A369993 -id:A369993 - cp's OEIS frontend

A369990 Numerator of canonical iterated stribolic area Integral_{t=0..1} h_n(t) dt (of order 1).

1, 1, 1, 3, 2, 161, 24941, 49675943612, 3267335346149361824147, 2507700451651989905962493021537936733790431031, 39058362193701767718721504578116138158143785410766642680982462728116470023287868511995843

Offset: 0

Roland Miyamoto, Feb 07 2024

a(n) = numerator of Integral_{t=0..1} h_n(t) dt, where h_0 = 1, h_1 = T(h_0), h_2 = T(h_1), ...:[0,1]->[0,1], the operator T is given by T(g)(x) := Integral_{y=x..1} g^*(y) dy / Integral_{y=0..1} g(y) dy and g^*(y) := sup g^{-1}[y,1] (pseudo-inverse).
Geometrically speaking, T rotates by 90 degrees before integrating, which is why we call h_0, h_1, h_2,... the canonical stribolic iterates (from Greek stribo=turn/twist).
Alternatively, a(n) can be calculated from the polynomial q_n := h_n ° ... ° h_1. Cf. alternative formula below.
The sequence (a(n)/A369991(n)) is strictly decreasing and converges to the stribolic constant kappa=A369988.
We observe that a(n) and a(n+1) are coprime for n = 0..22 with the sole exception of gcd(a(5),a(6)) = 7.

			h_2(x) = (1-x)^2, h_2^*(x) = 1 - sqrt(x) = -h_3'(x)/3, h_3(x) = 1 - 3x + 2x^(3/2), hence Integral_{t=0..1} h_2(t) dt = 1/3 and Integral_{t=0..1} h_3(t) dt = 3/10. Therefore a(2)=1 and a(3)=3.

Roland Miyamoto, Table of n, a(n) for n = 0..14
Roland Miyamoto, Table of n, a(n) for n = 0..23
Roland Miyamoto, Polynomial parametrisation of the canonical iterates to the solution of -gamma*g' = g^{-1}, arXiv:2402.06618 [math.CO], 2024.

Cf. A369988 (decimal expansion of limit), A369991 (denominator).

Cf. A369992, A369993.

from functools import cache; from sympy.abc import x
@cache
def kappa(n): return (1-(n%2)*2) * Q(n).subs(x,1) if n else 1
@cache
def Q(n): return (q(n).diff() * q(n-1)).integrate()
@cache
def q(n): return (1-x if n==1 else n%2-Q(n-1)/kappa(n-1)) if n else x
def numer(c): return c.numerator if c%1 else c
print([numer(kappa(n)) for n in range(15)])

a(0)=1, a(n) is the numerator of kappa_n := Integral_{t=0..1} h_n(t) dt where h_1(x):=1-x and h_{n+1}(x) := Integral_{t=x..1} h_n^*(t) dt / kappa_n for n=1,2,...; here, h_n^* denotes the compositional inverse of h_n.

Alternatively, the rational sequence (kappa_n) := (a(n)/A369991(n)) and the two polynomial sequences (q_n), (Q_n) together are determined by the following equations for n=1,2,...: kappa_0=1, q_0=X, q_1=1-X, Q_n(0)=0, Q_n' = q_n'*q_{n-1}, kappa_n = (-1)^n * Q_n(1), q_{n+1} = (n+1) mod 2 - Q_n / kappa_n.

A369991 Denominator of canonical iterated stribolic area Integral_{t=0..1} h_n(t) dt (of order 1).

Original entry on oeis.org

1, 2, 3, 10, 7, 572, 89148, 177918244665, 11711158115225119429452, 8990773234863161759100003096510729982749072312, 140048278006628885452600904137492554179859017924910241263151850844470542993943699969398879

Offset: 0

Roland Miyamoto, Feb 08 2024

a(n) = denominator of Integral_{t=0..1} h_n(t) dt, where h_0 = 1, h_1 = T(h_0), h_2 = T(h_1), ...:[0,1]->[0,1], the operator T is given by T(g)(x) := Integral_{y=x..1} g^*(y) dy / Integral_{y=0..1} g(y)dy and g^*(y) := sup g^{-1}[y,1] (pseudo-inverse).
Geometrically speaking, T rotates by 90 degrees before integrating, which is why we call h_0, h_1, h_2, ... the canonical stribolic iterates (from Greek stribo=turn/twist).
Alternatively, a(n) can be calculated from the polynomial q_n := h_n ° ... ° h_1. Cf. alternative formula below.
The sequence (a(n)/A369991(n)) is strictly decreasing and converges to the stribolic constant kappa=A369988.

			h_2(x) = (1-x)^2, h_2^*(x) = 1 - sqrt(x) = - h_3'(x)/3, h_3(x) = 1 - 3x + 2x^(3/2), hence Integral_{t=0..1} h_2(t) dt = 1/3 and Integral_{t=0..1} h_3(t) dt = 3/10. Therefore a(2)=3 and a(3)=10.

