A145005 Values of k at which the number of roots of the function x + k*sin(x) increases.

5, 11, 18, 24, 30, 37, 43, 49, 55, 62, 68, 74, 81, 87, 93, 99, 106, 112, 118, 125, 131, 137, 143, 150, 156, 162, 169, 175, 181, 187, 194, 200, 206, 213, 219, 225, 231, 238, 244, 250, 257, 263, 269, 275, 282, 288, 294, 301, 307, 313, 319, 326, 332, 338, 345

Offset

1,1

Comments

First differs from A004083 at n = 19564195 where a(n) = 122925461 and A004083(n) = 122925462. - Martin Fuller, Apr 10 2025

Links

Table of n, a(n) for n=1..55.

Martin Fuller, Proof for formula

Anthony G. Shannon, Bahar Kuloğlu, and Engin Özkan, Rhaly terraced sequences their generalizations, properties and applications, Comp. Appl. Math., Vol. 44 (2025), Article No. 226. See p. 4.

Formula

a(n) = ceiling(sqrt(x^2+1)) where x is the root of 1+sqrt(x^2+1)*cos(x) in the range [(2n-1)Pi, (2n-1/2)Pi]. - Martin Fuller, Apr 10 2025

Example

For k = 0..4, there is 1 root; for k = 5..10, there are 5 roots; for k = 11..17, there are 9 roots; etc... .

Prog

(PARI) a(n)=my(x=solve(x=(2*n-1)*Pi, (2*n-1/2)*Pi, 1+sqrt(x^2+1)*cos(x))); ceil(sqrt(x^2+1)); \\ Martin Fuller, Apr 10 2025

Crossrefs

Cf. A004083.

Sequence in context: A314272 A184822 A004083 * A190365 A056000 A080566

Adjacent sequences: A145002 A145003 A145004 * A145006 A145007 A145008

Keyword

nonn

Author

Sébastien Dumortier, Sep 28 2008

Extensions

a(0)=0 removed, and a(16) onwards from Martin Fuller, Apr 10 2025

Status

approved