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A074584
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Esanacci (hexanacci or "6-anacci") numbers.
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18
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6, 1, 3, 7, 15, 31, 63, 120, 239, 475, 943, 1871, 3711, 7359, 14598, 28957, 57439, 113935, 225999, 448287, 889215, 1763832, 3498707, 6939975, 13766015, 27306031, 54163775, 107438335, 213112838, 422726969, 838513963, 1663261911
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OFFSET
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0,1
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COMMENTS
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These esanacci numbers follow the same pattern as Lucas, generalized tribonacci (A001644), generalized tetranacci (A073817), and generalized pentanacci (A074048) numbers.
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.
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
E. Weisstein, Fibonacci n-Step
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1).
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FORMULA
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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.
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).
a(n) = 2*a(n-1) - a(n-7) for n >= 7. - Vincenzo Librandi, Dec 20 2010
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MATHEMATICA
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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]
LinearRecurrence[{1, 1, 1, 1, 1, 1}, {6, 1, 3, 7, 15, 31}, 40] (* Harvey P. Dale, Nov 08 2011 *)
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PROG
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(PARI) polsym(polrecip(1-x-x^2-x^3-x^4-x^5-x^6), 40) \\ G. C. Greubel, Apr 22 2019
(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
(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
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CROSSREFS
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Cf. A000078, A001630, A001644, A000032, A073817, A074048.
Sequence in context: A296476 A296504 A188859 * A195478 A259731 A176399
Adjacent sequences: A074581 A074582 A074583 * A074585 A074586 A074587
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KEYWORD
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easy,nonn
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AUTHOR
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Mario Catalani (mario.catalani(AT)unito.it), Aug 26 2002
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STATUS
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approved
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