The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A215466 Expansion of x*(1-4*x+x^2) / ( (x^2-7*x+1)*(x^2-3*x+1) ). 11
 0, 1, 6, 38, 252, 1705, 11628, 79547, 544824, 3733234, 25585230, 175356611, 1201893336, 8237850373, 56462937882, 387002396990, 2652553009008, 18180866487757, 124613506702404, 854113665498719, 5854182112700460 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Peter Bala, Aug 05 2019: (Start) Let  U(n;P,Q), where P and Q are integer parameters, denote the Lucas sequence of the first kind. Then, excluding the case P = -1, the sequence ( U(n;P,1) + U(2*n;P,1) )/(P + 1) is a fourth-order linear divisibility sequence with o.g.f. x*(1 - 2*(P - 1)*x + x^2)/((1 - P*x + x^2)*(1 - (P^2 - 2)*x + x^2)). This is the case P = 3. See A000027 (P = 2), A165998 (P = -2) and A238536 (P = -3). More generally, the sequence U(n;P,1) + U(2*n;P,1) + ... + U(k*n;P,1) is a linear divisibility sequence of order 2*k. As an example, see A273625 (P = 3, k = 3 and then sequence normalized with initial term 1). (End) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Wikipedia, Lucas sequence E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5. H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277. H. C. Williams and R. K. Guy, Odd and even linear divisibility sequences of order 4, INTEGERS, 2015, #A33. Index entries for linear recurrences with constant coefficients, signature (10,-23,10,-1). FORMULA a(n) = L(n)*F(3n)/4 if n even, = F(n)*L(3n)/4 if n odd, where L=A000032, F=A000045. a(n) = 3*A004187(n)/4 + A001906(n)/4. a(n) = 10*a(n-1) - 23*a(n-2) + 10*a(n-3) - a(n-4), a(0)=0, a(1)=1, a(2)=6, a(3)=38. - Harvey P. Dale, Nov 02 2015 a(n) = (1/4)*(Fibonacci(2*n) + Fibonacci(4*n)) = (1/4)*(A001906(n) + A033888(n)). - Peter Bala, Aug 05 2019 E.g.f.: exp(5*x/2)*(cosh(x)+exp(x)*cosh(sqrt(5)*x))*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Aug 17 2019 MAPLE A215466 := proc(n)     if type(n, 'even') then         A000032(n)*combinat[fibonacci](3*n)/4 ;     else         combinat[fibonacci](n)*A000032(3*n)/4 ;     end if; end proc: MATHEMATICA CoefficientList[Series[x*(1 - 4*x + x^2)/((x^2 - 7*x + 1)*(x^2 - 3*x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 23 2012 *) LinearRecurrence[{10, -23, 10, -1}, {0, 1, 6, 38}, 30] (* Harvey P. Dale, Nov 02 2015 *) PROG (MAGMA) I:=[0, 1, 6, 38]; [n le 4 select I[n] else 10*Self(n-1)-23*Self(n-2)+10*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 23 2012 (MAGMA) /* By definition: "/ m:=20; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1-4*x+x^2)/((x^2-7*x+1)*(x^2-3*x+1)))); // Bruno Berselli, Dec 24 2012 (PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 10, -23, 10]^n*[0; 1; 6; 38])[1, 1] \\ Charles R Greathouse IV, Nov 13 2015 CROSSREFS Cf. A000032, A000045, A001906, A033888, A165998, A238536, A273625. Sequence in context: A135030 A217633 A162558 * A147957 A098410 A079949 Adjacent sequences:  A215463 A215464 A215465 * A215467 A215468 A215469 KEYWORD nonn,easy AUTHOR R. J. Mathar, Aug 11 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 5 21:22 EDT 2022. Contains 355102 sequences. (Running on oeis4.)