This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A074584 #72 Jun 01 2025 13:16:44
%S A074584 6,1,3,7,15,31,63,120,239,475,943,1871,3711,7359,14598,28957,57439,
%T A074584 113935,225999,448287,889215,1763832,3498707,6939975,13766015,
%U A074584 27306031,54163775,107438335,213112838,422726969,838513963,1663261911,3299217791,6544271807
%N A074584 Esanacci (hexanacci or "6-anacci") numbers.
%C A074584 These esanacci numbers follow the same pattern as Lucas, generalized tribonacci (A001644), generalized tetranacci (A073817), and generalized pentanacci (A074048) numbers.
%C A074584 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 A074584 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 A074584 T. D. Noe, <a href="/A074584/b074584.txt">Table of n, a(n) for n=0..200</a>
%H A074584 Martin Burtscher, Igor Szczyrba, and Rafał 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 A074584 Mario Catalani, <a href="http://arxiv.org/abs/math/0210201">Polymatrix and Generalized Polynacci Numbers</a>, arXiv:math/0210201 [math.CO], 2002.
%H A074584 Spiros D. Dafnis, Andreas N. Philippou, and Ioannis E. Livieris, <a href="https://doi.org/10.3390/math8091487">An Alternating Sum of Fibonacci and Lucas Numbers of Order k</a>, Mathematics (2020) Vol. 9, 1487.
%H A074584 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 A074584 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>.
%H A074584 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1,1).
%F A074584 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 A074584 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 A074584 a(n) = 2*a(n-1) - a(n-7) for n >= 7. - _Vincenzo Librandi_, Dec 20 2010
%t A074584 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 A074584 LinearRecurrence[{1,1,1,1,1,1},{6,1,3,7,15,31},40] (* _Harvey P. Dale_, Nov 08 2011 *)
%o A074584 (PARI) polsym(polrecip(1-x-x^2-x^3-x^4-x^5-x^6), 40) \\ _G. C. Greubel_, Apr 22 2019
%o A074584 (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 A074584 (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
%o A074584 (Python)
%o A074584 def aupton(nn):
%o A074584 alst = [6, 1, 3, 7, 15, 31]
%o A074584 for n in range(6, nn+1): alst.append(sum(alst[n-6:n]))
%o A074584 return alst[:nn+1]
%o A074584 print(aupton(33)) # _Michael S. Branicky_, Jun 01 2021
%Y A074584 Cf. A000078, A001630, A001644, A000032, A073817, A074048.
%K A074584 easy,nonn
%O A074584 0,1
%A A074584 Mario Catalani (mario.catalani(AT)unito.it), Aug 26 2002