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 A074584 Esanacci (hexanacci or "6-anacci") numbers. 18
 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, 3299217791, 6544271807 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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 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). LINKS T. D. Noe, Table of n, a(n) for n=0..200 Martin Burtscher, Igor Szczyrba, and 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. Spiros D. Dafnis, Andreas N. Philippou, and Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487. 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. Eric Weisstein's World of Mathematics, Fibonacci n-Step. Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1). 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.: (6-5*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 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}, 40] (* Harvey P. Dale, Nov 08 2011 *) PROG (PARI) polsym(polrecip(1-x-x^2-x^3-x^4-x^5-x^6), 40) \\ G. C. Greubel, Apr 22 2019 (Magma) R:=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 (Python) def aupton(nn):   alst = [6, 1, 3, 7, 15, 31]   for n in range(6, nn+1): alst.append(sum(alst[n-6:n]))   return alst[:nn+1] print(aupton(33)) # Michael S. Branicky, Jun 01 2021 CROSSREFS Cf. A000078, A001630, A001644, A000032, A073817, A074048. Sequence in context: A296476 A296504 A188859 * A195478 A259731 A176399 Adjacent sequences:  A074581 A074582 A074583 * A074585 A074586 A074587 KEYWORD easy,nonn AUTHOR Mario Catalani (mario.catalani(AT)unito.it), Aug 26 2002 STATUS approved

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Last modified November 26 07:57 EST 2022. Contains 358354 sequences. (Running on oeis4.)