A128429 - OEIS

%I #31 Feb 16 2025 08:33:04

%S 1,1,1,1,1,1,4,7,10,16,25,40,67,109,175,283,457,739,1198,1939,3136,

%T 5074,8209,13282,21493,34777,56269,91045,147313,238357,385672,624031,

%U 1009702,1633732,2643433,4277164,6920599,11197765,18118363,29316127,47434489

%N A linear recurrence sequence: a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-6).

%C The characteristic polynomial of this recurrence is x^6 - x^5 - x^3 - x - 1 = (x^2 - x - 1)*(x^6 - 1)/(x^2 - 1), so the sequence can be written as the sum of a Fibonacci sequence and a sequence of period 6; see the formula line. Hence the ratio a(n+1)/a(n) has the same limit as the Fibonacci sequence does, namely the golden ratio, (1+sqrt(5))/2, about 1.61803398874989484820.

%D Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002.

%H Jinyuan Wang, <a href="/A128429/b128429.txt">Table of n, a(n) for n = 0..1000</a>

%H Bruce Rawles, <a href="http://www.intent.com/sg/">Sacred Geometry </a>

%H Kelley L. Ross, <a href="http://www.friesian.com/golden.htm">The Golden Ratio and The Fibonacci Numbers</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Golden_ratio">Golden Ratio</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 1, 0, 1, 1).

%F a(n) = (1/4)*(3F(n-1) + b(n mod 6)), where F(n) = A000045(n) is the n-th Fibonacci number and b(0)=b(2)=b(3)=1, b(1)=4, b(4)=-2 and b(5)=-5.

%F G.f.: (-1 + x^3 + x^4 + 2*x^5)/((x^2 + x - 1)*(1 + x + x^2)*(x^2 - x + 1)). - _R. J. Mathar_, Nov 16 2007

%t LinearRecurrence[{1, 0, 1, 0, 1, 1}, {1, 1, 1, 1, 1, 1}, 41] (* _Jean-François Alcover_, Sep 25 2017 *)

%Y Cf. Fibonacci numbers A000045; Lucas numbers A000032.

%K nonn

%O 0,7

%A Luis A Restrepo (luisiii(AT)mac.com), Mar 05 2007

%E Edited by _Dean Hickerson_ and _Don Reble_, Mar 09 2007