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A074584 Esanacci (hexanacci or "6-anacci") numbers. 18

%I

%S 6,1,3,7,15,31,63,120,239,475,943,1871,3711,7359,14598,28957,57439,

%T 113935,225999,448287,889215,1763832,3498707,6939975,13766015,

%U 27306031,54163775,107438335,213112838,422726969,838513963,1663261911

%N Esanacci (hexanacci or "6-anacci") numbers.

%C These esanacci numbers follow the same pattern as Lucas, generalized tribonacci (A001644), generalized tetranacci (A073817), and generalized pentanacci (A074048) numbers.

%C The closed form is a(n) = r1^n + r^2^n + r3^n + r4^n + r5^n + r6^n, with r1, r2, r3, r4, r5, r6 roots of the characteristic polynomial.

%C a(n) is also the trace of A^n, where A is the matrix ((1, 1, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0), (1, 0, 0, 1, 0, 0), (1, 0, 0, 0, 1, 0), (1, 0, 0, 0, 0, 1), (1, 0, 0, 0, 0, 0).

%H T. D. Noe, <a href="/A074584/b074584.txt">Table of n, a(n) for n=0..200</a>

%H Martin Burtscher, Igor Szczyrba, RafaƂ Szczyrba, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

%H Mario Catalani, <a href="http://arxiv.org/abs/math/0210201">Polymatrix and Generalized Polynacci Numbers</a>, arXiv:math/0210201 [math.CO], 2002.

%H Tony D. Noe and Jonathan Vos Post, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Noe/noe5.html">Primes in Fibonacci n-step and Lucas n-step Sequences,</a> J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

%H E. Weisstein, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1,1).

%F a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6), a(0)=6, a(1)=1, a(2)=3, a(3)=7, a(4)=15, a(5)=31.

%F G.f.: (6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6).

%F a(n) = 2*a(n-1) - a(n-7) for n >= 7. - _Vincenzo Librandi_, Dec 20 2010

%t CoefficientList[Series[(6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6), {x, 0, 40}], x]

%t LinearRecurrence[{1,1,1,1,1,1},{6,1,3,7,15,31},40] (* _Harvey P. Dale_, Nov 08 2011 *)

%o (PARI) polsym(polrecip(1-x-x^2-x^3-x^4-x^5-x^6), 40) \\ _G. C. Greubel_, Apr 22 2019

%o (MAGMA) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6) )); // _G. C. Greubel_, Apr 22 2019

%o (Sage) ((6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 22 2019

%Y Cf. A000078, A001630, A001644, A000032, A073817, A074048.

%K easy,nonn

%O 0,1

%A Mario Catalani (mario.catalani(AT)unito.it), Aug 26 2002

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Last modified April 3 19:43 EDT 2020. Contains 333198 sequences. (Running on oeis4.)