OFFSET
0,2
COMMENTS
See example and the link for more explanation and limits of validity.
The offset is motivated by the special status of the initial term a(0)=0.
REFERENCES
Alexandre Moatti, Récréations mathéphysiques, Editions le Pommier. ISBN: 9782746504875.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Charles R Greathouse IV, Re: Super-birthdays, seqfan list, Jan 2011.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
From Alexander R. Povolotsky, Jan 18 2011: (Start)
G.f.: x*(11 + 6*x + 5*x^2 + 6*x^3)/((1 - x)*(1 - x^4)). [corrected by Georg Fischer, May 10 2019]
a(n) = +1*a(n-1) +1*a(n-4) -1*a(n-5). (End)
EXAMPLE
A standard year has 365 = 350+14+1 = 1 (mod 7) days, and a leap year has 366 = 2 (mod 7) days.
A super-birthday occurs when this sums up to a multiple of 7.
If you are born between 2 and 3 years after a Feb. 29:
1+2+1+1+1+2+1+1 +1+2+1 = 14, after 11 years,
1+1+2+1+1+1 = 7, 6 years later, age of 17,
2+1+1+1+2 = 7, 5 years later: age of 22,
1+1+1 +2+1+1 = 7, 6 years later, i.e. age of 28,
and then the same cycles repeat.
MATHEMATICA
CoefficientList[Series[x*(11+6*x+5*x^2+6*x^3)/((1-x)*(1-x^4)), {x, 0, 50}], x] (* G. C. Greubel, Feb 22 2017 *)
PROG
(PARI) a(n)=[0, 11, 17, 22][n%4+1]+n\4*28
(PARI) my(x='x+O('x^50)); concat([0], Vec(x*(11+6*x+5*x^2+6*x^3)/((1-x)*(1-x^4)))) \\ G. C. Greubel, May 10 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); [0] cat Coefficients(R!( x*(11+6*x+5*x^2+6*x^3)/((1-x)*(1-x^4)) )); // G. C. Greubel, May 10 2019
(Sage) (x*(11+6*x+5*x^2+6*x^3)/((1-x)*(1-x^4))).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, May 10 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric Angelini and M. F. Hasler, Jan 16 2011
STATUS
approved