A000044 - OEIS

%I M0691 N0255 #81 Jan 05 2025 19:51:30

%S 1,1,1,2,3,5,8,13,21,34,55,89,144,232,375,606,979,1582,2556,4130,6673,

%T 10782,17421,28148,45480,73484,118732,191841,309967,500829,809214,

%U 1307487,2112571,3413385,5515174,8911138,14398164,23263822,37588502,60733592,98130253,158553878,256183302,413927966,668803781,1080619176,1746009572,2821113574,4558212008

%N Dying rabbits: a(0) = 1; for 1 <= n <= 12, a(n) = Fibonacci(n); for n >= 13, a(n) = a(n-1) + a(n-2) - a(n-13).

%C A107358 is a more satisfactory version, but I have left the present sequence unchanged (except for making the definition clearer) since it has been in the OEIS so long.

%C Number of compositions of n into parts 1, 3, 5, 7, 9, and 11. - _Joerg Arndt_, Sep 05 2014

%C If a(0) = 1 then it is not clear why a(2) = 1, it should be equal to a(1) + a(0) = 2. Does the first comment mean that a(0) is erroneous and should read a(0) = 0? In contrast to A107358, the term a(13) = 232 = 144 + 89 - 1 seems correct, since in this month the first and oldest pair of rabbits die. But a(14) should be equal to a(13) + a(12) = 232 + 144 because the first pair (which was also the only one present in month 2) has already died and there is no other pair aged 12 months. In general, the number of pairs which die in month n because they are aged exactly 12 months, equals a(n-14): this is the number of newborn pairs in month n - 12, viz. a(n-12) = a(n-13) [those from preceding month] + a(n-14) [the newborn ones] - #(those which die). - _M. F. Hasler_, Oct 06 2017

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Harvey P. Dale, <a href="/A000044/b000044.txt">Table of n, a(n) for n = 0..1000</a>

%H J. H. E. Cohn, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.

%H V. E. Hoggatt, Jr. and D. A. Lind, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.

%H Diyath Pannipitiya, <a href="https://arxiv.org/abs/2402.07015">To Symbolic Dynamics Through The Thue-Morse Sequence</a>, arXiv:2402.07015 [math.DS], 2024.

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

%F G.f.: 1/(1 - z - z^3 - z^5 - z^7 - z^9 -z^11).

%F G.f. A(x) = 1 / (1 - x / (1 - x^2 / (1 + x^10 / (1 + x^2 / (1 - x^2 / (1 + x^6 / (1 + x^2 / (1 - x^2 / (1 + x^2))))))))). - _Michael Somos_, Jan 04 2013

%F For n >= 11, a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-7) + a(n-9) + a(n-11). - _Eric M. Schmidt_, Sep 04 2014

%e G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 13*x^7 + 21*x^8 + 34*x^9 + ...

%p with(combinat); f:=proc(n) option remember; if n=0 then RETURN(1); fi; if n <= 12 then RETURN(fibonacci(n)); fi; f(n-1)+f(n-2)-f(n-13); end;

%t CoefficientList[Series[1/(1 - z - z^3 - z^5 - z^7 - z^9 - z^11), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 10 2011 *)

%t LinearRecurrence[{1,1,0,0,0,0,0,0,0,0,0,0,-1},{1,1,1,2,3,5,8,13,21,34,55,89,144},100] (* _Harvey P. Dale_, Mar 24 2012 *)

%o (Magma) [ n eq 1 select 1 else n le 13 select Fibonacci(n-1) else Self(n-1)+Self(n-2)-Self(n-13): n in [1..50] ]; // _Klaus Brockhaus_, Dec 21 2010

%o (PARI) Vec(1/(1-z-z^3-z^5-z^7-z^9-z^11)+O(z^50)) \\ _Charles R Greathouse IV_, Jun 10 2011

%Y Cf. A107358. See A000045 for the Fibonacci numbers.

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_; entry revised May 25 2005

%E G.f. corrected by _Charles R Greathouse IV_, Jun 10 2011