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A327035 An unbounded sequence consisting solely of Fibonacci numbers with the property that for any four consecutive terms the maximum term is the sum of the two minimum terms.

%I A327035 #26 Mar 04 2024 00:41:57
%S A327035 1,1,0,1,1,1,2,2,1,3,3,2,5,5,3,8,8,5,13,13,8,21,21,13,34,34,21,55,55,
%T A327035 34,89,89,55,144,144,89,233,233,144,377,377,233,610,610,377,987,987,
%U A327035 610,1597,1597,987,2584,2584,1597,4181,4181,2584,6765,6765,4181
%N A327035 An unbounded sequence consisting solely of Fibonacci numbers with the property that for any four consecutive terms the maximum term is the sum of the two minimum terms.
%C A327035 This sequence was constructed to show that there are many sequences, besides those merging with multiples of the Padovan sequence A000931, with the property that for any four consecutive terms the maximum term is the sum of the two minimum terms. This refutes a conjecture that was formerly in that entry.
%H A327035 Michael De Vlieger, <a href="/A327035/b327035.txt">Table of n, a(n) for n = 0..10000</a>
%H A327035 David Nacin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Nacin/nacin5.html">Van der Laan Sequences and a Conjecture on Padovan Numbers</a>, J. Int. Seq., Vol. 26 (2023), Article 23.1.2.
%H A327035 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 1, 0, 0, 1).
%F A327035 G.f.: (x^5 + x + 1)/(-x^6 - x^3 + 1).
%F A327035 a(3*n) = A000045(n+1), a(3*n+1) =  A000045(n+1), a(3*n+2) =  A000045(n).
%F A327035 a(n) = a(n-3) + a(n-6).
%e A327035 For n=7, as n is 3(2)+1, a(n) = A000045(2+1) = 2.
%t A327035 LinearRecurrence[{0, 0, 1, 0, 0, 1}, {1,1,0,1,1,1}, 50]
%o A327035 (Python)
%o A327035 a = lambda x:[1,1,0,1,1,1][x] if x<6 else a(x-3)+a(x-6)
%o A327035 (Racket)
%o A327035 (define (a x) (cond [(< x 6) (list-ref (list 1 1 0 1 1 1) x)]
%o A327035 [else (+ (a (- x 3)) (a (- x 6)))]))
%o A327035 (Sage)
%o A327035 s=((x^5 + x + 1)/(-x^6 - x^3 + 1)).series(x, 23); s.coefficients(x, sparse=False)
%Y A327035 Cf. A321664, A321341, A328943.
%Y A327035 Exhibits a property shared with multiples of A000931.
%K A327035 nonn
%O A327035 0,7
%A A327035 _David Nacin_, Nov 28 2019