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A075676 Sequences A001644 and A000073 interleaved. 1

%I

%S 3,1,3,2,11,7,39,24,131,81,443,274,1499,927,5071,3136,17155,10609,

%T 58035,35890,196331,121415,664183,410744,2246915,1389537,7601259,

%U 4700770,25714875,15902591,86992799,53798080,294294531,181997601

%N Sequences A001644 and A000073 interleaved.

%H G. C. Greubel, <a href="/A075676/b075676.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(n) = A000073(n) if n odd, a(n) = A001644(n) if n even.

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

%F a(n) = 3*a(n-2) + a(n-4) + a(n-6), a(0)=3, a(1)=1, a(2)=3, a(3)=2, a(4)=11, a(5)=7.

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

%F a(n) = T(n) + (1+(-1)^n)*(T(n-1) + (3/2)*T(n-2)).

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

%t LinearRecurrence[{0,3,0,1,0,1},{3,1,3,2,11,7},40] (* _Harvey P. Dale_, May 01 2014 *)

%o (PARI) my(x='x+O('x^40)); Vec((3+x-6*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); Coefficients(R!( (3+x- 6*x^2-x^3-x^4)/(1-3*x^2-x^4-x^6) )); // _G. C. Greubel_, Apr 21 2019

%o (Sage) ((3+x-6*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, A075536.

%K easy,nonn

%O 0,1

%A Mario Catalani (mario.catalani(AT)unito.it), Sep 24 2002

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Last modified June 17 17:43 EDT 2021. Contains 345085 sequences. (Running on oeis4.)