OFFSET
0,7
REFERENCES
R. K. Guy, "Anyone for Twopins?" in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Natasha Blitvić, Vicente I. Fernandez, A Combinatorial Model for Heterogeneous Microbial Growth, arXiv:1901.04080 [math.CO], 2019.
S. Falcon, Generalized (k,r)-Fibonacci Numbers, Gen. Math. Notes, 25(2), 2014, 148-158.
R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
I. Wloch, U. Bednarz, D. Bród, A Wloch and M. Wolowiec-Musial, On a new type of distance Fibonacci numbers, Discrete Applied Math., Volume 161, Issues 16-17, November 2013, Pages 2695-2701.
Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 0, 1).
FORMULA
a(n) = Sum_{k=0..floor(n/2)} binomial(floor((n+3k-3)/5), k). - Paul Barry, Jul 10 2004
G.f.: (x+x^2)/(1-x^2-x^5). - Ralf Stephan, Apr 21 2004
a(n) = a(n-2) + a(n-5). - Michael Somos, Jul 15 2004
a(n+1) = Sum_{k=0..floor(n/5)} A065941(n-4*k, n-5*k). - Johannes W. Meijer, Aug 05 2013
MAPLE
A005686 := -(z+1)*(z**3+z+1)/(-1+z**2+z**5); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for the initial 1's
a := proc(n): if n = 0 then 0 else add(binomial(floor((n+3*k-4)/5), k), k=0..floor((n-1)/2)) fi: end: seq(a(n), n=0..54); # Johannes W. Meijer, Aug 05 2013
MATHEMATICA
nn=54; CoefficientList[Series[(x+x^2)/(1-x^2-x^5), {x, 0, nn}], x] (* Geoffrey Critzer, Apr 28 2013 *)
m = 5; For[n = 0, n < m, n++, a[n] = 1]; For[n = m, n < 51, n++, a[n] = a[n - m] + a[n - 2]]; Table[a[n], {n, 0, 50}] (*Sergio Falcon, Nov 12 2015 *)
Join[{0}, LinearRecurrence[{0, 1, 0, 0, 1}, {1, 1, 1, 1, 1}, 60]] (* Vincenzo Librandi, Jan 19 2016 *)
PROG
(PARI) a(n)=if(n<0, polcoeff((x^3+x^4)/(1+x^3-x^5)+x^-n*O(x), -n), polcoeff((x+x^2)/(1-x^2-x^5)+x^n*O(x), n)) /* Michael Somos, Jul 15 2004 */
(PARI) a(n)=sum(k=0, (n-1)\2, binomial((n+3*k-4)\5, k))
(Magma) I:=[1, 1, 1, 1, 1]; [0] cat [n le 5 select I[n] else Self(n-2)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jan 19 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Paul Barry, Jul 10 2004
STATUS
approved