cp's OEIS Frontend

A104621 Heptanacci-Lucas numbers.

%I A104621 #53 Jan 05 2025 19:51:38
%S A104621 7,1,3,7,15,31,63,127,247,493,983,1959,3903,7775,15487,30847,61447,
%T A104621 122401,243819,485679,967455,1927135,3838783,7646719,15231991,
%U A104621 30341581,60439343,120393007,239818559,477709983,951581183,1895515647,3775799303,7521257025
%N A104621 Heptanacci-Lucas numbers.
%C A104621 This 7th-order linear recurrence is a generalization of the Lucas sequence A000032. Mario Catalani would refer to this is a generalized heptanacci sequence, had he not stopped his series of sequences after A001644 "generalized tribonacci", A073817 "generalized tetranacci", A074048 "generalized pentanacci", A074584 "generalized hexanacci." _T. D. Noe_ and I have noted that each of these has many more primes than the corresponding tribonacci A000073 (see A104576), tetranacci A000288 (see A104577), pentanacci, hexanacci and heptanacci (see A104414). For primes in Heptanacci-Lucas numbers, see A104622. For semiprimes in Heptanacci-Lucas numbers, see A104623.
%H A104621 G. C. Greubel, <a href="/A104621/b104621.txt">Table of n, a(n) for n = 0..3300</a> (terms 0..200 from T. D. Noe)
%H A104621 Martin Burtscher, Igor Szczyrba, 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 A104621 Mario Catalani, <a href="http://arxiv.org/abs/math/0210201">Polymatrix and Generalized Polynacci Numbers</a>, arXiv:math/0210201 [math.CO], 2002.
%H A104621 C. A. Charalambides, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/29-4/charalambides.pdf">Lucas numbers and polynomials of order k and the length of the longest circular success run</a>, The Fibonacci Quarterly, 29 (1991), 290-297.
%H A104621 Spiros D. Dafnis, Andreas N. Philippou, 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 A104621 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 A104621 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1,1,1).
%F A104621 a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7); a(0) = 7, a(1) = 1, a(2) = 3, a(3) = 7, a(4) = 15, a(5) = 31, a(6) = 63.
%F A104621 From _R. J. Mathar_, Nov 16 2007: (Start)
%F A104621 G.f.: (7 - 6*x - 5*x^2 - 4*x^3 - 3*x^4 - 2*x^5 - x^6)/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7).
%F A104621 a(n) = 7*A066178(n) - 6*A066178(n-1) - 5*A066178(n-2) - ... - 2*A066178(n-5) - A066178(n-6) if n >= 6. (End)
%p A104621 A104621 := proc(n)
%p A104621     option remember;
%p A104621     if n <=6 then
%p A104621         op(n+1,[7, 1, 3, 7, 15, 31, 63])
%p A104621     else
%p A104621         add(procname(n-i),i=1..7) ;
%p A104621     end if;
%p A104621 end proc: # _R. J. Mathar_, Mar 26 2015
%t A104621 a[0]=7; a[1]=1; a[2]=3; a[3]=7; a[4]=15; a[5]=31; a[6]=63; a[n_]:= a[n]= a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]+a[n-7]; Table[a[n], {n,0,40}] (* _Robert G. Wilson v_, Mar 17 2005 *)
%t A104621 LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {7, 1, 3, 7, 15, 31, 63}, 40] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2012 *)
%o A104621 (PARI) my(x='x+O('x^40)); Vec((-7+6*x+5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6+x^7)) \\ _G. C. Greubel_, Dec 18 2017
%o A104621 (PARI) polsym(polrecip(1-x-x^2-x^3-x^4-x^5-x^6-x^7), 40) \\ _G. C. Greubel_, Apr 22 2019
%o A104621 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (-7+6*x+ 5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6+x^7) )); // _G. C. Greubel_, Apr 22 2019
%o A104621 (Sage) ((-7+6*x+5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6 +x^7)).series(x, 41).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 22 2019
%Y A104621 Cf. A000032, A001644, A073817, A074048, A074584, A104414, A104576, A104577.
%K A104621 easy,nonn
%O A104621 0,1
%A A104621 _Jonathan Vos Post_, Mar 17 2005