------
sum 4
So a(5) = 5! * 4 = 480.
For n = 6,
partition | |
--------------------------------------------------------------------
6 -> one 6 -> 1/(1!) (= 1 )
= 5 + 1 -> one 5 and one 1 -> 1/(1!*1!) (= 1 )
= 4 + 2 -> one 4 and one 2 -> 1/(1!*1!) (= 1 )
= 4 + 1 + 1 -> one 4 and two 1 -> 1/(1!*2!) (= 1/2)
= 3 + 3 -> two 3 -> 1/(2!) (= 1/2)
= 3 + 2 + 1 -> one 3, one 2 and one 1 -> 1/(1!*1!*1!) (= 1 )
= 2 + 2 + 1 + 1 -> two 2 and two 1 -> 1/(2!*2!) (= 1/4)
--------------------------------------------------------------------
sum 21/4
So a(6) = 6! * 21/4 = 3780.
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)/j!, j=0..min(2, n/i))))
end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23); # Alois P. Heinz, Oct 02 2017
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1]/j!, {j, 0, Min[2, n/i]}]]];
a[n_] := n! b[n, n];
a /@ Range[0, 23] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)
CROSSREFS
Column k=2 of A293135.
Cf. A000726, A162891, A263401, A293141.
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 01 2017
STATUS
approved
A293204 G.f.: Product_{m>0} (1+x^m+2!*x^(2*m)). +10
6
1, 1, 3, 2, 6, 7, 12, 13, 22, 26, 42, 46, 73, 80, 116, 139, 194, 226, 306, 358, 482, 558, 735, 856, 1108, 1300, 1657, 1926, 2426, 2834, 3530, 4110, 5082, 5898, 7234, 8409, 10216, 11860, 14304, 16568, 19891, 22990, 27470, 31670, 37630, 43382, 51274, 58982, 69450 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 501 terms from Seiichi Manyama)
FORMULA
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4 * sqrt(Pi) * n^(3/4)), where c = Pi^2/3 - arctan(sqrt(7))^2 + log(2)^2/4 + polylog(2, -1/4 - I*sqrt(7)/4) + polylog(2, -1/4 + I*sqrt(7)/4) = 1.323865936864425754643630663383779192757247984691212163137... - Vaclav Kotesovec, Oct 02 2017
Equivalently, c = -polylog(2, -1/2 + I*sqrt(7)/2) - polylog(2, -1/2 - I*sqrt(7)/2). - Vaclav Kotesovec, Oct 05 2017
EXAMPLE
Let's consider the partitions of n where no positive integer appears more than twice. (See A000726)
For n = 5,
partition | |
--------------------------------------------------------------
5 -> one 5 -> 1! (= 1)
= 4 + 1 -> one 4 and one 1 -> 1!*1! (= 1)
= 3 + 2 -> one 3 and one 2 -> 1!*1! (= 1)
= 3 + 1 + 1 -> one 3 and two 1 -> 1!*2! (= 2)
= 2 + 2 + 1 -> two 2 and one 1 -> 2!*1! (= 2)
--------------------------------------------------------------
a(5) = 7.
For n = 6,
partition | |
--------------------------------------------------------------
6 -> one 6 -> 1! (= 1)
= 5 + 1 -> one 5 and one 1 -> 1!*1! (= 1)
= 4 + 2 -> one 4 and one 2 -> 1!*1! (= 1)
= 4 + 1 + 1 -> one 4 and two 1 -> 1!*2! (= 2)
= 3 + 3 -> two 3 -> 2! (= 2)
= 3 + 2 + 1 -> one 3, one 2 and one 1 -> 1!*1!*1! (= 1)
= 2 + 2 + 1 + 1 -> two 2 and two 1 -> 2!*2! (= 4)
--------------------------------------------------------------
a(6) = 12.
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*j!, j=0..min(2, n/i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Alois P. Heinz, Oct 02 2017
MATHEMATICA
nmax = 100; CoefficientList[Series[Product[1 + x^k + 2*x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 02 2017 *)
CROSSREFS
Column k=2 of A293202.
Cf. A000726, A263401, A293138, A293182.
Cf. A293072.
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 02 2017
STATUS
approved
A275821 Expansion of Product_{k>=1} (1 + x^(2*k) - x^(3*k)). +10
4
1, 0, 1, -1, 1, 0, 1, -1, 1, -1, 3, -2, 3, -3, 2, -1, 4, -3, 4, -4, 7, -7, 7, -7, 9, -6, 11, -10, 10, -11, 15, -14, 18, -19, 21, -17, 24, -22, 26, -29, 35, -34, 42, -43, 43, -39, 52, -52, 59, -59, 74, -72, 79, -87, 93, -87, 107, -108, 118, -126, 149, -146 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,11
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)
FORMULA
a(n) ~ (-1)^n * c^(1/4) * exp(sqrt(c*n)) / (2^(3/2)*sqrt(Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + 2*exp(-x) + exp(-2*x) - exp(-3*x)) dx = 1.522848148277623680909526566...