Roland Miyamoto, Table of n, a(n) for n = 0..14
Roland Miyamoto, Table of n, a(n) for n = 0..23
Roland Miyamoto, Polynomial parametrisation of the canonical iterates to the solution of -gamma*g' = g^{-1}, arXiv:2402.06618 [math.CO], 2024.

Cf. A369988 (decimal expansion of limit), A369990 (numerator).

Cf. A369992, A369993.

from functools import cache; from sympy.abc import x
@cache
def kappa(n): return (1-(n%2)*2) * Q(n).subs(x,1) if n else 1
@cache
def Q(n): return (q(n).diff() * q(n-1)).integrate()
@cache
def q(n): return (1-x if n==1 else n%2-Q(n-1)/kappa(n-1)) if n else x
def denom(c): return c.denominator if c%1 else 1
print([denom(kappa(n)) for n in range(15)])

a(0)=1, a(n) is the denominator of kappa_n := Integral_{t=0..1} h_n(t) dt where h_1(x):=1-x and h_{n+1}(x) := Integral_{t=x..1} h_n^*(t) dt / kappa_n for n=1,2,...; here, h_n^* denotes the compositional inverse of h_n.

Alternatively, the rational sequence (kappa_n) := (A369990(n)/a(n)) and the two polynomial sequences (q_n), (Q_n) together are determined by the following equations for n=1,2,...: kappa_0=1, q_0=X, q_1=1-X, Q_n(0)=0, Q_n' = q_n'*q_{n-1}, kappa_n = (-1)^n * Q_n(1), q_{n+1} = (n+1) mod 2 - Q_n / kappa_n.

A369992 Irregular triangle read by rows: T(n,k) = (2^floor(n/2)+k)-th numerator coefficient of the polynomial q_n used to parametrize the canonical stribolic iterates h_n (of order 1), for n=0,1,2,... and 0 <= k <= A000045(n+1) - 2^floor(n/2).

Original entry on oeis.org

1, -1, 1, -3, 2, 5, -4, -35, 28, 70, -100, 35, 3575, -5720, -6292, 19240, -14300, 3520, -13856700, 22170720, 24387792, -74574240, 217088300, -401631120, -382444920, 2019752592, -1656568485, -1470440400, 3671101720, -2832601200, 1025395800, -147804800

Offset: 0

Roland Miyamoto, Mar 01 2024

The n-th row of the triangle contains 1 + A000045(n+1) - 2^floor(n/2) integers c_{2^floor(n/2)},...,c_{A000045(n+1)} forming a polynomial q_n = (n mod 2) + Sum_{i} c_i*X^i / A369993(n) that is related to A369990 and A369991 as follows: q_n = h_n ° ... ° h_1 (function composition), that is, h_n maps q_{n-1}(t) to q_n(t) for 0 <= t <= 1, and h_n has Integral_{x=0..1} h_n(x) dx = A369990(n)/A369991(n).
The gcd of each row in the triangle equals 1.
All previous statements are proved in the arXiv article, see link below.
Observation: In each of the 25 rows computed so far, there are no zeros and at most two consecutive entries of the same sign.

			q_5 = 1 + (-35*X^4 + 28*X^5 + 70*X^6 - 100*X^7 + 35*X^8) / 2 gives rise to row 5 (counting from 0) of the triangle (rows 0 to 7 are given):
          1;
         -1;
          1;
         -3,        2;
          5,       -4;
        -35,       28,       70,      -100,        35;
       3575,    -5720,    -6292,     19240,    -14300,       3520;
  -13856700, 22170720, 24387792, -74574240, 217088300, -401631120, -382444920, 2019752592, -1656568485, -1470440400, 3671101720, -2832601200, 1025395800, -147804800;

Roland Miyamoto, Table of n, a(n) for n = 0..1228
Roland Miyamoto, Polynomial parametrisation of the canonical iterates to the solution of -gamma*g' = g^{-1}, arXiv:2402.06618 [math.CO], 2024.

A369993 (denominator).

Cf. A369990, A369991, A369988.

from functools import cache, reduce; from sympy.abc import x; from sympy import lcm, fibonacci
@cache
def kappa(n): return (1-(n%2)*2) * Q(n).subs(x,1) if n else 1
@cache
def Q(n): return (q(n).diff() * q(n-1)).integrate()
@cache
def q(n): return (1-x if n==1 else n%2-Q(n-1)/kappa(n-1)) if n else x
def denom(c): return c.denominator if c%1 else 1
def row(n): qn = q(n); k0 = 1<<(n>>1); k1 = 1+fibonacci(n+1); dn = reduce(lcm,(denom(qn.coeff(x,k)) for k in range(k0,k1))); return [qn.coeff(x,k)*dn for k in range(k0,k1)]
for n in range(15): print(row(n))

The polynomials q_n = (n mod 2) + Sum_{k>=0} T(n,k)*X^(2^floor(n/2)+k) / A369993(n) are determined by the equations q_0=X, q_1=1-X, q_n(0) = n mod 2 and (A369990(n) / A369991(n)) * q_{n+1}' = -q_n' * q_{n-1} for n=1,2,...

