Quoting Neil Sloane <njasloane@gmail.com>:
Stern's diatomic array (A049456) is a triangle in which row n lists the denominators in the n-th row of the table of Farey fractions. Another version of A049456 is the Stern-Brocot sequence (A002487) which has the recurrence a(2n)=a(n), a(2n+1)=a(n)+a(n+1), with a(0)=0, a(1)=1. There are a huge number of references. Someone recently asked me about a sequence which I was able to show was equal to the number of distinct terms in the n-th row of Stern's A049456. Surprisingly, this was not in the OEIS, although it is now - see A293160: 1, 2, 3, 5, 7, 13, 20, 31, 48, 78, 118, 191, 300, 465, 734, 1175, 1850, 2926, 4597, 7296, 11552, 18278, 28863, 45832, 72356, 114742, ... Can someone find a recurrence? (The smallest positive missing number from row n is A135510, which has an 1850 reference to Eisenstein.)
Some preliminaries to look for clues ... The ratios of successive terms 2.0 1.5 1.6666666 1.4 1.8571428 1.5384616 1.55 1.548387 1.625 1.5128205 1.6186441 1.5706806 1.55 1.5784947 1.6008174 1.5744681 1.5816216 1.5710868 1.5871221 1.5833334 1.5822369 1.5791115 1.5879153 1.5787222 1.585798 These are mostly an up & down sequence, but with exceptions. The ratios appear to be converging, but slowly, somewhat erraticly. The ratio of term21:term20 is exactly 19:12, rather simple. A simple linear recurrence would have roughly a geometric convergence to the term ratio, similar to the Fibonacci sequence. This sequence doesn't. The remainders mod 2 and 3 are (1 0 1 1 1 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 0) (1 2 0 2 1 1 2 1 0 0 1 2 0 0 2 2 2 1 1 0 2 2 0 1 2 1) with no obvious periodicity. The factorizations are "nothing special", with perhaps a minor excess of small factors 2,3,5, and maybe 19. (1 (1)) (2 (2)) (3 (3)) (5 (5)) (7 (7)) (13 (13)) (20 (2 2 5)) (31 (31)) (48 (2 2 2 2 3)) (78 (2 3 13)) (118 (2 59)) (191 (191)) (300 (2 2 3 5 5)) (465 (3 5 31)) (734 (2 367)) (1175 (5 5 47)) (1850 (2 5 5 37)) (2926 (2 7 11 19)) (4597 (4597)) (7296 (2 2 2 2 2 2 2 3 19)) (11552 (2 2 2 2 2 19 19)) (18278 (2 13 19 37)) (28863 (3 3 3 1069)) (45832 (2 2 2 17 337)) (72356 (2 2 18089)) (114742 (2 103 557)) The balanced remainder of the Nth term mod N looks a little unusual. Random would be |average| of N/4; these are a little less. No obvious pattern of the signs. (1 1 0) (2 2 0) (3 3 0) (4 5 1) (5 7 2) (6 13 1) (7 20 -1) (8 31 -1) (9 48 3) (10 78 -2) (11 118 -3) (12 191 -1) (13 300 1) (14 465 3) (15 734 -1) (16 1175 7) (17 1850 -3) (18 2926 -8) (19 4597 -1) (20 7296 -4) (21 11552 2) (22 18278 -4) (23 28863 -2) (24 45832 -8) (25 72356 6) (26 114742 4) Rich