OFFSET
0,7
COMMENTS
This sequence is the case r = 5 in the solution to an r-th order probability difference equation that can be found in Eqs. (4) and (3) on p. 356 of Dunkel (1925). (Equation (3) follows equation (4) in the paper!) For r = 2, we get a shifted version of A000071. For r = 3, we get a shifted version of A008937. For r = 4, we get a shifted version of A107066. For r = 6, we get a shifted version of A172316. See also the table in A172119. - Petros Hadjicostas, Jun 15 2019
For n>0, 2*a(n+4) is the number of direct lineal ancestors possessed by an individual n generations in the past, if all mating pairs in the individual's pedigree are fifth-cousin pairs; 2*a(5)=2 is the individual's two parents in generation 1, 2*a(6)=4 is the individual's four grandparents in generation 2, and so on. - Noah A Rosenberg, Jun 09 2026
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
O. Dunkel, Solutions of a probability difference equation, Amer. Math. Monthly, 32 (1925), 354-370; see pp. 356 and 369.
Joseph Lachance, Inbreeding, pedigree size, and the most recent common ancestor of humanity. J. Theor. Biol. 261 (2009), 238-247. See Table 1.
T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011), Article #11.4.2.
Elisa Heinrich Mora and Noah A. Rosenberg, An nth-cousin mating model and the n-anacci numbers. Fibonacci Quart. 64 (2026), 270-280. See Table 1.
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
Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,-1)
FORMULA
For n >= 6, a(n+1) = 2*a(n) - a(n-5).
G.f.: x^5 / ( (x-1)*(x^5 + x^4 + x^3 + x^2 + x - 1) ).
a(n) = Sum_{k=1..n-4} Sum_{j=0..floor((n-k-4)/5)} (-1)^j*binomial(n-5*j-5, k-1)*binomial(n-k-5*j-4, j). - Vladimir Kruchinin, Oct 19 2011
4*a(n) = A000322(n+1) - 1. - R. J. Mathar, Aug 16 2017
From Petros Hadjicostas, Jun 15 2019: (Start)
a(n) = 1 + a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) for n >= 5. (See Eq. (4) and the Theorem with r = 5 on p. 356 of Dunkel (1925).)
a(n) = T(n - 5, 5) for n >= 5, where T(n, k) = Sum_{j = 0..floor(n/(k+1))} (-1)^j * binomial(n - k*j, n - (k+1)*j) * 2^(n - (k+1)*j) for 0 <= k <= n. This is Richard Choulet's formula in A172119.
(End)
MAPLE
A001949:=1/(z-1)/(z**5+z**4+z**3+z**2+z-1); # Simon Plouffe in his 1992 dissertation
MATHEMATICA
t={0, 0, 0, 0, 0}; Do[AppendTo[t, t[[-5]]+t[[-4]]+t[[-3]]+t[[-2]]+t[[-1]]+1], {n, 40}]; t (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
(* Alternative: *)
LinearRecurrence[{2, 0, 0, 0, 0, -1}, {0, 0, 0, 0, 0, 1}, 40] (* Harvey P. Dale, Jan 17 2015 *)
PROG
(Maxima)
a(n):=sum(sum((-1)^j*binomial(n-5*j-5, k-1)*binomial(n-k-5*j-4, j), j, 0, (n-k-4)/5), k, 1, n-4); /* Vladimir Kruchinin, Oct 19 2011 */
(PARI) x='x+O('x^99); concat(vector(5), Vec(x^5/((x-1)*(x^5+x^4+x^3+x^2+x-1)))) \\ Altug Alkan, Oct 04 2017
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
Name edited by Petros Hadjicostas, Jun 15 2019
STATUS
approved