Sum_k T(n,k) = (-1)^n * A369993(n) for n=0,1,2,...

A369988 Decimal expansion of Mallows's constant or stribolic constant kappa (of order 1).

Original entry on oeis.org

2, 7, 8, 8, 7, 7, 0, 6, 1

Offset: 0

Roland Miyamoto, Feb 07 2024

This constant is the area under the unique bijective, differentiable function h:[0,1]->[0,1] satisfying -c*h' = h^{-1} (compositional inverse) for some c > 0. That is, kappa = Integral_{t=0..1} h(t) dt, and then we also have kappa = c = -1/h'(0).
Equivalently, 1/kappa = 3.5858... is the only a > 0 such that there exists a differentiable function g:[0,a]->[0,a] which becomes its own derivative when rotated 90 degrees clockwise about the origin (into the fourth quadrant; whence the names "stribola" for g and h and "stribolic constant" for kappa, from Greek stribo=turn/twist), namely g(x):=h(kappa*x)/kappa for 0 <= x <= a = 1/kappa.
In 1997, Colin Mallows observed and conjectured that the rows in Levine's triangle A012257 take on stribolic shape and that A011784(n+1)/(A011784(n)*A011784(n-1)) converges as n->oo. Presuming his conjecture, the limit would equal kappa, while Mallows estimated it to be "approximately ... 0.277", see A011784. Later, in 2006, Martin Fuller suggested 0.27887706... for the limit, based on a numerical iteration, see A012257.
Set kappa_n := A369990(n) / A369991(n) and theta_n := (kappa_n-kappa_{n+1}) / (kappa_{n-1}-kappa_n). Under the hypothesis that theta_{2m} < theta_{2m+2} < theta_{2*m+3} < theta_{2*m+1} for m=1,2,... (verified for all values known so far), we would obtain 0.27887706136895087 < kappa_{21}' < kappa < kappa_{22}' < 0.27887706136898083, which is sharper than formula (3) below. Here, the transformed sequence (kappa_n') = G(kappa_n) is defined via kappa_n' := (kappa_{n-1}*kappa_{n+1} - kappa_n^2) / (kappa_{n-1} - 2*kappa_n + kappa_{n+1}). (See first arXiv article for a proof of this conjecture-dependent statement.) Feeling even more adventurous, we could apply the transformation G four times and would obtain 0.278877061368975064775 < kappa_{19}'''' < kappa < kappa_{18}'''' < 0.278877061368975064815.
It is an open question whether kappa is rational or irrational, algebraic or transcendental.

			0.278877061...

N. J. A. Sloane, My Favorite Integer Sequences, in: C. Ding, T. Helleseth, H. Niederreiter (editors), Sequences and their Applications, Discrete Mathematics and Theoretical Computer Science, Springer, London (1999) 103-130.

Roland Miyamoto, Polynomial parametrisation of the canonical iterates to the solution of -gamma*g' = g^{-1}, arXiv:2402.06618 [math.CO], 2024.
Roland Miyamoto, Solution to the iterative differential equation -gamma*g' = g^{-1}, arXiv:2404.11455 [math.CA], 2024.
Roland Miyamoto and J. W. Sander, Solving the iterative differential equation -gamma*g' = g^{-1}, in: H. Maier, J. & R. Steuding (eds.), Number Theory in Memory of Eduard Wirsing, Springer, 2023, pp. 223-236.
N. J. A. Sloane, My favorite integer sequences, in: Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Colin Mallows, and Bjorn Poonen, Discussion of A011784. [Scans of pages 150-155 and 164 of Sloane's notebook "Lattices 77", from June-July 1997.]

Cf. A369990, A369991, A369992, A369993, A011784, A012257, A014621.

Set kappa_n := A369990(n) / A369991(n).  Then
(1) kappa = lim_{n->oo}kappa_n = inf{kappa_n: n >= 0},
(2) kappa_n - 1 + kappa_n/kappa_{n-1} < kappa < kappa_n for n=1,2,...,
(3) 0.2788770612338 < kappa_{23} - 1 + kappa_{23}/kappa_{22} < kappa < kappa_{23} < 0.2788770613941.

cp's OEIS Frontend

A369990 Numerator of canonical iterated stribolic area Integral_{t=0..1} h_n(t) dt (of order 1).

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A369991 Denominator of canonical iterated stribolic area Integral_{t=0..1} h_n(t) dt (of order 1).

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A369992 Irregular triangle read by rows: T(n,k) = (2^floor(n/2)+k)-th numerator coefficient of the polynomial q_n used to parametrize the canonical stribolic iterates h_n (of order 1), for n=0,1,2,... and 0 <= k <= A000045(n+1) - 2^floor(n/2).

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Python

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A369988 Decimal expansion of Mallows's constant or stribolic constant kappa (of order 1).

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