Search: a263401 -id:a263401
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A002390
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Decimal expansion of natural logarithm of golden ratio.
(Formerly M3318 N1334)
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+0
72
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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Also equals arcsinh(1/2).
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((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
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
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EXAMPLE
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0.481211825059603447497758913424368423135184334385660519661...
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MAPLE
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arcsinh(1/2);
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A162891
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Expansion of 1 / Product_{k>=1} (1-x^k-x^(2*k)).
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10
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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
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0,3
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LINKS
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FORMULA
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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
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MAPLE
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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):
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MATHEMATICA
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nmax = 50; CoefficientList[Series[1/Product[1-x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2016 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A275820
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Expansion of Product_{k>=1} (1 + x^(2*k) + x^(3*k)).
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+0
6
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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
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OFFSET
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0,7
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LINKS
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FORMULA
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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...
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A293138
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E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!).
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6
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1, 1, 3, 12, 72, 480, 3780, 35280, 372960, 4263840, 54432000, 758419200, 11436163200, 185253868800, 3214699488000, 59172265152000, 1163830187520000, 24097823253504000, 525794940582912000, 12073276215576576000, 290883846352619520000, 7318777466097377280000
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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):
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A293204
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G.f.: Product_{m>0} (1+x^m+2!*x^(2*m)).
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+0
6
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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):
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Product[1 + x^k + 2*x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 02 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A275821
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Expansion of Product_{k>=1} (1 + x^(2*k) - x^(3*k)).
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+0
4
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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
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OFFSET
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0,11
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LINKS
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FORMULA
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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...
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A100847
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Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity.
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+0
3
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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EXAMPLE
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a(3) = 7 because we have 6, 42, 411, 33, 222, 21111 and 111111.
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MAPLE
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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):
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A276527
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Expansion of Product_{k>=1} 1 / (1 + x^k - x^(2*k)).
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+0
3
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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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...
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MATHEMATICA
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nmax = 50; CoefficientList[Series[1/Product[1+x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A293182
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Expansion of Product_{k>=1} (1 + 2*x^k - x^(2*k)).
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+0
2
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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...
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MAPLE
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N:= 100:
P:= mul(1+2*x^m- x^(2*m), m=1..N):
S:= series(P, x, N+1):
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Product[1+2*x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A293253
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Expansion of Product_{k>=1} (1 + x^k + x^(k^2)).
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+0
2
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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
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OFFSET
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0,2
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LINKS
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Product[(1+x^k+x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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