seqs needed (monotonous approx to Pi)

[Olivier Gerard]

At 18:29 +0200 99.04.18, N. J. A. Sloane wrote:
->anyway, what i am writing about is to request that someone
->compute some more sequences of this type.  Namely,
->the sequence of n's such that:
->
->cos n falls monotoniaclly from cos(1) to 0
->|cos(n)| """""""""""""""""""""""""""""""""
->same thing for tan, sec, csc, cot, cosh, sinh, tanh, etc
->
->then
->
->same thing for
->sin n rises monotonically from sin(1)
->- and same for cos tan etc etc
->

Since apparently, nobody has started to do that, I will.

(note to Neil: decrease has a typo in the sequence you listed)

I don't think this makes sense for cosh, sinh and tanh since they
are not periodic or oscillatory in the real domain. But it would be
interesting for Bessel functions (compute them, any one ?).

As you will see in the list appended to this message, the
32 cases reduce by pairs to 16, then partially to 8, some of the remaining ones
differing only by the first few terms or being subsequences of others.

For the four cases where I give only a few terms, perhaps someone
will take time to make a clever algorithm (and not the brute force
approach I took) to expand them to three lines in the database.

Olivier


Sin decreasing == Cosec increasing :  (case known)
{1,3,44,311,377,688,710}
Next term is superior to 100,000

(|Sec| ~ |Tan| ~ |Sin| decreasing)  ==  (|Cotan| ~ |Cosec| ~ |Cos| increasing)
{1,3,22,333,355}
Next term is superior to 100,000

Sec decreasing == Cos increasing:
{1,6,19,25,44,333,710}
Next term is superior to 100,000

Tan decreasing == Cotan increasing
{1,4,7,10,13,16,19,22,355}
Next term is superior to 100,000


(Cos decreasing) == (Sec increasing) :
{1,5,11,699,1409,2119,2829,3539,4249,4959,5669,6379,7089,7799,8509,9219,9929,
  10639,11349,12059,12769,13479,14189,14899,15609,16319,17029,17739,18449,
  19159,19869,20579,21289,21999,22709,23419,24129,24839,25549,26259,26969,
  27679,28389,29099,29809,30519,31229,31939,32649,33359,34069,34779,35489,
  36199,36909,37619,38329,39039,39749,40459,41169,41879,42589,43299,44009,
  44719,45429,46139,46849,47559,48269,48979,49689,50399,51109,51819,52174}

(|Cos| ~ |Cosec| ~ |CoTan|) decreasing == |Tan| ~ |Sec| ~ |Sin| increasing :
{1,2,5,8,11,344,699,1054,1409,1764,2119,2474,2829,3184,3539,3894,4249,4604,
  4959,5314,5669,6024,6379,6734,7089,7444,7799,8154,8509,8864,9219,9574,9929,
  10284,10639,10994,11349,11704,12059,12414,12769,13124,13479,13834,14189,
  14544,14899,15254,15609,15964,16319,16674,17029,17384,17739,18094,18449,
  18804,19159,19514,19869,20224,20579,20934,21289,21644,21999,22354,22709,
  23064,23419,23774,24129,24484,24839,25194,25549,25904,26259,26614,26969,
  27324,27679,28034,28389,28744,29099,29454,29809,30164,30519,30874,31229,
  31584,31939,32294,32649,33004,33359,33714,34069,34424,34779,35134,35489,
  35844,36199,36554,36909,37264,37619,37974,38329,38684,39039,39394,39749,
  40104,40459,40814,41169,41524,41879,42234,42589,42944,43299,43654,44009,
  44364,44719,45074,45429,45784,46139,46494,46849,47204,47559,47914,48269,
  48624,48979,49334,49689,50044,50399,50754,51109,51464,51819,52174}


Cosec decreasing == Sin increasing:
{1,2,8,14,33,322,366,699,1409,2119,2829,3539,4249,4959,5669,6379,7089,7799,
  8509,9219,9929,10639,11349,12059,12769,13479,14189,14899,15609,16319,17029,
  17739,18449,19159,19869,20579,21289,21999,22709,23419,24129,24839,25549,
  26259,26969,27679,28389,29099,29809,30519,31229,31939,32649,33359,34069,
  34779,35489,36199,36909,37619,38329,39039,39749,40459,41169,41879,42589,
  43299,44009,44719,45429,46139,46849,47559,48269,48979,49689,50399,51109,
  51819}

Cotan decreasing == Tan increasing:
{1,14,36,58,80,102,124,146,168,190,212,234,256,278,300,322,344,699,1054,1409,
  1764,2119,2474,2829,3184,3539,3894,4249,4604,4959,5314,5669,6024,6379,6734,
  7089,7444,7799,8154,8509,8864,9219,9574,9929,10284,10639,10994,11349,11704,
  12059,12414,12769,13124,13479,13834,14189,14544,14899,15254,15609,15964,
  16319,16674,17029,17384,17739,18094,18449,18804,19159,19514,19869,20224,
  20579,20934,21289,21644,21999,22354,22709,23064,23419,23774,24129,24484,
  24839,25194,25549,25904,26259,26614,26969,27324,27679,28034,28389,28744,
  29099,29454,29809,30164,30519,30874,31229,31584,31939,32294,32649,33004,
  33359,33714,34069,34424,34779,35134,35489,35844,36199,36554,36909,37264,
  37619,37974,38329,38684,39039,39394,39749,40104,40459,40814,41169,41524,
  41879,42234,42589,42944,43299,43654,44009,44364,44719,45074,45429,45784,
  46139,46494,46849,47204,47559,47914,48269,48624,48979,49334,49689,50044,
  50399,50754,51109,51464,51819}