A184551 Super-birthdays (falling on the same weekday), version 3 (birth within 2 and 3 years after a February 29).

0, 11, 17, 22, 28, 39, 45, 50, 56, 67, 73, 78, 84, 95, 101, 106, 112, 123, 129, 134, 140, 151, 157, 162, 168, 179, 185, 190, 196, 207, 213, 218, 224, 235, 241, 246, 252, 263, 269, 274, 280, 291, 297

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

Cf. A184549, A184550, A184552.

Sequence in context: A325902 A050715 A006618 * A190039 A066074 A106563

Adjacent sequences: A184548 A184549 A184550 * A184552 A184553 A184554

Keyword

nonn

Author

Eric Angelini and M. F. Hasler, Jan 16 2011

Status

approved