A321664 - OEIS

%I #40 Mar 21 2023 15:41:28

%S 0,1,1,1,2,1,2,3,2,4,5,3,7,8,5,12,13,8,20,21,13,33,34,21,54,55,34,88,

%T 89,55,143,144,89,232,233,144,376,377,233,609,610,377,986,987,610,

%U 1596,1597,987,2583,2584,1597,4180,4181,2584,6764,6765,4181,10945

%N A sequence consisting of three disjoint copies of the Fibonacci sequence, one shifted, with the property that for any four consecutive terms the maximum term is the sum of the two minimum terms.

%C 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 Michael De Vlieger, <a href="/A321664/b321664.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,0,0,0,-1).

%F G.f.: (1 + x + x^2 + x^3 + x^4)/(1 - x^3 - x^6) - 1/(1 - x^3).

%F G.f.: (x + x^2 + x^3 - x^5 - x^7)/(1 - 2*x^3 + x^9).

%F a(3*n) = A000045(n+2)-1, a(3*n+1) =  A000045(n+2), a(3*n+2) =  A000045(n+1).

%F a(n) = 2*a(n-3) - a(n-9). - _G. C. Greubel_, Dec 04 2018

%e For n=13, as n is 1 (mod 3), we find a(3*4+1) is the 4+2=6th Fibonacci number which is 8.

%p seq(coeff(series(((x^4+x^3+x^2+x+1)/(1-x^3-x^6))-(1/(1-x^3)),x,n+1), x, n), n = 0 .. 60); # _Muniru A Asiru_, Nov 29 2018

%t CoefficientList[Series[(x+x^2+x^3-x^5-x^7)/(1-2x^3+x^9), {x, 0, 20}], x] (* or *)

%t LinearRecurrence[{0,0,2,0,0,0,0,0,-1}, {0,1,1,1,2,1,2,3,2}, 50] (* _G. C. Greubel_, Dec 04 2018 *)

%o (Python)

%o def a(n):

%o     if n<6:

%o         return [0,1,1,1,2,1][n]

%o     return a(n-3)+a(n-6)+[1,0,0][n%3]

%o (Racket)

%o (define (f x) (cond [(< x 6) (list-ref (list 0 1 1 1 2 1) x)]

%o [else (+ (f (- x 3)) (f (- x 6)) (list-ref (list 1 0 0) (remainder x 3)))]))

%o (Magma) m:=70; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((x+x^2+x^3-x^5-x^7)/(1-2*x^3+x^9))); // _Vincenzo Librandi_, Nov 29 2018

%o (PARI) my(x='x+O('x^70)); Vec((x+x^2+x^3-x^5-x^7)/(1-2*x^3+x^9)) \\ _G. C. Greubel_, Dec 04 2018

%o (Sage) s=((x+x^2+x^3-x^5-x^7)/(1-2*x^3+x^9)).series(x, 70); s.coefficients(x, sparse=False) # _G. C. Greubel_, Dec 04 2018

%Y Exhibits a property shared with multiples of A000931.

%K nonn

%O 0,5

%A _David Nacin_, Nov 23 2018