Reply from On-Line Encyclopedia of Integer Sequences (fwd)
Richard Guy
rkg at cpsc.ucalgary.ca
Tue Jan 22 00:43:36 CET 2002
In connexion with the following, I append part of a
message sent to a graduate student, which may contain
items which you could use.
The student was encouraged to try to find a relationship
between the two sequences submitted to `lookup'. The
first one is the sequence `r' in the attached message.
Warning. Mostly hand calculations and transcriptions,
so there are probably several numerical errors.
If some hero wants to do this for other values of 7
(any squarefree number) then there would be another
countable infinity of sequences for the OEIS.
Best, R.
---------- Forwarded message ----------
Date: Mon, 14 Jan 2002 17:31:03 -0500 (EST)
From: sequences-reply at research.att.com
To: rkg at cpsc.ucalgary.ca
Subject: Reply from On-Line Encyclopedia of Integer Sequences
Matches (up to a limit of 50) found for 2 1 2 3 2 7 2 3 6 6 2 7 14 3 2 3 6 9 6 7 6 6 2 :
Matches (up to a limit of 50) found for 4 2 6 6 4 16 2 6 16 10 4 14 32 6 2 4 16 14 12 24 :
%I A064860
%S A064860 4,4,2,6,6,4,16,2,6,16,10,4,14,32,6,2,4,16,14,12,24,10,14,2,26,24,22,
%T A064860 36,38,12,24,8,14,8,52,8,34,32,14,16,52,32,46,14,2,14,52,2,120,64,4,32,
%U A064860 50,44,12,32,18,34,58,12,70,52,48,12,50,18,66,8,14,96,18,8,64,42,30,36
%N A064860 Period of continued fraction of sqrt(7)*n.
%K A064860 nonn
%O A064860 1,1
%A A064860 Richard Guy, Oct 26, 2001
Even though there are a large number of sequences in the table, at least
one of yours is not there! Please send it to me.
Date: Wed, 16 Jan 2002 12:41:12 -0700 (MST)
From: Richard Guy <rkg at cpsc.ucalgary.ca>
Examine [and correct where necessary!] the
following table for simple relationships,
prove any that are true and find counterexamples
for the others. Do the same thing for other
values of 7. R.
r is the rank of apparition of the number n
(i.e., the least r for which n divides u_r)
in the recurring sequence u_k+1 = 16 u_k - u_k-1,
u_0 = 0, u_1= 3. (u_2 = 48, u_3 = 765, u_4 = 12192)
[Lucas-Lehmer tell us that if n is a prime, p,
then p divides u _ p-(7|p) and r divides
p-(7|p) where (7|p) is the Legendre symbol
\pm1 according as p = \pm(1, 3, 9) or
\pm(5, 11, 13) mod 28. u_k is the coefficient
of sqrt(7) in the k-th power of the
fundamental unit in Q(sqrt(7)). One expects
that r and m (see next para.) have the
same order of magnitude. Can this be, has this
been, quantified?]
m (don't like typing ell) is the length of the
period of the continued fraction for sqrt(7*n^2)
a_m = 2 * a_0 = 2 * floor(sqrt(7*n^2)) is the
m-th partial quotient.
a_m/2 is the partial quotient in the middle
of the palindrome.
I've added three more rows to the table:
the length of period, q, of the neg(ative)
continued fraction; the sum, s, of the
partial quotients of the neg; and the
difference, d = s - q. The partial quotient
b_q will be a_m + 2
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
r 1 2 1 2 3 2 7 2 3 6 6 2 7 14 3
m 4 4 2 6 6 4 16 2 6 16 10 4 14 32 6
q 2 6 1 4 9 2 36 6 5 12 30 2 53 80 7
a_m 4 10 14 20 26 30 36 42 46 52 58 62 68 74 78
a_m/2 1 2 1 2 1 6 17 6 3 12 8 2 3 18 5
s 9 24 16 30 48 40 135 54 64 102 132 68 192 258 96
d 7 18 15 26 39 38 99 48 59 90 102 66 139 178 89
n 16 17 18 19 20 21 22 23 24 25 26 27
r 2 3 6 9 6 7 6 6 2 15 14 9
m 2 4 16 14 12 24 10 14 2 26 24 22
q 3 2 8 89 20 18 34 24 2 81 128 27
a_m 84 88 94 100 104 110 116 120 126 132 136 142
a_m/2 3 43 22 7 6 5 16 16 2 9 34 9
s 90 135 154 294 292 159 192 186 130 360 474 238
d 87 133 146 105 272 141 158 162 128 279 346 211
n 28 29 30 31 32 33 34 35 36 37 38
