A000044 -id:A000044 - cp's OEIS frontend

A191869 First differences of the dying rabbits sequence A000044.

0, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 88, 143, 231, 373, 603, 974, 1574, 2543, 4109, 6639, 10727, 17332, 28004, 45248, 73109, 118126, 190862, 308385, 498273, 805084, 1300814, 2101789, 3395964, 5487026, 8865658, 14324680, 23145090, 37396661, 60423625

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 18 2011

Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1).

Cf. A000044.

A000044 = CoefficientList[Series[1/(1 - z - z^3 - z^5 - z^7 - z^9 - z^11), {z, 0, 200}], z]; GetDiff[seq_List] := Drop[seq, 1] - Drop[seq, -1]; A191869 = GetDiff[A000044]

A191869_list=Vec((-x^11-x^9-x^7-x^5-x^3)/(x^11+x^9+x^7+x^5+x^3+x-1)+O(x^99)) /* returns a list of the first 96 nonzero terms, a(3)...a(99) */

A191869(n)=polcoeff((1+x^2+x^4+x^6+x^8)/(1-x-x^3-x^5-x^7-x^9-x^11+O(x^max(1,n-2))),n-3)  \\ M. F. Hasler, Jun 19 2011

G.f.: x^3(1 + x + x^2 + x^3 + x^4)(1 - x + x^2 - x^3 + x^4)/(1 - x - x^3 - x^5 - x^7 - x^9 - x^11). - Charles R Greathouse IV, Jun 19 2011

A107358 Dying rabbits: a(n) = Fibonacci(n) for n <= 12; for n >= 13, a(n) = a(n-1) + a(n-2) - a(n-13).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 376, 608, 982, 1587, 2564, 4143, 6694, 10816, 17476, 28237, 45624, 73717, 119108, 192449, 310949, 502416, 811778, 1311630, 2119265, 3424201, 5532650, 8939375, 14443788, 23337539, 37707610, 60926041, 98441202, 159056294

Offset: 0

N. J. A. Sloane, May 25 2005

In the limit, the growth rate is 1.61575... per generation as opposed to 1.61803... for Fibonacci numbers. - T. D. Noe, Jan 22 2009

If the rabbits die after 12 months, then those that were there in month 1 should die in month 13, whence a(13) = 144 + 89 - 1 = 232 and not 233. In month 14, no rabbits die because the only pair which was there in month 2 already dies. Then in month 15, the one pair born in month 3 will die. In general, the number of rabbits which die in month n (because they are aged 12 months) is equal to the number of newborn rabbits in month n - 12, which is the number of rabbits present in month n - 14. (Recall that a(n - 12) = a(n - 13) + a(n - 14) - #(dying rabbits) = #(rabbits from previous month) + #(newborn rabbits) - #(dying rabbits).) So the recurrence should read a(n) = a(n - 1) + a(n - 2) - a(n - 14). - M. F. Hasler, Oct 06 2017

T. D. Noe, Table of n, a(n) for n = 0..500
J. H. E. Cohn, Letter to the editor, Fib. Quart. 2 (1964), 108.
V. E. Hoggatt, Jr. and D. A. Lind, The dying rabbit problem, Fib. Quart. 7 (1969), 482-487.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,0,0,0,0,0,0,0,0,-1).

See A000045 for the Fibonacci numbers. This is a better version of A000044.

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

LinearRecurrence[{1,1,0,0,0,0,0,0,0,0,0,0,-1},Fibonacci[Range[0,12]],50] (* Harvey P. Dale, Feb 28 2013 *)

Vec(x/(x^13-x^2-x+1)+O(x^99)) \\ Charles R Greathouse IV, Jun 10 2011

G.f.: x/((x-1)*(1+x)*(x^11+x^9+x^7+x^5+x^3+x-1)). - R. J. Mathar, Jul 27 2009

A171997 a(n) = a(n-1) + a(n-2) - floor(a(n-2)/2) - floor(a(n-5)/2); initial terms are 1, 1, 2, 3, 4.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 13, 16, 20, 24, 29, 35, 42, 50, 59, 70, 83, 97, 114, 134, 156, 182, 212, 246, 285, 330, 382, 441, 509, 588, 678, 781, 900, 1037, 1193, 1373, 1580, 1817, 2089, 2402, 2761, 3172, 3645, 4187, 4809, 5523, 6342, 7282, 8360

Offset: 1

Roger L. Bagula, Nov 22 2010

nonn

lim_{n -> infinity} a(n+1)/a(n) = 1.14710876512065387719410850648860644150605499412513....

a(n) = A062435(n+2) for n < 15.

Cf. A062435 (integer part of log(n!)^log(log(1 + n))), A023434 (a(n)=a(n-1)+a(n-2)-a(n-4)), A023435 (a(n)=a(n-1)+a(n-2)-a(n-5)), A023436 (a(n)=a(n-1)+a(n-2)-a(n-6)), A023437 (a(n)=a(n-1)+a(n-2)-a(n-7)), A023438 (a(n)=a(n-1)+a(n-2)-a(n-8)), A023439 (a(n)=a(n-1)+a(n-2)-a(n-9)), A023440 (a(n)=a(n-1)+a(n-2)+a(n-10)), A023441 (a(n)=a(n-1)+a(n-2)-a(n-11)), A023442 (a(n)=a(n-1)+a(n-2)-a(n-12)), A000044 (a(n)=a(n-1)+a(n-2)-a(n-13)), A173199 (a(n)=a(n-1)+a(n-2)-floor(a(n-3)/2)-floor(a(n-8)/2)).

I:=[1,1,2,3,4]; [n le 5 select I[n] else Self(n-1) + Self(n-2) - Floor(Self(n-2)/2) - Floor(Self(n-5)/2): n in [1..60]]; // Vincenzo Librandi, Jun 24 2015

f[-3] = 0; f[-2] = 0; f[-1] = 0; f[0] = 1; f[1] = 1;
f[n_] := f[n] = f[n - 1] + f[n - 2] - Floor[f[n - 2]/2] - Floor[f[n - 5]/2]
Table[f[n], {n, 0, 50}]

Offset changed from 0 to 1 by Klaus Brockhaus, Nov 29 2010

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A191869 First differences of the dying rabbits sequence A000044.

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A107358 Dying rabbits: a(n) = Fibonacci(n) for n <= 12; for n >= 13, a(n) = a(n-1) + a(n-2) - a(n-13).

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A171997 a(n) = a(n-1) + a(n-2) - floor(a(n-2)/2) - floor(a(n-5)/2); initial terms are 1, 1, 2, 3, 4.

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