Hello Math-Fun, I've sent this 5 days ago to SeqFans but it seems that it never reached the list. I try here, Best, É. ---------------------------------------------------- Hello (SeqFans) Math-Fun, I was looking for a seq S (of distinct integers) where one could pick any 4 successive terms a, b, c, d and have a/b = c/d. I quickly found https://oeis.org/A038754 [Original definition of the seq is between brackets] [a(2n) = 3^n, a(2n+1) = 2*3^n]. S = 1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486,... The above seq starts with 1,2,3. The start 1,2,4 produces the powers of 2. The start 1,2,5 produces https://oeis.org/A026383 [a(n) = 5a(n-2), starting 1,2]. The start 1,2,6, produces https://oeis.org/A026549 [Ratios of successive terms are 2,3,2,3,2,3,2,3...] The start 1,2,7 produces https://oeis.org/A123752 [a(n) = 7*a(n-2), a(0) = 1, a(1) = 2]. The start 1,2,8 produces https://oeis.org/A098232 [Largest power of 2 <= 3^n]. The start 1,2,9 produces https://oeis.org/A083423 [a(n) = (5*3^n + (-3)^n)/6]. The start 1,2,10 produces https://oeis.org/A004643 [Powers of 2 written in base 4]. etc. (I love the rainbow of definitions!) The start 1,3,2 produces https://oeis.org/A164073 [a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 3]. The start 1,3,4 produces https://oeis.org/A084221 [a(n+2) = 4*a(n), with a(0)=1, a(1)=3]. The start 1,3,5 produces https://oeis.org/A056487 [a(n) = 5^(n/2) for n even, a(n) = 3*5^((n-1)/2) for n odd]. ....(!?) and produces also https://oeis.org/A111386 [a(1) = 1, a(2) = 3; for n >= 3, take a(n) to be the smallest odd number not occurring earlier such that a(n-1) divides the concatenation a(n-2)a(n)]. The start 1,3,6 produces https://oeis.org/A026532 [Ratios of successive terms are 3,2,3,2,3,2,3,2...] The start 1,3,7 produces 1,3,7,21,49,147,343,1029,... _not in the OEIS_. The start 1,3,8 produces https://oeis.org/A096886 [Expansion of (1+3*x)/(1-8*x^2)]. The start 1,3,9 produces the powers of 3 The start 1,3,10 produces https://oeis.org/A004663 [Powers of 3 written in base 9]. etc. The start 1,4,2 produces https://oeis.org/A143095 [(1, 2, 4, 8,...) interleaved with (4, 8, 16, 32,...)]. But we have repeated terms there. The start 1,4,3 produces https://oeis.org/A166552 [a(n) = 3*a(n-2) for n > 2; a(1) = 1; a(2) = 4] The start 1,4,5 produces https://oeis.org/A133632 [a(1)=1, a(n)=(p-1)*a(n-1), if n is even, else a(n)=p*a(n-2), where p=5]. The start 1,4,6 produces https://oeis.org/A164532 [a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 4]. The start 1,4,7 produces 1,4,7,28,49,196,343,1372,... _nt in the OEIS_ The start 1,4,8 produces https://oeis.org/A094015 [Expansion of (1+4x)/(1-8x^2)]. The start 1,4,9 produces https://oeis.org/A133125 [A133080 * A000244]. The start 1,4,10 produces https://oeis.org/A136859 [Numbers n such that n and the square of n use only the digits 0, 1, 4 and 6]. etc. (another rainbow!) The start 1,5,2 produces 1,5,2,10,4,20,8,40,16,80,32,160,... _not in the OEIS_. The start 1,5,3 produces https://oeis.org/A166465 [a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 5]. The start 1,5,4 produces 1,5,4,20,16,80,64,320,256,... _not in the OEIS_. The start 1,5,6 produces https://oeis.org/A166023 [a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 5] The start 1,5,7 produces 1,5,7,35,49,245,343,... _not in the OEIS_. The start 1,5,8 produces 1,5,8,40,64,320,512,... _not in the OEIS_. The start 1,5,9 produces 1,5,9,45,81,405,729,... _not in the OEIS_. The start 1,5,10 produces https://oeis.org/A268100 [a(n) = 2^((n-1) mod 2)*5*10^floor((n-1)/2)] etc. The start 1,6,2 produces https://oeis.org/A163864 [a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 6]. The start 1,6,3 produces https://oeis.org/A166450 [a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 6]. The start 1,6,4 produces https://oeis.org/A081631 [2*2^n-(-2)^n]. The start 1,6,5 produces 1,6,5,30,25,150,125,750,... _not in the OEIS_. The start 1,6,7 produces 1,6,7,42,49,... _not in the OEIS_. The start 1,6,8 produces https://oeis.org/A164640 [a(n) = 8*a(n-2) for n > 2; a(1) = 1, a(2) = 6]. The start 1,6,9 produces 1,6,9,54,81,... _not in the OEIS_. The start 1,6,10 produces 1,6,10,60,100,600,1000,6000,10000 _not in the OEIS_. etc. The start 1,7,2 produces 1,7,2,14,4,28,8,56,16,112,32,224,... _not in the OEIS_. The start 1,7,3 produces https://oeis.org/A166481 [a(n) = 3*a(n-2) for n > 2; a(1) = 1; a(2) = 7]. The start 1,7,4 produces 1,7,4,28,16,96,64,... _not in the OEIS_. The start 1,7,5 produces 1,7,5,35,25,175,... _not in the OEIS_. The start 1,7,6 produces 1,7,6,42,36,252,... _not in the OEIS_. The start 1,7,8 produces 1,7,8,56,64,... _not in the OEIS_. The start 1,7,9 produces 1,7,9,63,81,... _not in the OEIS_. The start 1,7,10 produces 1,7,10,70,100,700,1000,7000,... _not in the OEIS_. etc. The start 1,8,2 produces https://oeis.org/A164587 [a(n) = 2*a(n - 2) for n > 2; a(1) = 1, a(2) = 8]. The start 1,8,3 produces 1,8,3,24,9,72,27,216,81,648,243,... _not in the OEIS_. The start 1,8,4 produces almost https://oeis.org/A135520 [erase the 1st term, then a(n) = 4*a(n-2)]. The start 1,8,5 produces 1,8,5,40,25,200,125,... _not in the OEIS_. The start 1,8,6 produces 1,8,6,48,36,... _not in the OEIS_. The start 1,8,7 produces 1,8,7,56,49,... _not in the OEIS_. The start 1,8,9 produces 1,8,9,72,81,... _not in the OEIS_. The start 1,8,10 produces 1,8,10,80,100,800,1000,... _not in the OEIS_. etc. The start 1,9,2 produces 1,9,2,18,4,36,8,72,16,144,32,288,... _not in the OEIS_. The start 1,9,3 produces almost https://oeis.org/A162852 [a(n) = 3*a(n-2) for n > 2; a(1) = 3, a(2) = -1]. But, again, a lot of not-distinct terms. The start 1,9,4 produces 1,9,4,36,16,144,... _not in the OEIS_. The start 1,9,5 produces 1,9,5,45,25,225,... _not in the OEIS_. The start 1,9 6 produces 1,9,6,54,36,... _notin the OEIS_. The start 1,9,7 produces 1,9,7,63,49,... _not in the OEIS_. The start 1,9,8 produces 1,9,8,72,64,... _not in the OEIS_. The start 1,9,10 produces 1,9,10,90,100,900,1000,... _not in the OEIS_. etc. The start 1,10,2 produces 1,10,2,20,4,40,8,80,16,160,32,320,... _not in the OEIS_. The start 1,10,3 produces 1,10,3,30,9,90,27,270,... _not in the OEIS_. etc. The start 2,1,6 produces 2,1,6,3,18,9,54,27,... _not in the OEIS_. As 19 is prime, the start 20,19 produces 20,19,40,38,80,72,... _not in the OEIS_ (and never in the OEIS, I guess). etc. Note that the remark here https://oeis.org/A164073, by a friend of mine ("Absolute second differences are the sequence itself"), is always true for some seqs above that follow the pattern: 1, a, 2, 2a, 4, 4a, 8, 8a, 16, 16a... for a = 0 to k. Oh, and what about picking 5 (instead of 4) successive integers a, b, c, d, e in a seq of distinct terms such that a/b is always equal to d/e? Never mind, this has been already submitted by Paul Curtz: https://oeis.org/A133464 Too good, the OEIS'authors!-)) My two,three, four cents. É. P.-S.#1 Many thanks to Klaus Brockhaus who computed 10 years ago a lot of the above, and forgive the errors I left in this mail, despite my carefull rerereading). P.-S.#2 Let me pick again (in a similar seq T) 5 successive integers a, b, c, d, e -- but such that a/c is now equal to d/e: T = 1, 2, 3, 4, 12, 24, 96, 576,... Ah ah, _T is not in the OEIS_!-)) Best, É.