A263401 -id:A263401 - cp's OEIS frontend

A002390 Decimal expansion of natural logarithm of golden ratio.

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

Offset: 0

N. J. A. Sloane

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

			0.481211825059603447497758913424368423135184334385660519661...

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).

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

Cf. A000108, A001622, A013661, A086463, A086466, A263401.

Maple
```
arcsinh(1/2); evalf(%, 120);
```

RealDigits[N[Log[GoldenRatio],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

asinh(1/2) \\ Charles R Greathouse IV, Jan 04 2016

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
Equals Sum_{n>=0} ((-1)^(n)*binomial(2*n, n))/(2^(4*n + 1)*(2*n + 1)). - Antonio Graciá Llorente, Nov 13 2024

A162891 Expansion of 1 / Product_{k>=1} (1-x^k-x^(2*k)).

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Jul 16 2009

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000045, A000041, A001156, A003105, A263401, A276527.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/(&*[(1-x^k-x^(2*k)): k in [1..100]]))); // G. C. Greubel, Oct 24 2018

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

nmax = 50; CoefficientList[Series[1/Product[1-x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2016 *)

al(n)=Vec(1/prod(k=1,n,1-x^k-x^(2*k)+x*O(x^n)))

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

A275820 Expansion of Product_{k>=1} (1 + x^(2k) + x^(3k)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Nov 15 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000726, A263401, A264905, A266686, A275821.

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 *)

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...

A293138 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!).

Original entry on oeis.org

1, 1, 3, 12, 72, 480, 3780, 35280, 372960, 4263840, 54432000, 758419200, 11436163200, 185253868800, 3214699488000, 59172265152000, 1163830187520000, 24097823253504000, 525794940582912000, 12073276215576576000, 290883846352619520000, 7318777466097377280000

Offset: 0

Seiichi Manyama, Oct 01 2017

nonn

			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.

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=2 of A293135.

Cf. A000726, A162891, A263401, A293141.

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

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 *)

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

A293204 G.f.: Product_{m>0} (1+x^m+2!x^(2m)).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Oct 02 2017

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 501 terms from Seiichi Manyama)

Column k=2 of A293202.
Cf. A000726, A263401, A293138, A293182.
Cf. A293072.

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

nmax = 100; CoefficientList[Series[Product[1 + x^k + 2*x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 02 2017 *)

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

A275821 Expansion of Product_{k>=1} (1 + x^(2k) - x^(3k)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Nov 15 2016

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

Cf. A000726, A263401, A264905, A266686, A275820.

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 *)

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...

A100847 Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity.

Original entry on oeis.org

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

Offset: 0

Vladeta Jovovic, Aug 16 2007

			a(3) = 7 because we have 6, 42, 411, 33, 222, 21111 and 111111.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A055922, A117958, A130126, A131942, A102247, A263401.

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

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 *)

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

More terms from Emeric Deutsch, Aug 25 2007

A276527 Expansion of Product_{k>=1} 1 / (1 + x^k - x^(2*k)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Nov 16 2016

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A003105, A162891.

Cf. A000726, A109389, A263401.

nmax = 50; CoefficientList[Series[1/Product[1+x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]

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...

A293182 Expansion of Product_{k>=1} (1 + 2x^k - x^(2k)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Oct 01 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A162891, A263401, A276527, A293138.

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

nmax = 100; CoefficientList[Series[Product[1+2*x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]

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...

A293253 Expansion of Product_{k>=1} (1 + x^k + x^(k^2)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Oct 04 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A033461, A263401, A293204, A293254.

nmax = 100; CoefficientList[Series[Product[(1+x^k+x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

A002390 Decimal expansion of natural logarithm of golden ratio.

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A162891 Expansion of 1 / Product_{k>=1} (1-x^k-x^(2*k)).

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A275820 Expansion of Product_{k>=1} (1 + x^(2*k) + x^(3*k)).

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A293138 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!).

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A293204 G.f.: Product_{m>0} (1+x^m+2!*x^(2*m)).

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A275821 Expansion of Product_{k>=1} (1 + x^(2*k) - x^(3*k)).

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A100847 Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity.

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A276527 Expansion of Product_{k>=1} 1 / (1 + x^k - x^(2*k)).

A275820 Expansion of Product_{k>=1} (1 + x^(2k) + x^(3k)).

A293204 G.f.: Product_{m>0} (1+x^m+2!x^(2m)).

A275821 Expansion of Product_{k>=1} (1 + x^(2k) - x^(3k)).

A293182 Expansion of Product_{k>=1} (1 + 2x^k - x^(2k)).