|
| |
|
|
A008466
|
|
a(n) = 2^n-Fibonacci(n+2).
|
|
16
| |
|
|
0, 0, 1, 3, 8, 19, 43, 94, 201, 423, 880, 1815, 3719, 7582, 15397, 31171, 62952, 126891, 255379, 513342, 1030865, 2068495, 4147936, 8313583, 16655823, 33358014, 66791053, 133703499, 267603416, 535524643, 1071563515
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| Toss a fair coin n times; a(n) is number of tosses having a run of 2 or more heads.
Also the number of binary words of length n with at least two neighboring 1 digits. For example, a(4)=8 because 8 binary words of length 4 have two or more neighboring 1 digits: 0011, 0110, 0111, 1011, 1100, 1101, 1110, 1111. - Alois P. Heinz, Jul 18 2008
Equivalently, number of solutions (x_1, ..., x_n) to the equation x_1*x_2 + x_2*x_3 + x_3*x_4 + ... +x_{n-1}*x_n = 1 in base 2 dismal arithmetic. - N. J. A. Sloane, Apr 23 2011.
Row sums of triangle A153281 = (1, 3, 8, 19, 43,...). [From Gary W. Adamson, Dec 23 2008
|
|
|
REFERENCES
| Feller, W.; An Introduction to Probability Theory and Its Application, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.
B. E. Merkel, Probabilities of Consecutive Events in Coin Flipping, Master's Thesis, Univ. Cincinatti, May 11 2011; http://etd.ohiolink.edu/view.cgi/Merkel%20Benjamin%20E.pdf?ucin1307442290
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..300
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1020
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for sequences related to dismal arithmetic
|
|
|
FORMULA
| a(1)=0, a(2)=1, a(3)=3, a(n)=3*a(n-1)-a(n-2)-2*a(n-3). E.g. a(8)=201=3*a(7)-a(6)-2*a(5)=3*94-43-2*19. - Miklos Kristof, Nov 24 2003
G.f.: x^2/((1-2x)(1-x-x^2)) - Paul Barry, Feb 16 2004
Convolution of Fibonacci(n) and (2^n-0^n)/2. a(n)=sum{k=0..n, (2^k-0^k)Fib(n-k)/2}; a(n+1)=sum{k=0..n, Fib(k)2^(n-k)}=2^n*sum{k=0..n, Fib(k)/2^k}. - Paul Barry, May 19 2004
a(n) = a(n-1)+a(n-2)+2^(n-2) E.g. a(7)=a(6)+a(5)+2^5=43+19+32=94 - Jon Stadler (jstadler(AT)capital.edu), Aug 21 2006
a(n) = 2*a(n-1) + Fib(n-1) E.g. a(7) = 2*a(6) + Fib(6) = 2*43 + 8 = 94 - Thomas M. Green (tgreen(AT)astound.net), Aug 21 2007
a(n) = term (1,3) in the 3x3 matrix [3,1,0; -1,0,1; -2,0,0]^n. - Alois P. Heinz, Jul 18 2008
a(n) = 2*a(n-1)-a(n-3)+2^(n-3) - Carmine Suriano, Mar 08 2011.
|
|
|
MAPLE
| a:= n-> (Matrix([[3, 1, 0], [ -1, 0, 1], [ -2, 0, 0]])^(n))[1, 3]; seq ((a(n)), n=0..50); # Alois P. Heinz, Jul 18 2008
|
|
|
MATHEMATICA
| Table[2^n-Fibonacci[n+2], {n, 0, 20}] (Vladimir Orlovsky, Jul 22 2008)
Find Boolean values of variables that satisfy the formula x1 x2 + x2 x3 + ... + xn-1 xn = 1 (from N. J. A. Sloane, Apr 23 2011):
MMM = 30;
For[ M=2, M <= MMM, M++,
vlist = Array[x, M];
cl[i_] := And[ x[i], x[i+1] ];
cl2 = False; For [ i=1, i <= M-1, i++, cl2 = Or[cl2, cl[i]] ];
R[M] = SatisfiabilityCount[ cl2, vlist ] ]
Table[ R[M], {M, 2, MMM}]
|
|
|
CROSSREFS
| Cf. A050227.
a(n) = A101220(2, 2, n-1), for n > 0.
Cf. A050231, A050232, A050233.
Cf. A153281 - Gary W. Adamson, Dec 23 2008
Sequence in context: A099050 A065352 A161993 * A102712 A054480 A191787
Adjacent sequences: A008463 A008464 A008465 * A008467 A008468 A008469
|
|
|
KEYWORD
| nonn,nice,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jack Kennedy (kennedy(AT)oldnews.org), Eric Weisstein (eric(AT)weisstein.com)
|
| |
|
|
