cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A192918 Decimal expansion of the real root of r^3 + r^2 + r - 1.

Original entry on oeis.org

5, 4, 3, 6, 8, 9, 0, 1, 2, 6, 9, 2, 0, 7, 6, 3, 6, 1, 5, 7, 0, 8, 5, 5, 9, 7, 1, 8, 0, 1, 7, 4, 7, 9, 8, 6, 5, 2, 5, 2, 0, 3, 2, 9, 7, 6, 5, 0, 9, 8, 3, 9, 3, 5, 2, 4, 0, 8, 0, 4, 0, 3, 7, 8, 3, 1, 1, 6, 8, 6, 7, 3, 9, 2, 7, 9, 7, 3, 8, 6, 6, 4, 8, 5, 1, 5, 7, 9, 1, 4, 5, 7, 6, 0, 5, 9, 1, 2, 5, 4, 6, 2
Offset: 0

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Author

Frank M Jackson, Aug 26 2011

Keywords

Comments

The real solution r of the cubic equation r^3 + r^2 + r - 1 = 0 is the reciprocal of the tribonacci constant A058265. If the four sides of a quadrilateral form a geometric progression 1:r:r^2:r^3 where r is the common ratio then r is limited to the range 1/t < r < t where t is the tribonacci constant. More generally if f(n) is the n-th step Fibonacci constant then a polygon of n+1 sides can have sides in a geometric progression 1:r:r^2:...:r^n if the common ratio r is limited to the range 1/f(n) < r < f(n).
From Wolfdieter Lang, Aug 22 2022: (Start)
The roots of this cubic are obtained from the roots of y^3 + (2/3)*y - 34/27 after subtracting 1/3. The y-roots are y1 = (u_p^(1/3) + u_m^(1/3)*e_m)/3, y2 = (e_m*u_p^(1/3) + u_m^(1/3))/3 and y3 = e_p*(u_p^(1/3) + u_m^(1/3))/3. Here u_p = 17 + 3*sqrt(33), u_m = 17 - 3*sqrt(33), e_p = -(1 + sqrt(3)*i) and e_m = -(1 - sqrt(3)*i), where i = sqrt(-1).
The roots of the x-cubic are then x1, the present real solution, and x2 = y2 - 1/3 = -0.771844506... + 1.11514250...*i and the complex conjugate x3 = y3 - 1/3. (End)

Examples

			0.543689012692076361570855971801747986525203297650983935240...

Links

Crossrefs

Reciprocal of A058265.
Cf. A376841.

Programs

  • Magma
    SetDefaultRealField(RealField(100)); (1/3)*(-1 -2/(17 +3*Sqrt(33))^(1/3) +(17+3*Sqrt(33))^(1/3)); // G. C. Greubel, Feb 06 2019
  • Mathematica
    N[Reduce[r+r^2+r^3==1, r], 100]
RealDigits[(1/3)*(-1 -2/(17+3*Sqrt[33])^(1/3) +(17+3*Sqrt[33])^(1/3)), 10, 100][[1]] (* G. C. Greubel, Feb 06 2019 *)
RealDigits[Root[r^3+r^2+r-1,1],10,120][[1]] (* Harvey P. Dale, May 18 2023 *)
  • PARI
    polrootsreal(r^3 + r^2 + r - 1)[1] \\ Charles R Greathouse IV, Apr 14 2014
  • Sage
    numerical_approx((1/3)*(-1 -2/(17+3*sqrt(33))^(1/3) +(17+ 3*sqrt(33))^(1/3)), digits=100) # G. C. Greubel, Feb 06 2019

Formula

Equals (1/3)*(-1-2/(17+3*sqrt(33))^(1/3) + (17+3*sqrt(33))^(1/3)).
Equals (1/3)*(u_p^(1/3) + u_m^(1/3)*e_m - 1), with u_p = 17 + 3*sqrt(33), u_m = 17 - 3*sqrt(33), and e_m = -(1 - sqrt(3)*i), with i = sqrt(-1). - Wolfdieter Lang, Aug 22 2022
Equals hypergeom([1/4,1/2,3/4],[2/3,4/3],16/27)/2. - Gerry Martens, Jul 13 2023

