A075536 - OEIS

%I #13 Sep 08 2022 08:45:07

%S 0,1,1,7,4,21,13,71,44,241,149,815,504,2757,1705,9327,5768,31553,

%T 19513,106743,66012,361109,223317,1221623,755476,4132721,2555757,

%U 13980895,8646064,47297029,29249425,160004703,98950096,541292033,334745777

%N a(n) = ((1+(-1)^n)*T(n+1) + (1-(-1)^n)*S(n))/2, where T(n) = tribonacci numbers A000073, S(n) = generalized tribonacci numbers A001644.

%H G. C. Greubel, <a href="/A075536/b075536.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 3, 0, 1, 0, 1).

%F a(2n) = A073717(n) = A000073(2n+1).

%F a(2n+1) = A001644(2n+1).

%F a(n) = 3*a(n-2) + a(n-4) + a(n-6), a(0)=0, a(1)=1, a(2)=1, a(3)=7, a(4)=4, a(5)=21.

%F O.g.f.: x*(1 + x + 4*x^2 + x^3 - x^4)/(1 - 3*x^2 - x^4 - x^6).

%p A075536 := proc(n)

%p     if type(n,'even') then

%p         A000073(n+1) ;

%p     else

%p         A001644(n) ;

%p     end if;

%p end proc:

%p seq(A075536(n),n=0..80) ; # _R. J. Mathar_, Aug 05 2021

%t CoefficientList[Series[(x+x^2+4x^3+x^4-x^5)/(1-3x^2-x^4-x^6), {x, 0, 40}], x]

%t LinearRecurrence[{0,3,0,1,0,1},{0,1,1,7,4,21},40] (* _Harvey P. Dale_, Jul 10 2012 *)

%o (PARI) my(x='x+O('x^40)); concat([0], Vec(x*(1+x+4*x^2+x^3-x^4)/(1-3*x^2-x^4-x^6))) \\ _G. C. Greubel_, Apr 21 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x+4*x^2+x^3-x^4)/(1-3*x^2-x^4-x^6) )); // _G. C. Greubel_, Apr 21 2019

%o (Sage) (x*(1+x+4*x^2+x^3-x^4)/(1-3*x^2-x^4-x^6)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 21 2019

%Y Cf. A000073, A001644, A005013, A005247.

%K easy,nonn

%O 0,4

%A Mario Catalani (mario.catalani(AT)unito.it), Sep 23 2002

%E Index in definition corrected. - _R. J. Mathar_, Aug 05 2021