Mikhail Gaichenkov - cp's OEIS frontend

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

Offset: 0

Mikhail Gaichenkov, Oct 03 2023

The root appears in the problem of minimizing the area of self-intersection of a folded rectangle. A rectangle with sides a, b (a

The unique ratio of sides a/b=T=0.81502370129163... is derived based on the real root of the quintic. If a/b

			0.45913372331020753...

Mikhail Gaichenkov, Quintic equation with integer coefficients, Math Stackexchange, 2023.
Mikhail Gaichenkov, Folded rectangle
Index entries for algebraic numbers, degree 5

First[RealDigits[Root[#^5 + 3*#^4 + 4*#^3 + # - 1 &, 1], 10, 100]] (* Paolo Xausa, Jun 25 2024 *)

polrootsreal(x^5 + 3*x^4 + 4*x^3 + x-1)[1]

A202408 Denominator of series coefficients for Archimedes' spiral which transforms into Galileo's spiral.

Original entry on oeis.org

1, 2, 32, 768, 49152, 1, 56623104, 3170893824, 541165879296, 175337744891904, 28054039182704640, 29389945810452480, 2844006276185865584640, 147888326361665010401280, 25765432859454526256578560, 83480002464632665071314534400

Offset: 0

Mikhail Gaichenkov, Dec 19 2011

Numerators are listed in A202407 which is the main entry for these sequences.

km = 16; a[0] = 0; r[t_] = Sum[a[k] t^(2 k), {k, 0, km}]; coes = CoefficientList[Series[r'[t]^2 + r[t]^2 - t^2 , {t, 0, 2 km}], t] // Union // Rest; Table[a[k], {k, 0, km}] /. Solve[Thread[coes == 0] ] // Last // Most // Denominator (* Jean-François Alcover, Jan 18 2013 *)

Corrected and extended by Max Alekseyev, Dec 19 2011

A202407 Numerators of series coefficients for Archimedes's spiral that transforms into Galileo's spiral.

Original entry on oeis.org

0, 1, -1, 1, -1, 0, -1, -1, 17, 587, 3151, -173, -2641109, -6343201, 29002301, 24753572807, 6013935944287, -979056822493, -11395219462649, -4313800586682649, -2178360615103441, 74893762899375939059, 5307412498351127900521

Offset: 0

Mikhail Gaichenkov, Dec 19 2011

The curve defined by the differential equation in polar coordinates r'(t)^2 + r(t)^2 = t^2 with r(0)=0, r"(0) > 0. Solution is represented by a power series in z=t^2 (satisfying the differential equation 4*z*r'(z)^2 + r(z)^2 = z). The sequence lists coefficients of t^(2*n) (or z^n) in this series.
For large t, the curve represents Archimedes's spiral. As t vanishes, the curve transforms into a Galileo spiral. The junction point of the curve and the ray uniformly rotated in the origin coordinates moves uniformly accelerated.
Let L_{A} and L_{AG} are the lengths of Archimedean spiral and the spiral defined by the differential equation, then lim_{t -> oo} L_{A}/L_{AG} = 1. In other words, the lengths of Archimedean spiral and the spiral defined by the differential equation are equivalent for large t. - Mikhail Gaichenkov, Jan 08 2013
According to Robert Bryant, the key to understanding the solutions of the ODE near the singular points is the Briot-Bouquet normal form for dealing with singular points, and, fortunately, it is just the right thing both at the origin and along the lines theta^2 - r^2 = 0. - Mikhail Gaichenkov, Feb 18 2013

			The first ten terms of this expansion are: r(t) = 0 + 1/2*t^2 - 1/32*t^4 + 1/768*t^6 - 1/49152*t^8 + 0*t^10 - 1/56623104*t^12 - 1/317893824*t^14 + 17/541165879296*t^16 + 587/175337744891904*t^18 + ...
The radius of the convergence is about 7/2.

Robert Bryant, MathOverflow: What is symmetry group of non-linear equation?
A. Pichugin, MathOverflow: Analytical solutions of a differential equation (from Archimedes' Spiral)

Denominators are listed in A202408.

Order:=60: dsolve( { diff(r(t),t)^2 + r(t)^2 = t^2, r(0)=0 }, r(t), series ); # Max Alekseyev, Dec 19 2012

km = 23; a[0] = 0; r[t_] = Sum[a[k] t^(2 k), {k, 0, km}]; coes = CoefficientList[Series[r'[t]^2 + r[t]^2 - t^2 , {t, 0, 2 km}], t] // Union // Rest; Table[a[k], {k, 0, km}] /. Solve[Thread[coes == 0] ] // Last // Most // Numerator (* Jean-François Alcover, Jan 18 2013 *)

Corrected and extended by Max Alekseyev, Dec 19 2011

A181609 Kendell-Mann numbers in terms of Mahonian distribution.

