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 A321664 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. 1
 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, 89, 55, 143, 144, 89, 232, 233, 144, 376, 377, 233, 609, 610, 377, 986, 987, 610, 1596, 1597, 987, 2583, 2584, 1597, 4180, 4181, 2584, 6764, 6765, 4181, 10945 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. LINKS Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,0,0,0,-1). FORMULA G.f.: (1 + x + x^2 + x^3 + x^4)/(1 - x^3 - x^6) - 1/(1 - x^3). G.f.: (x + x^2 + x^3 - x^5 - x^7)/(1 - 2*x^3 + x^9). a(3*n) = A000045(n+2)-1, a(3*n+1) =  A000045(n+2), a(3*n+2) =  A000045(n+1). a(n) = 2*a(n-3) - a(n-9). - G. C. Greubel, Dec 04 2018 EXAMPLE 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. MAPLE 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 MATHEMATICA CoefficientList[Series[(x+x^2+x^3-x^5-x^7)/(1-2x^3+x^9), {x, 0, 20}], x] 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 *) PROG (Python) def a(n):     if n<6:         return [0, 1, 1, 1, 2, 1][n]     return a(n-3)+a(n-6)+[1, 0, 0][n%3] (Racket) (define (f x) (cond [(< x 6) (list-ref (list 0 1 1 1 2 1) x)] [else (+ (f (- x 3)) (f (- x 6)) (list-ref (list 1 0 0) (remainder x 3)))])) (MAGMA) m:=70; R:=PowerSeriesRing(Integers(), m);  cat Coefficients(R!((x+x^2+x^3-x^5-x^7)/(1-2*x^3+x^9))); // Vincenzo Librandi, Nov 29 2018 (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 (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 CROSSREFS Exhibits a property shared with multiples of A000931. Sequence in context: A087154 A029839 A082304 * A250099 A241949 A288126 Adjacent sequences:  A321661 A321662 A321663 * A321665 A321666 A321667 KEYWORD nonn AUTHOR David Nacin, Nov 23 2018 STATUS approved

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Last modified December 7 22:50 EST 2021. Contains 349590 sequences. (Running on oeis4.)