A066447 Number of basis partitions (or basic partitions) of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 8, 10, 13, 16, 20, 26, 32, 40, 50, 61, 74, 90, 108, 130, 156, 186, 222, 264, 313, 370, 436, 512, 600, 702, 818, 952, 1106, 1282, 1484, 1715, 1978, 2278, 2620, 3008, 3448, 3948, 4512, 5150, 5872, 6684, 7600, 8632, 9791, 11094, 12558, 14198, 16036, 18096, 20398
Offset: 0

Views

Author

Herbert S. Wilf, Dec 29 2001

Keywords

Comments

The k-th successive rank of a partition pi = (pi_1, pi_2, ..., pi_s) of the integer n is r_k = pi_k - pi'_k, where pi' denotes the conjugate partition. A partition pi is basic if the number of dots in its Ferrers diagram is the least among all the Ferrers diagrams of partitions with the same rank vector.
The g.f. Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) = Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 + x^k - 2*x^k) == Sum_{n >= 0} x^(n^2) (mod 2). It follows that a(n) is odd iff n is a square (Nolan et al., equation 6, p. 282). - Peter Bala, Jan 08 2025

Links

Crossrefs

Row sums of A066448.
Cf. A001130, A001935, A003114, A306734, A333374.
Cf. A192918, A376841.

Programs

  • Maple
    b := proc(n,d); option remember; if n=0 and d=0 then RETURN(1) elif n<=0 or d<=0 then RETURN(0) else RETURN(b(n-d,d)+b(n-2*d+1,d-1)+b(n-3*d+1,d-1)) fi: end: A066447 := n->add(b(n,d),d=0..n);
  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[x^(n^2)*Product[(1 + x^k)/(1 - x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 17 2020 *)
nmax = 60; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k - 1), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Mar 17 2020 *)
  • PARI
    N=66; x='x+O('x^N); s=sum(n=0,N,x^(n^2)*prod(k=1,n,(1+x^k)/(1-x^k))); Vec(s) /* Joerg Arndt, Apr 07 2011 */

Formula

G.f.: Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) [Given in Nolan et al. reference]. - Joerg Arndt, Apr 07 2011
Limit_{n->infinity} a(n) / A333374(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... - Vaclav Kotesovec, Mar 17 2020
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.193340468476900308848561788251945... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 + cosh(arccosh(53*sqrt(11/2)/64)/3) / (3*sqrt(22))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 19 2020, updated Oct 10 2024

A333374 G.f.: Sum_{k>=1} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)/(1 - x^j)).

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 3, 4, 6, 8, 10, 12, 15, 18, 22, 28, 34, 42, 52, 62, 75, 90, 106, 126, 150, 176, 208, 246, 288, 338, 397, 462, 538, 626, 724, 838, 968, 1114, 1282, 1474, 1690, 1936, 2217, 2532, 2890, 3296, 3750, 4264, 4844, 5492, 6222, 7042, 7958, 8986, 10138
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 17 2020

Keywords

Comments

The g.f. Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j) = Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j + 2*x^j) == Sum_{k >= 1} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

Links

Crossrefs

Cf. A001935, A001936, A003106, A053261, A066447, A003114, A306734, A333179.
Cf. A192918, A376841.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[(1+x^k)/(1-x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

Limit_{n->infinity} A066447(n) / a(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... (the tribonacci constant).
Compare with: A306734(n) / A333179(n) -> A060006 (the plastic constant) and A003114(n) / A003106(n) -> A001622 (golden ratio).
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.10511708841962944170826735560432... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 - sinh(arcsinh(sqrt(11)/4)/3) / (12*sqrt(11))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 17 2020, updated Oct 10 2024
Showing 1-3 of 3 results.

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