r 14 14 6 15 4 6 6 21 6 18 18
m 36 38 12 24 8 14 8 52 8 34 32
q 50 92 10 178 4 14 6 116 10 196 202
a_m 148 152 158 164 168 174 178 184 190 194 200
a_m/2 8 20 38 81 41 2 44 3 10 26 50
s 312 390 228 657 219 216 276 513 260 662 720
d 262 298 218 479 215 202 270 397 250 466 518
n 39 40 41 42 43 44 45 46 47 48 49
r 7 6 21 14 22 6 3 6 23 2 49
m 14 16 52 32 46 14 2 14 52 2 120
q 39 8 46 42 242 42 17 42 76 1 406
a_m 206 210 216 222 226 232 238 242 248 252 258
a_m/2 13 2 107 54 32 32 17 34 123 1 129
s 292 252 539 404 816 324 272 336 663 254 1293
d 253 244 493 362 574 282 255 294 587 253 887
n 50 51 52 53 54 55 56 57 58 59 60
r 30 3 14 26 18 6 14 9 14 29 6
m 64 4 32 50 44 12 32 18 34 58 12
q 122 2 72 198 46 38 34 53 90 199 6
a_m 264 268 274 280 284 290 296 300 306 312 316
a_m/2 66 13 16 40 70 2 4 21 42 21 18
s 744 285 510 816 556 372 426 426 528 880 372
d 622 283 438 618 510 334 392 373 438 681 366
n 61 62 63 64 65 66 67 68 69 70 71
r 31 30 21 8 21 6 34 6 6 42 9
m 70 52 48 12 50 18 66 8 14 96 18
q 345 368 68 10 73 16 750 10 28 202 55
a_m 322 328 332 338 342 348 354 358 364 370 374
a_m/2 23 82 1 83 23 4 50 22 4 92 25
s 1164 1350 613 489 642 390 1410 420 426 1104 510
d 819 982 545 479 569 374 660 410 398 902 455
n 72 73 74 75 76 77 78 79 80 81 82
r 6 37 18 15 18 42 14 20 6 27 42
m 8 64 42 30 36 96 32 34 12 58 96
q 6 512 148 67 124 274 62 258 10 127 122
a_m 380 386 390 396 402 406 412 418 422 428 432
a_m/2 4 193 54 27 24 2 102 58 1 29 108
s 478 1815 810 612 786 1206 664 1062 477 862 1164
d 472 1303 662 545 662 932 602 804 467 735 1042
n 83 84 85 86 87 88 89 90 91 92 93
r 41 14 3 22 14 6 45 6 7 6 15
m 94 28 4 42 30 10 112 8 16 10 32
q 321 38 2 434 82 70 120 18 22 78 76
a_m 438 444 448 454 460 464 470 476 480 486 492
a_m/2 31 26 7 64 6 66 235 118 1 68 27
s 1344 608 459 1458 644 618 1287 632 537 648 705
d 1023 570 457 1024 562 548 1167 614 515 570 629
n 94 95 96 97 98 99 100 101 102 103 104
r 46 9 4 49 98 6 30 17 6 51 14
m 104 18 4 108 228 8 72 46 8 108 28
q 178 71 2 690 872 14 108 85 6 740 50
a_m 496 502 506 512 518 522 528 534 538 544 550
a_m/2 124 35 125 255 128 4 32 37 134 271 8
s 1402 654 635 2431 2932 580 1026 774 676 2615 714
d 1224 583 633 1741 2060 566 918 689 670 1875 664
n 105 106 107 109 113 127 131 137 139
r 21 26 18 54 28 4 13 34 23
m 44 58 42 114 58 2 34 66 58
q 66 186 164 692 188 96 79 202 163
a_m 554 560 566 576 596 672 692 724 734
a_m/2 1 80 80 82 84 96 49 102 51
s 825 1068 930 2424 1128 864 894 1414 1140
d 759 882 766 1732 940 768 815 1212 977
m = 2r for n = 2, 3, 5, 6, 9, 12, 13, 15, 20,
30, 32, 39, 55, 57, 59, 60, 71,
75, 76, 80, 84, 95, 104 and
what other values?
m = r for n = 8, 16, 24, 48, 96 and any other
values?
is it true that r \leq m \leq 4r ? [No! n = 127.
Note also n = 96,
numerologists will note that 127 = 2^7 - 1 ]
a_m/2 = 1 for n = 1, 3, 5, 48, 63, 80, 91,
105 and what other values?
a_m/2 = 2 for n = 2, 4, 12, 24, 33, 40, 55,
77 and any others?
a_m / a_m/2 is close to an integer i for
n = 3 7 8 10 11 14 17 19 22 26 30 31 32 34
i = 14 2 7 4 7 4 2 14 7 4 4 2 4 4
n 37 38 41 42 43 44 45 46 47 49 50 53 54 57 58
i 7 4 2 4 7 7 14 7 2 2 4 7 4 14 7
n 60 61 62 64 67 70 73 78 79 82 83 86 88 89 90
1 14 14 4 4 7 4 2 4 7 4 14 7 7 2 4
n 92 94 96 97 98 102 103 106 107 109 113 127 137
i 7 4 4 2 4 4 2 7 7 7 7 7 7
Is the average size, s/q, of the partial quotients
of the negs more often exactly an integer, j,
than you'd expect?
n 2 3 6 8 12 16 24 34 36 45 46 48 60 68 84
j 4 16 20 9 34 30 65 46 26 16 8 254 62 42 16
n 113 127 137 These last three seem
j 6 9 7 especially spectacular
End.
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