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A000044
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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).
(Formerly M0691 N0255)
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4
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1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 232, 375, 606, 979, 1582, 2556, 4130, 6673, 10782, 17421, 28148, 45480, 73484, 118732, 191841, 309967, 500829, 809214, 1307487, 2112571, 3413385, 5515174, 8911138, 14398164, 23263822, 37588502, 60733592, 98130253, 158553878, 256183302, 413927966, 668803781, 1080619176, 1746009572, 2821113574, 4558212008
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OFFSET
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0,4
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COMMENTS
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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.
Number of compositions of n into parts 1, 3, 5, 7, 9, and 11. - Joerg Arndt, Sep 05 2014
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
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).
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FORMULA
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G.f.: 1/(1 - z - z^3 - z^5 - z^7 - z^9 -z^11).
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
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
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EXAMPLE
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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 + ...
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MAPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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