The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A116201 a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=1, a(3)=1. 8
 0, 1, 1, 1, 3, 4, 7, 13, 21, 37, 64, 109, 189, 325, 559, 964, 1659, 2857, 4921, 8473, 14592, 25129, 43273, 74521, 128331, 220996, 380575, 655381, 1128621, 1943581, 3347008, 5763829, 9925797, 17093053, 29435671, 50690692, 87293619, 150326929, 258875569 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS This is a divisibility sequence; that is, if n divides m then a(n) divides a(m). - T. D. Noe, Dec 22 2008 This is the case P1 = 1, P2 = -3, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 31 2014 Also, the inverse radii of a family of spheres defined as follows: the first three spheres have radius of 1 and touch each other and the common plane, while each subsequent sphere touches the three immediately preceding ones and the same plane. - Ivan Neretin, Sep 11 2018 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 A. D. Mednykh, I. A. Mednykh, The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic, arXiv preprint arXiv:1711.00175 [math.CO], 2017. See Section 4. Wikipedia, Soddy-Gosset theorem. 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, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1). FORMULA From R. J. Mathar, Mar 31 2008: (Start) O.g.f: -x*(x-1)*(x+1)/(1 - x - x^2 - x^3 + x^4). a(n) = A135431(n) - A135431(n-1). (End) From Peter Bala, Mar 31 2014: (Start) a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(13))/4 and beta = (1 - sqrt(13))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind. a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 3/4; 1, 1/2]. a(n) = U(n-1,(sqrt(3) + i)/4)*U(n-1,(sqrt(3) - i)/4), where U(n,x) denotes the Chebyshev polynomial of the second kind. See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End) a(n) = a(-n) = A116732(n+2) - A116732(n), 0 = a(n) - 2*a(n+1) + 2*a(n+4) - a(n+5) for all n in Z. - Michael Somos, Feb 26 2019 EXAMPLE G.f. = x + x^2 + x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 13*x^7 + 21*x^8 + ... - Michael Somos, Feb 26 2019 MAPLE a:=0: a:=1: a:=1: a:=1: for n from 4 to 35 do a[n]:= a[n-1]+a[n-2]+a[n-3]-a[n-4] end do: seq(a[n], n=0..35); # Emeric Deutsch, Apr 12 2008 MATHEMATICA a = {0, 1, 1, 1, 3}; Do[AppendTo[a, a[[ -1]]+a[[ -2]]+a[[ -3]]-a[[ -4]]], {80}]; a (* Stefan Steinerberger, Mar 24 2008 *) CoefficientList[Series[(- x^3 + x)/(x^4 - x^3 - x^2 - x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Apr 02 2014 *) a[ n_] := 1 - SeriesCoefficient[ (1 - 2 x) / (1 - 2 x + 2 x^4 - x^5), {x, 0, Abs@n}]; (* Michael Somos, Feb 26 2019 *) LinearRecurrence[{1, 1, 1, -1}, {0, 1, 1, 1}, 50] (* Harvey P. Dale, Mar 26 2019 *) PROG (PARI) {a(n) = n=abs(n); 1 - polcoeff( (1 - 2*x) / (1 - 2*x + 2*x^4 - x^5) + x * O(x^n), n)}; /* Michael Somos, Feb 26 2019 */ CROSSREFS Cf. A100047, A116732, A135431. Sequence in context: A125118 A310008 A299024 * A280224 A282718 A092406 Adjacent sequences:  A116198 A116199 A116200 * A116202 A116203 A116204 KEYWORD nonn,easy AUTHOR R. K. Guy, Mar 23 2008 STATUS approved

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

Last modified July 9 22:46 EDT 2020. Contains 335570 sequences. (Running on oeis4.)