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%I #11 Mar 31 2012 13:21:30
%S 1,2,5,14,37,102,279,756,2070,5609,15198,41530,114049,315447,876513,
%T 2446326,6861432,19315953,54556553
%N 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 It appears that a(n+1)/a(n) may be converging slowly to 3, but even that it converges is not obvious.
%e 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 (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 Some (infinite) examples of such sequences: A000079, A007051, A076883, A001519, A024537, A024576, A057960.
%K more,nonn
%O 2,2
%A _Franklin T. Adams-Watters_, Apr 26 2006
%E More terms from _Joshua Zucker_, Jun 05 2006
%E Comment edited by _Franklin T. Adams-Watters_, May 14 2010
%E Ambiguous terms a(0), a(1) removed by _Max Alekseyev_, Jan 18 2012
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