A099925 - OEIS

%I #18 Sep 17 2020 06:19:02

%S 3,0,4,3,8,10,19,28,48,75,124,198,323,520,844,1363,2208,3570,5779,

%T 9348,15128,24475,39604,64078,103683,167760,271444,439203,710648,

%U 1149850,1860499,3010348,4870848,7881195,12752044,20633238,33385283,54018520,87403804

%N a(n) = Lucas(n) + (-1)^n.

%C Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio and put c = sum {n = 1..inf} 1/2^floor(n*(phi + 2)). The bicimal expansion of the constant c begins 0.001000100100010001001.... The binary digits are the generalized Fibonacci word A221150.

%C The sequence 2^a(n) for n >= 1 gives the partial quotients, apart from the first, in the simple continued fraction expansion of the constant 1/2*c = 0.06692 72114 83804 90296 ... = 1/(14 + 1/(2^0 + 1/(2^4 + 1/(2^3 + 1/(2^8 + 1/(2^10 + 1/(2^19 + ...))))))). Cf. A008346. - _Peter Bala_, Nov 06 2013

%H Colin Barker, <a href="/A099925/b099925.txt">Table of n, a(n) for n = 0..1000</a>

%H Yüksel Soykan, <a href="https://doi.org/10.9734/AJARR/2020/v11i230259">Generalized Pell-Padovan Numbers</a>, Asian Journal of Advanced Research and Reports (2020) Vol. 11, No. 2, 8-28, Article No. 57839.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1).

%F G.f.: (3-2*x^2)/((1+x)*(1-x-x^2)).

%F a(0) = 3, a(1) = 0, a(2) = 4 and a(n) = 2*a(n-2) + a(n-3) for n >= 3. - _Peter Bala_, Nov 06 2013

%F a(n) = A068397(n) - 1 for n>2.

%F a(n) = ((-1)^n+(1/2*(1-sqrt(5)))^n+(1/2*(1+sqrt(5)))^n). - _Colin Barker_, Jun 03 2016

%t CoefficientList[Series[(3 - 2 x^2)/((1 + x) (1 - x - x^2)), {x, 0, 38}], x] (* _Michael De Vlieger_, Sep 16 2020 *)

%o (PARI) Vec((3-2*x^2)/((1+x)*(1-x-x^2)) + O(x^40)) \\ _Colin Barker_, Jun 03 2016

%Y Cf. A000032, A008346, A098600, A068397.

%K nonn,easy

%O 0,1

%A _Ralf Stephan_, Nov 02 2004