A001687 - OEIS

%I M0147 N0059 #63 Apr 13 2022 13:25:16

%S 0,1,0,1,0,1,1,1,2,1,3,2,4,4,5,7,7,11,11,16,18,23,29,34,45,52,68,81,

%T 102,126,154,194,235,296,361,450,555,685,851,1046,1301,1601,1986,2452,

%U 3032,3753,4633,5739,7085,8771,10838,13404,16577,20489,25348,31327

%N a(n) = a(n-2) + a(n-5).

%C a(n+1) is the number of compositions of n into parts 2 and 5. [_Joerg Arndt_, Mar 15 2013]

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001687/b001687.txt">Table of n, a(n) for n = 0..1000</a>

%H Dorota Bród, <a href="https://doi.org/10.1007/s40590-021-00369-5">On trees with unique locating kernels</a>, Boletín de la Sociedad Matemática Mexicana (2021) Vol. 27, Art. No. 61.

%H T. M. Green, <a href="http://www.jstor.org/stable/2687953">Recurrent sequences and Pascal's triangle</a>, Math. Mag., 41 (1968), 13-21.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=405">Encyclopedia of Combinatorial Structures 405</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H E. Wilson, <a href="http://www.anaphoria.com/meruone.PDF">The Scales of Mt. Meru</a>

%H R. Yanco, <a href="/A007380/a007380.pdf">Letter and Email to N. J. A. Sloane, 1994</a>

%H R. Yanco and A. Bagchi, <a href="/A007380/a007380_1.pdf">K-th order maximal independent sets in path and cycle graphs</a>, Unpublished manuscript, 1994. (Annotated scanned copy)

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 0, 0, 1).

%F G.f.: x/(1-x^2-x^5).

%F G.f. A(x) satisfies 1+x^4*A(x) = 1/(1-x^5-x^7-x^9-....). - _Jon Perry_, Jul 04 2004

%F G.f.: -x/( x^5 - 1 + 5*x^2/Q(0) ) where Q(k) = x + 5 + k*(x+1) - x*(k+1)*(k+6)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Mar 15 2013

%p A001687:=-z/(-1+z**2+z**5); # _Simon Plouffe_ in his 1992 dissertation

%t CoefficientList[Series[x/(1-x^2-x^5),{x,0,60}],x] (* or *) Nest[ Append[#,#[[-5]]+#[[-2]]]&, {0,1,0,1,0}, 60]  (* _Harvey P. Dale_, Apr 06 2011 *)

%t LinearRecurrence[{0, 1, 0, 0, 1}, {0, 1, 0, 1, 0}, 100] (* _T. D. Noe_, Aug 09 2012 *)

%o (PARI) a(n)=if(n<0,polcoeff(x^4/(1+x^3-x^5)+x^-n*O(x),-n),polcoeff(x/(1-x^2-x^5)+x^n*O(x),n)) /* _Michael Somos_, Jul 15 2004 */

%o (Maxima)

%o a(n):=sum(if mod(n-5*k,3)=0 then binomial(k,(5*k-n)/3) else 0,k,1,n); /* _Vladimir Kruchinin_, May 24 2011 */

%Y Cf. A005686.

%K nonn

%O 0,9

%A _N. J. A. Sloane_, following a suggestion from _Robert G. Wilson v_