A377080 -id:A377080 - cp's OEIS frontend

A230152 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=4.

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

Offset: 0

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-4.

			0.8566748838545028748523248153124343698313999454937526255...

Paolo P. Lava, Table of n, a(n) for n = 0..1000 (corrected by _Sean A. Irvine_)
Index entries for algebraic numbers, degree 5

Cf. A075778, A230151, A230153-A230158, A377080.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-4);

Root[x^5 + x^4 - 1, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

A377081 G.f.: Sum_{k>=1} x^(3k^2) Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2

Offset: 0

Vaclav Kotesovec, Oct 15 2024

In general, if m > 0 and g.f. = Sum_{k>=1} x^(m*k^2) * Product_{j=1..k} (1 + x^j), then a(n) ~ (1+r) * exp(sqrt((4*m*(2*m+1)*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((r + 2*m*(1+r))*n)), where r is the smallest positive real root of the equation r^(2*m)*(1+r) = 1.

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

Cf. A306734 (m=1), A377080 (m=2).

Cf. A230154, A350890, A350894.

nmax = 200; CoefficientList[Series[Sum[x^(3*k^2)*Product[1+x^j, {j, 1, k}], {k, 1, Sqrt[nmax/3]}], {x, 0, nmax}], x]

a(n) ~ (1+r) * exp(sqrt((84*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((6 + 7*r)*n)), where r = A230154 = 0.898653712628699293260875722... is the real root of the equation r^6*(1+r) = 1.

cp's OEIS Frontend

A230152 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=4.

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A377081 G.f.: Sum_{k>=1} x^(3k^2) Product_{j=1..k} (1 + x^j).

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cp's OEIS Frontend

A230152 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=4.

Views

Author

Keywords

Comments

Examples

Links

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Programs

Maple

Mathematica

A377081 G.f.: Sum_{k>=1} x^(3*k^2) * Product_{j=1..k} (1 + x^j).

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Author

Keywords

Comments

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Mathematica

Formula

A377081 G.f.: Sum_{k>=1} x^(3k^2) Product_{j=1..k} (1 + x^j).