cp's OEIS Frontend

A117294 Number of sequences of length n starting with 1,2 which satisfy a recurrence a(k+1) = floor(c*a(k)) for some constant c.

%I A117294 #23 Jul 28 2025 09:01:20
%S A117294 1,2,5,14,37,102,279,756,2070,5609,15198,41530,114049,315447,876513,
%T A117294 2446326,6861432,19315953,54556553,154591186,439307113,1251678183,
%U A117294 3574777087,10231666185,29343549576,84309936418,242651784699,699476361863,2019289119525,5837355573611,16896103820563,48963682959055,142051622347551
%N A117294 Number of sequences of length n starting with 1,2 which satisfy a recurrence a(k+1) = floor(c*a(k)) for some constant c.
%C A117294 It appears that a(n+1)/a(n) may be converging slowly to 3, but even that it converges is not obvious.
%C A117294 From _Martin Fuller_, Apr 05 2025: (Start)
%C A117294 Each finite sequence corresponds to a range of c, e.g. (1,2,4,9) has c in [9/4,5/2).
%C A117294 Let x=(maximum element)*(range width of c), e.g. x(1,2,4,9)=9*(5/2 - 9/4)=9/4.
%C A117294 Then the number of extensions of this sequence in a(n+1) is either floor(x)+1 or ceil(x)+1.  Non-rigorously, the expected number is x+1.
%C A117294 Also x is bounded 0<x<=c for all finite sequences with the same c.
%C A117294 Does X have a distribution for large n and approximate c? Can this be used to work out the growth rate of a(n)? (End)
%H A117294 Martin Fuller, <a href="/A117294/a117294.html.txt">Graph of sequences counted by A117294</a>
%H A117294 Martin Fuller, <a href="/A117294/a117294.cpp.txt">C++ program</a>
%e A117294 a(4) = 5: length 4 sequences are 1,2,4,8; 1,2,4,9; 1,2,5,12; 1,2,5,13; and 1,2,5,14.
%o A117294 (Racket) (define (A117294 n) (local ((define (get-ratios seq add?) (cond [(empty? (rest seq)) empty] [else (cons (/ (cond [add? (add1 (first seq))] [else (first seq)]) (second seq)) (get-ratios (rest seq) add?))])) (define (extend-one seq) (local ((define startnext (floor (* (apply max (get-ratios seq false)) (first seq)))) (define endnext (ceiling (* (apply min (get-ratios seq true )) (first seq)))) (define ltodo (build-list (- endnext startnext) (lambda (n) (cons (+ startnext n) seq))))) (cond [(>= (length seq) (sub1 n)) (length ltodo)] [else (apply + (map extend-one ltodo))])))) (extend-one (list 2 1)))) ;; _Joshua Zucker_, Jun 05 2006
%Y A117294 Some (infinite) examples of such sequences: A000079, A007051, A076883, A001519, A024537, A024576, A057960.
%K A117294 nonn
%O A117294 2,2
%A A117294 _Franklin T. Adams-Watters_, Apr 26 2006
%E A117294 More terms from _Joshua Zucker_, Jun 05 2006
%E A117294 Comment edited by _Franklin T. Adams-Watters_, May 14 2010
%E A117294 Ambiguous terms a(0), a(1) removed by _Max Alekseyev_, Jan 18 2012
%E A117294 a(21)-a(24) from _Jinyuan Wang_, Mar 20 2025
%E A117294 a(25)-a(34) from _Martin Fuller_, Apr 05 2025