A387730 Indices where the cumulative sum of sin(2k+1)^(2k+1) reaches a record low value.

0, 2, 5, 8, 84118, 84121, 84124, 84125, 84473, 84476, 84479, 84480, 84828, 84831, 84834, 84835, 85183, 85186, 85189, 85190, 85538, 85541, 85544, 85545, 85893, 85896, 85899, 85900, 86248, 86251, 86254, 86255, 86603, 86606, 86609, 86610, 86958, 86961, 86964, 86965

Offset

0,2

Comments

The sequence has consecutive differences of 3, 3, 1, 348 after the 4th 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 232 terms with the last term being 104005.

Sum_{k=0..104005} sin(2k+1)^(2k+1) = -27.592119...

Links

Jwalin Bhatt, Table of n, a(n) for n = 0..231

Example

-----------------------------------------
|   k   | Sum_{n=0..k} sin(2k+1)^(2k+1) |
-----------------------------------------
|   0   |           0.8414...           |
|   1   |           0.8442...           |
|   2   |           0.0334...           |
|   3   |           0.0862...           |
|   4   |           0.0866...           |
|   5   |          -0.9132...           |
-----------------------------------------

Prog

(Python)
from mpmath import iv
def a_upto(limit):
    sum_, lows = iv.mpf('0'), [0]
    for i in range(1, 1+limit):
        t = 2*i + 1
        sum_ += iv.sin(t)**t
        if sum_ < 0:
            lows.append(i)
            sum_ = 0
    return lows
print(a_upto(85000)) # Jwalin Bhatt, Nov 09 2025

Crossrefs

Cf. A387706, A389560.

Sequence in context: A242875 A222897 A282455 * A323457 A011039 A346045

Adjacent sequences: A387727 A387728 A387729 * A387731 A387732 A387733

Keyword

nonn

Author

Jwalin Bhatt, Sep 06 2025

Status

approved