A323768 -id:A323768 - cp's OEIS frontend

A323777 Decimal expansion of the root of the equation (1-2r)^(3-4r) = (1-r)^(2-2r) r^(1-2*r).

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

Offset: 0

Vaclav Kotesovec, Jan 27 2019

			0.2206760413237406963128222699980501671876810310275740395417335127215630565...

Cf. A166895, A209331, A209428.

Cf. A220359, A237421, A245259, A323768, A323773, A323778.

RealDigits[r/.FindRoot[(1-2*r)^(3-4*r) == (1-r)^(2-2*r) * r^(1-2*r), {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A323773 Decimal expansion of the root of the equation (1-2r)^(4r-1) * (1-r)^(1-2r) = r^(2r).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 27 2019

			0.3663201503052830964087236563781171194011826607210994595491823160184...

Cf. A220359, A228833, A237421, A245259, A323768, A323777, A323778.

RealDigits[r/.FindRoot[(1-2*r)^(4*r-1) * (1-r)^(1-2*r) == r^(2*r), {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A323778 Decimal expansion of the root of the equation (1-r)^(2-2r) r^(2r) = 1-2r.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 27 2019

			0.36549849821985804457973687544629908832275880696346029501595516768211883674...

Cf. A123165, A206851.

Cf. A220359, A237421, A245259, A323768, A323773, A323777.

RealDigits[r/.FindRoot[(1-r)^(2-2*r) * r^(2*r) == 1-2*r, {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A323769 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n.

Original entry on oeis.org

1, 1, 2, 9, 83, 1268, 62283, 10296321, 2668655428, 1306416217435, 3055324257386077, 17213278350960504924, 137320554100797006975445, 3087543920644806918694851647, 335732238884967561227813578781572, 61125387696211835948801235842204794881

Offset: 0

Seiichi Manyama, Jan 27 2019

nonn

The limit a(n) / (5^(n/4) * phi^(n*(n+1)) / (2*Pi*n)^(n/2)) does not exist but oscillates between 2 attractors. The value is dependent on the fractional part of n/(sqrt(5)*phi), see graph. - Vaclav Kotesovec, Jan 28 2019

Seiichi Manyama, Table of n, a(n) for n = 0..71
Vaclav Kotesovec, Graph - the asymptotic ratio
Vaclav Kotesovec, Graph - dependence of the limit on the fractional part of n/(sqrt(5)*phi)

Main diagonal of A323767.

Cf. A011973, A051286, A181545, A181546, A181547, A209428, A323768.

Table[Sum[Binomial[n-k,k]^n, {k, 0, n/2}], {n, 0, 15}] (* Vaclav Kotesovec, Jan 27 2019 *)

{a(n) = sum(k=0, n\2, binomial(n-k, k)^n)}

a(n)^(1/n) ~ 5^(1/4) * phi^(n+1) / sqrt(2*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 27 2019

log(a(n)) ~ n*(n*v + w - log(n))/2 with v = 2*log((1 + sqrt(5))/2) and w = log((35 + 15*sqrt(5))/(8*Pi^2))/2, preceding formula recast. - Peter Luschny, Jan 28 2019

cp's OEIS Frontend

A323777 Decimal expansion of the root of the equation (1-2r)^(3-4r) = (1-r)^(2-2r) r^(1-2*r).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A323773 Decimal expansion of the root of the equation (1-2r)^(4r-1) * (1-r)^(1-2r) = r^(2r).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A323778 Decimal expansion of the root of the equation (1-r)^(2-2r) r^(2r) = 1-2r.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A323769 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

A323777 Decimal expansion of the root of the equation (1-2*r)^(3-4*r) = (1-r)^(2-2*r) * r^(1-2*r).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A323773 Decimal expansion of the root of the equation (1-2*r)^(4*r-1) * (1-r)^(1-2*r) = r^(2*r).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A323778 Decimal expansion of the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A323769 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A323777 Decimal expansion of the root of the equation (1-2r)^(3-4r) = (1-r)^(2-2r) r^(1-2*r).

A323773 Decimal expansion of the root of the equation (1-2r)^(4r-1) * (1-r)^(1-2r) = r^(2r).

A323778 Decimal expansion of the root of the equation (1-r)^(2-2r) r^(2r) = 1-2r.