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A060455 7th-order Fibonacci numbers with a(0)=...=a(6)=1.

%I A060455 #40 Jul 02 2025 16:02:01
%S A060455 1,1,1,1,1,1,1,7,13,25,49,97,193,385,769,1531,3049,6073,12097,24097,
%T A060455 48001,95617,190465,379399,755749,1505425,2998753,5973409,11898817,
%U A060455 23702017,47213569,94047739,187339729,373174033,743349313,1480725217
%N A060455 7th-order Fibonacci numbers with a(0)=...=a(6)=1.
%C A060455 a(n) = number of runs in polyphase sort using 8 tapes and n-6 phases.
%D A060455 N. Wirth, Algorithmen und Datenstrukturen, 1975 (table 2.15 chapter 2.3.4).
%H A060455 Indranil Ghosh, <a href="/A060455/b060455.txt">Table of n, a(n) for n = 0..3339</a> (terms 0..200 from T. D. Noe)
%H A060455 R. L. Gilstad, <a href="http://www.computer.org/csdl/proceedings/afips/1960/5057/00/50570143.pdf">Polyphase Merge Sort - Advanced Technique</a>, Proc. AFIPS Eastern Jt. Comp. Conf. 18 (1960) 143-148.
%H A060455 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1,1,1).
%F A060455 a(n) = a(n-1) + a(n-2) + ... + a(n-7) for n > 6, a(0)=a(1)=...=a(6)=1.
%F A060455 G.f.: (-1 + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6)/(-1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7). - _R. J. Mathar_, Oct 11 2011
%e A060455 General formula for k-th order numbers: f(n,k) = f(n-1,k) + ... + f(n-1-k,k) for n > k, otherwise f(n,k) = 1.
%p A060455 A060455 := proc(n) option remember: if n >=0 and n<=6 then RETURN(1) fi: procname(n-1)+procname(n-2)+procname(n-3)+procname(n-4)+procname(n-5)+procname(n-6)+procname(n-7) end;
%t A060455 LinearRecurrence[{1,1,1,1,1,1,1},{1,1,1,1,1,1,1},40] (* _Harvey P. Dale_, Mar 17 2012 *)
%o A060455 (PARI) Vec((1-x^2-2*x^3-3*x^4-4*x^5-5*x^6)/(1-x-x^2-x^3-x^4-x^5-x^6-x^7) +O(x^40)) \\ _Charles R Greathouse IV_, Feb 03 2014
%o A060455 (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(  (1-x^2-2*x^3-3*x^4-4*x^5-5*x^6)/(1-x-x^2-x^3-x^4-x^5-x^6-x^7) )); // _G. C. Greubel_, Feb 03 2019
%o A060455 (Sage) ((1-x^2-2*x^3-3*x^4-4*x^5-5*x^6)/(1-x-x^2-x^3-x^4-x^5-x^6-x^7) ).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 03 2019
%Y A060455 For k=1..5 see A000045, A000213, A000288, A000322, A000383.
%Y A060455 Cf. A253333, A253318: primes and indices of primes in this sequence.
%Y A060455 Cf. A122189 Heptanacci numbers with a(0),...,a(6) = 0,0,0,0,0,0,1.
%K A060455 easy,nonn
%O A060455 0,8
%A A060455 _Frank Ellermann_, Apr 08 2001
%E A060455 More terms from _James Sellers_, Apr 11 2001