A198834
Number of sequences of n coin flips that win on the last flip, if the sequence of flips ends with (0,1,1) or (1,1,1).
Original entry on oeis.org
0, 0, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338
Offset: 1
For n=6 the a(6)=6 solutions are (0,0,0,0,1,1), (1,0,0,0,1,1); (0,1,0,0,1,1), (1,1,0,0,1,1), (0,0,1,0,1,1), (1,0,1,0,1,1) all for Abel.
- A. Engel, Wahrscheinlichkeit und Statistik, Band 2, Klett, 1978, pages 25-26.
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
- Index entries for linear recurrences with constant coefficients, signature (1,1).
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a(1):=0: a(2):=0: a(3):=2:
ml:=0.75: pot:=8:
for n from 4 to 100 do
pot:=2*pot:
a(n):=a(n-1)+a(n-2):
ml:=ml+n*a(n)/pot:
end do:
printf("%12.8f",ml);
seq(a(n),n=1..100);
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Join[{0, 0}, Table[2*Fibonacci[n], {n, 70}]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)
Join[{0},LinearRecurrence[{1,1},{0,2},50]] (* Vincenzo Librandi, Feb 19 2012 *)
A206474
Riordan array ((1+x-x^2)/(1-x^2), x/(1-x^2)).
Original entry on oeis.org
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 0, 3, 3, 4, 4, 1, 1, 1, 1, 6, 6, 5, 5, 1, 1, 0, 4, 4, 10, 10, 6, 6, 1, 1, 1, 1, 10, 10, 15, 15, 7, 7, 1, 1, 0, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1, 1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1
Offset: 0
Triangle begins :
1
1, 1
0, 1, 1
1, 1, 1, 1
0, 2, 2, 1, 1
1, 1, 3, 3, 1, 1
0, 3, 3, 4, 4, 1, 1
1, 1, 6, 6, 5, 5, 1, 1
0, 4, 4, 10, 10, 6, 6, 1, 1
1, 1, 10, 10, 15, 15, 7, 7, 1, 1
0, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1
1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1
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t[1, 0] = 1; t[2, 0] = 0; t[n_, n_] = 1; t[n_ /; n >= 0, k_ /; k >= 0] /; k <= n := t[n, k] = t[n-1, k-1] + t[n-2, k]; t[n_, k_] = 0; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)
A272632
Non-Fibonacci numbers that are both a sum and a difference of two Fibonacci numbers.
Original entry on oeis.org
4, 6, 7, 10, 11, 16, 18, 26, 29, 42, 47, 68, 76, 110, 123, 178, 199, 288, 322, 466, 521, 754, 843, 1220, 1364, 1974, 2207, 3194, 3571, 5168, 5778, 8362, 9349, 13530, 15127, 21892, 24476, 35422, 39603, 57314, 64079, 92736, 103682, 150050, 167761, 242786
Offset: 1
6 is a term because 6 = Fibonacci(1) + Fibonacci(5) = Fibonacci(6) - Fibonacci(3).
16 is a term because 16 = Fibonacci(6) + Fibonacci(6) = Fibonacci(8) - Fibonacci(5).
167761 is a term because it is not a Fibonacci number and 167761 = Fibonacci(24) + Fibonacci(26) = 46368 + 121393 and Fibonacci(24) + Fibonacci(26) = Fibonacci(27) - Fibonacci(23) by definition.
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mxf=30; {s,d} = Reap[Do[{a,b} = Fibonacci@{i,j}; Sow[a+b, 0]; Sow[a-b, 1], {i, mxf}, {j, i}]][[2]]; Complement[ Intersection[s, d], Fibonacci@ Range@ mxf] (* Giovanni Resta, May 04 2016 *)
A173388
a(n) = a(n - 3) + a(n - 4) if n is even, else a(n - 2) + a(n - 3).
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 2, 4, 4, 6, 6, 10, 10, 16, 16, 26, 26, 42, 42, 68, 68, 110, 110, 178, 178, 288, 288, 466, 466, 754, 754, 1220, 1220, 1974, 1974, 3194, 3194, 5168, 5168, 8362, 8362, 13530, 13530, 21892, 21892, 35422, 35422, 57314, 57314, 92736, 92736
Offset: 0
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a[0] = 1; a[1] = 1; a[2] = 1; a[3] = 1;
a[n_] := a[n] = If[Mod[n, 2] == 0, a[n - 3] + a[n - 4], a[n - 2] + a[n - 3]];
Table[a[n], {n, 0, 50}]
Formula added by the Assoc. Editors of the OEIS, Feb 24 2010
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