MATHEMATICA
nmax=100; CoefficientList[Series[Product[1+x^(2*k)-x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]
RootReduce[QPochhammer[Root[-1 + # + #^3 &, 1], x] QPochhammer[Root[-1 + # + #^3 &, 2], x] QPochhammer[Root[-1 + # + #^3 &, 3], x] + O[x]^70][[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[3]] = 1; p[[4]] = -1; Do[Do[p[[j + 1]] = p[[j + 1]] + If[j < 2 k, 0, p[[j - 2 k + 1]]] - If[j < 3 k, 0, p[[j - 3 k + 1]]], {j, nmax, k, -1}]; , {k, 2, nmax}]; p (* Vaclav Kotesovec, May 06 2018 *)
CROSSREFS
Cf. A000726, A263401, A264905, A266686, A275820.
KEYWORD
sign
AUTHOR
Vaclav Kotesovec, Nov 15 2016
STATUS
approved
A100847 Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity. +10
3
1, 2, 3, 7, 10, 17, 28, 42, 62, 93, 137, 193, 276, 383, 532, 734, 997, 1342, 1807, 2400, 3177, 4190, 5478, 7130, 9245, 11923, 15305, 19591, 24957, 31673, 40075, 50518, 63460, 79523, 99296, 123664, 153616, 190271, 235072, 289776, 356302, 437107, 535112, 653626 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: Product_{i>0} (1+x^i-x^(2*i))/(1-x^i).
a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((2*Pi^2/3 + 8*log(phi)^2)*n)) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016
EXAMPLE
a(3) = 7 because we have 6, 42, 411, 33, 222, 21111 and 111111.
MAPLE
g:=product((1+x^i-x^(2*i))/(1-x^i), i=1..50): gser:=series(g, x=0, 40): seq(coeff(gser, x, n), n=0..35); # Emeric Deutsch, Aug 25 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(`if`(irem(i+j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)
+b(n, i-1)))
end:
a:= n-> b(2*n$2):
seq(a(n), n=0..60); # Alois P. Heinz, May 31 2014
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1+x^k-x^(2*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)
CROSSREFS
Cf. A055922, A117958, A130126, A131942, A102247, A263401.
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Aug 16 2007
EXTENSIONS
More terms from Emeric Deutsch, Aug 25 2007
STATUS
approved
A276527 Expansion of Product_{k>=1} 1 / (1 + x^k - x^(2*k)). +10
3
1, -1, 1, -3, 5, -8, 12, -21, 37, -59, 92, -153, 256, -409, 654, -1073, 1754, -2824, 4552, -7394, 12010, -19406, 31337, -50782, 82306, -133072, 215152, -348346, 563939, -912217, 1475604, -2388075, 3864808, -6252750, 10115987, -16369340, 26488326, -42857128 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
FORMULA
a(n) ~ -p / (sqrt(5) * r^(n+1)), where r = -(sqrt(5)-1)/2 and p = Product_{n>1} 1/(1 + r^n - r^(2*n)) = 1.0964214808924344474065093...
MATHEMATICA
nmax = 50; CoefficientList[Series[1/Product[1+x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
Cf. A003105, A162891.
Cf. A000726, A109389, A263401.
KEYWORD
sign
AUTHOR
Vaclav Kotesovec, Nov 16 2016
STATUS
approved
A293182 Expansion of Product_{k>=1} (1 + 2*x^k - x^(2*k)). +10
2
1, 2, 1, 6, 3, 6, 16, 12, 16, 22, 51, 36, 60, 62, 91, 154, 148, 176, 236, 278, 328, 552, 508, 670, 771, 988, 1068, 1438, 1844, 1998, 2401, 2882, 3300, 4030, 4640, 5406, 7212, 7584, 9072, 10480, 12612, 13964, 17024, 18860, 22545, 27298, 30340, 34372, 41068 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Robert Israel, Table of n, a(n) for n = 0..10000
FORMULA
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2^(3/2) * sqrt(Pi) * n^(3/4)), where c = Pi^2/6 + log(1+sqrt(2))^2/2 + polylog(2, 3-2*sqrt(2))/2 - 2*polylog(2, sqrt(2)-1) = 1.18805291660775259061867850175092520191179528961165451864292...
MAPLE
N:= 100:
P:= mul(1+2*x^m- x^(2*m), m=1..N):
S:= series(P, x, N+1):
seq(coeff(S, x, n), n=0..N); # Robert Israel, Oct 01 2017
MATHEMATICA
nmax = 100; CoefficientList[Series[Product[1+2*x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
Cf. A162891, A263401, A276527, A293138.
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 01 2017
STATUS
approved
A293253 Expansion of Product_{k>=1} (1 + x^k + x^(k^2)). +10
2
1, 2, 1, 3, 4, 6, 6, 8, 12, 15, 20, 22, 30, 35, 46, 53, 67, 80, 97, 117, 138, 165, 195, 231, 272, 323, 378, 442, 514, 600, 696, 806, 931, 1078, 1240, 1431, 1638, 1881, 2147, 2461, 2802, 3197, 3632, 4131, 4685, 5310, 6009, 6790, 7670, 8652, 9749, 10968, 12336 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
MATHEMATICA
nmax = 100; CoefficientList[Series[Product[(1+x^k+x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
Cf. A033461, A263401, A293204, A293254.
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 04 2017
STATUS
approved
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Last modified August 29 18:55 EDT 2024. Contains 375518 sequences. (Running on oeis4.)