A231896 -id:A231896 - cp's OEIS frontend

A317451 a(n) = (n*A003500(n) - A231896(n))/2.

0, 2, 16, 92, 464, 2182, 9824, 42936, 183648, 772746, 3209968, 13196564, 53791408, 217700110, 875718080, 3504277360, 13959102912, 55383875346, 218965651152, 862983998924, 3391602170512, 13295446717334, 51999641009696, 202948920530728, 790569797639456, 3074179492922778

Offset: 0

Rigoberto Florez, Jul 28 2018

Derivative of Chebyshev second kind polynomials evaluated at 2.

R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Colin Barker, Table of n, a(n) for n = 0..1000
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind
Index entries for linear recurrences with constant coefficients, signature (8,-18,8,-1).

Cf. A003500, A231896, A133156 (Chebyshev polynomials of the second kind).

CoefficientList[ Series[2 x/(x^2 - 4x + 1)^2, {x, 0, 25}], x] (* Robert G. Wilson v, Aug 07 2018 *)

a(n) = subst(deriv(polchebyshev(n, 2)), x, 2); \\ Michel Marcus, Jul 29 2018.

concat(0, Vec(2*x / (1 - 4*x + x^2)^2 + O(x^40))) \\ Colin Barker, Aug 06 2018

From Colin Barker, Aug 06 2018: (Start)
G.f.: 2*x / (1 - 4*x + x^2)^2.
a(n) = (sqrt(3)*((2-sqrt(3))^n - (2+sqrt(3))^n) + 3*((2-sqrt(3))^(1+n) + (2+sqrt(3))^(1+n))*n) / 18.
a(n) = 8*a(n-1) - 18*a(n-2) + 8*a(n-3) - a(n-4) for n>3.
(End)

A231987 Decimal expansion of the side length (in radians) of the spherical square whose solid angle is exactly one steradian.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 17 2013

This is an inverse problem (but not an inverse value) to the one leading to A231986: what is the side s of a spherical square (in radians, rad) if it covers a given solid angle (in steradians, sr)? The solution (inverse of the formula in A231896) is s = 2*arcsin(sqrt(sin(Omega/4))). In this particular case, Omega = 1.

			1.041191803606873340234607533592568788900696676006087134915230281312997...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Solid angle, Section 3.3 (Pyramid).
Wikipedia, Steradian.

Cf. A072097 (rad/deg), A019685 (deg/rad), A231981 (sr/deg^2), A231982 (deg^2/sr), A231986 (inverse problem), A231896.

RealDigits[2*ArcSin[Sqrt[Sin[1/4]]], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

default(realprecision, 120);
2*asin(sqrt(sin(1/4))) \\ or
solve(x = 1, 2, 4*asin((sin(x/2))^2) - 1) \\ least positive solution - Rick L. Shepherd, Jan 28 2014

Equals 2*arcsin(sqrt(sin(1/4))).

Formula and comment corrected by Rick L. Shepherd, Jan 28 2014

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A317451 a(n) = (n*A003500(n) - A231896(n))/2.

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A231987 Decimal expansion of the side length (in radians) of the spherical square whose solid angle is exactly one steradian.

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