|
%I #32 Mar 17 2024 20:08:54
%S 0,6,17,23,28,34,45,51,56,62,73,79,84,90,101,107,112,118,129,135,140,
%T 146,157,163,168,174,185,191,196,202,213,219,224,230,241,247,252,258,
%U 269,275,280,286,297,303,308,314
%N Super-birthdays (falling on the same weekday), version 1 (birth within the year following a February 29).
%C See example and the link for more explanation and limits of validity.
%C The offset is motivated by the special status of the initial term a(0)=0.
%D Alexandre Moatti, Récréations mathéphysiques, Editions le Pommier. ISBN: 9782746504875.
%H Paolo Xausa, <a href="/A184549/b184549.txt">Table of n, a(n) for n = 0..10000</a>
%H Charles R Greathouse IV, <a href="http://list.seqfan.eu/oldermail/seqfan/2011-January/006812.html">Re: Super-birthdays</a>, seqfan list, Jan 2011.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F a(n) = a(n-1)+a(n-4)-a(n-5). G.f.: x*(5*x^3+6*x^2+11*x+6) / ((x-1)^2*(x+1)*(x^2+1)). - _Colin Barker_, Nov 04 2013
%e A standard year has 365 = 350+14+1 = 1 (mod 7) days,
%e and a leap year has 366 = 2 (mod 7) days.
%e A super-birthday occurs when this sums up to a multiple of 7.
%e If you are born in the year following a Feb 29:
%e 1+1+1+2+1+1 = 7 after 6 years,
%e 1+2+1+1+1+2+1+1 +1+2+1 = 14, 11 years later, i.e. age of 17,
%e 1+1+2+1+1+1 = 7, 6 years later: age of 23,
%e 2+1+1+1+2 = 7, 5 years later: age of 28,
%e and then the same cycles repeat.
%t LinearRecurrence[{1, 0, 0, 1, -1}, {0, 6, 17, 23, 28}, 100] (* _Paolo Xausa_, Mar 17 2024 *)
%o (PARI) a(n)=[0, 6, 17, 23][n%4+1]+n\4*28
%o (PARI) Vec(x*(5*x^3+6*x^2+11*x+6)/((x-1)^2*(x+1)*(x^2+1)) + O(x^100)) \\ _Colin Barker_, Nov 04 2013
%Y Cf. A184550, A184551, A184552.
%K nonn,easy
%O 0,2
%A _Eric Angelini_ and _M. F. Hasler_, Jan 16 2011
|
