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.: (6 + 5*x + 6*x^2 + 11*x^3)/((-1 + x)^2*(1 + x + x^2 + x^3)).
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 1 and 2 years after a Feb. 29:
1+1+2+1+1+1 = 7 after 6 years,
2+1+1+1+2 = 7, 5 years later: age of 11,
1+1+1 +2+1+1 = 7, 6 years later, i.e. age of 17,
1+2+1+1+1+2+1+1 +1+2+1 = 14, 11 years later: age of 28,
and then the same cycles repeat.
MATHEMATICA
Join[{0}, CoefficientList[Series[(6 + 5*x + 6*x^2 + 11*x^3)/((-1 + x)^2*(1 + x + x^2 + x^3)), {x, 0, 50}], x]] (* G. C. Greubel, Feb 22 2017 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 6, 11, 17, 28}, 50] (* Harvey P. Dale, Nov 21 2019 *)
PROG
(PARI) a(n)=[0, 6, 11, 17][n%4+1]+n\4*28
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric Angelini and M. F. Hasler, Jan 16 2011
STATUS
approved