|
| |
|
|
A000211
|
|
a(n) = a(n-1) + a(n-2) - 2.
(Formerly M2396 N0953)
|
|
7
| |
|
|
4, 3, 5, 6, 9, 13, 20, 31, 49, 78, 125, 201, 324, 523, 845, 1366, 2209, 3573, 5780, 9351, 15129, 24478, 39605, 64081, 103684, 167763, 271445, 439206, 710649, 1149853, 1860500, 3010351, 4870849, 7881198
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,1
|
|
|
COMMENTS
| Let I=I_n be the nXn identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n>=3, a(n) is the number of (0,1) nXn matrices A<=P^(-1)+I+P with exactly two 1's in every row and column. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 11 2010]
|
|
|
REFERENCES
| N. Metropolis et al., Permanents of cyclic (0,1) matrices, J. Combin. Theory, 7 (1969), 291-321.
H. Minc, Permanents of (0,1)-circulants, Canad. Math. Bull., 7 (1964), 253-263.
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 233.
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).
R. P. Stanley, Enumerative Combinatorics I, Example 4.7.15, p. 252.
K. Yamamoto, Structure polynomial of Latin rectangles and its application to a combinatorial problem, Memoirs of the Faculty of Science, Kyusyu University, Series A, 10 (1956), 1-13.
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..500
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.
Index to sequences with linear recurrences with constant coefficients, signature (2,0,-1).
|
|
|
FORMULA
| G.f.: 2/(1-x)+(2-x)/(1-x-x^2) =(4-5*x-x^2) / ((x-1)*(x^2+x-1)). a(n) = Lucas number A000032(n) + 2.
Binomial transform of [4, -1, 3, -4, 7, -11, 18,...], i.e. the series continues as a signed version of the Lucas series, A000204. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 08 2007
a(n)=F(n)+F(n+2)+2, n>=-1 {where F(n) is the n-th Fibonacci number} - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 01 2008
a(n)=per(I+P+P^2)=per(P^(-1)+I+P). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 11 2010]
|
|
|
MAPLE
| A000211:=-(1+z)*(4*z-3)/(z-1)/(z**2+z-1); [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence except for the leading 4.]
with(combinat): seq(fibonacci(n)+fibonacci(n+2)+2, n=-1..32); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 01 2008
(Maple) a := n -> (Matrix([[4, 1, 5]]). Matrix(3, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 0, -1][i] else 0 fi)^n)[1, 1] ; seq (a(n), n=0..33); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 01 2008]
|
|
|
MATHEMATICA
| Transpose[NestList[{Last[#], First[#]+Last[#]-2}&, {4, 3}, 40]] [[1]] (* From Harvey P. Dale, Mar 22 2011 *)
|
|
|
CROSSREFS
| Cf. A000204.
Sequence in context: A133981 A016701 A023829 * A059902 A068982 A171021
Adjacent sequences: A000208 A000209 A000210 * A000212 A000213 A000214
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
