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A074584
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Esanacci ("6-anacci") numbers.
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12
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| 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 esamatrix ((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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REFERENCES
| Mario Catalani, "Polymatrix and Generalized Polynacci Numbers", paper in progress.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
M. Catalani, Polymatrix and Generalized Polynacci Numbers
E. Weisstein, Fibonacci n-Step
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FORMULA
| 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.: (6-5x-4x^2-3x^3-2x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6)
For a(0)=6, a(1)=1, a(2)=3, a(3)=7, a(4)=15, a(5)=31, a(6)=63, a(n)=2*a(n-1)-a(n-7)[From Vincenzo Librandi, Dec 20 2010]
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MATHEMATICA
| 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}, 50] (* From Harvey P. Dale, Nov 08 2011 *)
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CROSSREFS
| Cf. A000078, A001630, A001644, A000032, A073817, A074048.
Sequence in context: A021167 A085677 A188859 * A195478 A176399 A181552
Adjacent sequences: A074581 A074582 A074583 * A074585 A074586 A074587
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KEYWORD
| easy,nonn
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AUTHOR
| Mario Catalani (mario.catalani(AT)unito.it), Aug 26 2002
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