Original entry on oeis.org

2, 3, 7, 23, 108, 604, 3980, 30186, 258969, 2479441, 26207604, 303119227, 3807956707, 51633582121, 751604592219, 11690365070546, 193492748067369, 3395655743755865, 62980031819261211, 1230967683216803500

Offset: 2

Mikhail Gaichenkov, Jan 30 2011

nonn

It is well known that the variance of the Mahonian distribution is equal to sigma^2=n(n-1)(2n+5)/72. It is possible to have the asymptotic expansion for Kendell-Mann numbers M(n)=n!/sigma * 1/sqrt(2*Pi) * (1 - 2/(3*n) + O(1/n^2)). This results in M(n+1)/M(n)=n-1/2+O(1/n) as n--> infinity. [corrected by Vaclav Kotesovec, May 17 2015]

			M(2)=2, M(3)=3, M(4)=7,...

Mikhail Gaichenkov, The property of Kendall-Mann numbers, answered by Richard Stanley, 2010
Mikhail Gaichenkov, A combinatorial proof for the property of KM numbers?

Cf. A000140.

M(n) = Round(n!/sqrt(Pi*n(n-1)(2n+5)/36)).

A181188 Primes at which the prime number race between the two prime classes with different sign of sin(prime(.)) changes leader.

Original entry on oeis.org

31, 101, 167, 229, 269, 271, 307, 311, 313, 317, 331, 359, 439, 479, 487, 491, 691, 787, 797, 3739, 3761, 3821, 4019, 4093, 4153, 4231, 4241, 4243, 4253, 5839, 5843, 5857, 5861, 6367, 6469, 6473, 6553, 6637, 6653, 6673, 6679, 7121, 7219, 7297, 7307, 7309, 7351, 7561, 7583, 7603, 7607, 7681, 8311

Offset: 1

Mikhail Gaichenkov, Oct 09 2010

nonn

Split the prime numbers into A070754 and A070753 according to the sign of the sine function:
 2,  3,  7, 13, 19| 47, 53, 59, 71, 83, 89, 97,101|103,107,109,127,139,151|179,191,197,223,...
 5, 11, 17, 23, 29| 31, 37, 41, 43, 61, 67, 73, 79|113,131,137,149,157,163|167,173,181,193,199,...
Comparison of A070754(i) with A070753(i) defines a prime number race. The leader chances at places i where sign( A070754(i)-A070753(i) ) <> sign( A070754(i+1)-A070753(i+1) ) indicated by the vertical bars above.
An equivalent observation is that the partial sum s(k) := sum_{i=1..k} A070748(i) has zeros at prime(k)= 29, 101, 163, 229, 263, 271,...
The sequence contains each prime(k+1) where s(k) >=0 and s(k+1)<0 or s(k) <0 and s(k+1)>=0. Cases where s(k) touches zero without actually flipping the sign are not relevant.

John Derbyshire and Mikhail Gaichenkov, The sign of the sine of p

isA070753 := proc(n) is(sin(ithprime(n))<0) ; end proc:
A070748 := proc(n) option remember; if isA070753(n) then -1 ; else 1; end if; end proc:
A070748s := proc(n) add( A070748(i),i=1..n) ; end proc:
for n from 1 to 10000 do if A070748s(n) >= 0 and A070748s(n+1) < 0 or A070748s(n) <0 and A070748s(n+1) >= 0 then printf("%d,",ithprime(n+1)) ; end if;end do:

s=0; p=0; while(1, p=nextprime(p+1); s+=(-1)^(p\Pi); if(s<=-7568,print1(p,", ")))

s=0;forprime(p=2,2000,s+=(-1)^(p\Pi);print1(s,", "))

cp's OEIS Frontend

User: Mikhail Gaichenkov

A366185 Decimal expansion of the real root of the quintic equation x^5 + 3*x^4 + 4*x^3 + x -1 = 0.

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A202408 Denominator of series coefficients for Archimedes' spiral which transforms into Galileo's spiral.

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A202407 Numerators of series coefficients for Archimedes's spiral that transforms into Galileo's spiral.

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Maple

Mathematica

Extensions

A181609 Kendell-Mann numbers in terms of Mahonian distribution.

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Formula

A181188 Primes at which the prime number race between the two prime classes with different sign of sin(prime(.)) changes leader.

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Author

Keywords

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Maple

PARI

PARI

A366185 Decimal expansion of the real root of the quintic equation x^5 + 3x^4 + 4x^3 + x -1 = 0.