|
Search: id:A005252
|
|
|
| A005252 |
|
Sum C(n-2k,2k), k = 0 . . floor(n/2).
(Formerly M1048)
|
|
+0
8
|
|
| 1, 1, 1, 1, 2, 4, 7, 11, 17, 27, 44, 72, 117, 189, 305, 493, 798, 1292, 2091, 3383, 5473, 8855, 14328, 23184, 37513, 60697, 98209, 158905, 257114, 416020, 673135, 1089155, 1762289, 2851443, 4613732, 7465176, 12078909, 19544085, 31622993
(list; graph; listen)
|
|
|
OFFSET
|
0,5
|
|
|
COMMENT
|
The Twopins/t sequence (see Guy).
Number of closed walks of length n at a vertex of the graph with adjacency matrix [1,1,0,0;0,0,0,1;1,0,0,0;0,0,1,1] - Paul Barry (pbarry(AT)wit.ie), Mar 15 2004
a(n+3) = number of n-bit sequences that avoid both 010 and 0110. Example: for n=3, there are 8 3-bit sequences and only 010 fails to qualify, so a(6)=7. - David Callan (callan(AT)stat.wisc.edu), Mar 25 2004
|
|
REFERENCES
|
R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 205 of first edition.
V. E. Hoggatt, Jr., and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
R. K. Guy, ``Anyone for Twopins?,'' in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
MacKay, Information Theory, Inference, and Learning Algorithms, CUP, 2003, p251.
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n=0..500
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 424
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
|
|
FORMULA
|
Second differences give sequence shifted twice - E. L. Tan, Univ. Phillipines.
G.f.: (1-x)/((1-x-x^2)(1-x+x^2)); a(n)=Fib(n+1)/2+A010892(n)/2; a(n)=(((1+sqrt(5))/2)^(n+1)/sqrt(5)-((1-sqrt(5))/2)^(n+1)/sqrt(5)+cos(pi*n/3)+sin(pi*n/3)/sqrt(3))/2. - Paul Barry (pbarry(AT)wit.ie), Mar 15 2004
a(n) = 2*a(n-1)-a(n-2)+a(n-4); a(0) = a(1) = a(2) = a(3) = 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 01 2006
|
|
MAPLE
|
A005252:=(-1+z)/(z**2-z+1)/(z**2+z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, X))), X = Sequence(b, card >= 2)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=2..40); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 26 2008
|
|
CROSSREFS
|
First differences of A024490.
Sequence in context: A117276 A035295 A006999 this_sequence A023430 A023429 A023428
Adjacent sequences: A005249 A005250 A005251 this_sequence A005253 A005254 A005255
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
njas
|
|
EXTENSIONS
|
More terms from (and formula corrected by) James A. Sellers (sellersj(AT)math.psu.edu), Feb 06 2000
|
|
|
Search completed in 0.002 seconds
|
