A387706 Indices where the cumulative sum of sin(2k+1)^(2k+1) reaches a record high value.

0, 1, 10999, 11002, 11005, 11006, 11354, 11357, 11360, 11361, 11709, 11712, 11715, 11716, 12064, 12067, 12070, 12071, 12419, 12422, 12425, 12426, 12774, 12777, 12780, 12781, 13129, 13132, 13135, 13136, 13484, 13487, 13490, 13491, 13839, 13842, 13845, 13846, 14194, 14197

Offset

0,3

Comments

The sequence has consecutive differences of 3, 3, 1, 348 after the 2nd term for as long as the sequence exists. The differences add up to 355 which is the numerator of a significant convergent to Pi.

It appears that the sequence has only 466 terms with the last term being 51831.

Sum_{k=0..51831} sin(2k+1)^(2k+1) = 60.2254500...

Links

Jwalin Bhatt, Table of n, a(n) for n = 0..465

Example

-----------------------------------------
|   k   | Sum_{n=0..k} sin(2k+1)^(2k+1) |
-----------------------------------------
|   0   |           0.8414...           |
|   1   |           0.8442...           |
|   2   |           0.0334...           |
| 10998 |           0.0101...           |
| 10999 |           0.9407...           |
-----------------------------------------

Prog

(Python)
from mpmath import iv
def a_upto(limit):
    sum_, highs = iv.mpf('0'), [0]
    for i in range(1, 1+limit):
        t = 2*i + 1
        sum_ += iv.sin(t)**t
        if sum_ > 0:
            highs.append(i)
            sum_ = 0
    return highs
print(a_upto(14200)) # Jwalin Bhatt, Nov 09 2025

Crossrefs

Cf. A387730, A389559.

Sequence in context: A104323 A252039 A252033 * A352751 A266983 A227865

Adjacent sequences: A387703 A387704 A387705 * A387707 A387708 A387709

Keyword

nonn

Author

Jwalin Bhatt, Sep 06 2025

Status

approved