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A052913
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a(0) = 1; a(1) = 4; a(n+2) = 5*a(n+1) - 2*a(n).
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1
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1, 4, 18, 82, 374, 1706, 7782, 35498, 161926, 738634, 3369318, 15369322, 70107974, 319801226, 1458790182, 6654348458, 30354161926, 138462112714, 631602239718, 2881086973162, 13142230386374
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Main diagonal of the array : m(1,j)=3^(j-1), m(i,1)=1; m(i,j)=m(i-1,j)+m(i,j-1): 1 3 9 27 81 ... / 1 4 13 40 ... / 1 5 18 58 ... / 1 6 24 82 ... - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 05 2002
a(n) is also the number of 3xn matrices of integers for which the upper-left hand corner is a 1, the rows and columns are weakly increasing, and two adjacent entries differ by at most 1. [From Richard Stanley (rstan(AT)math.mit.edu), Jun 06 2010]
a(n) is the number of compositions of n when there are 4 types of 1 and 2 types of other natural numbers. [From Milan R. Janjic (agnus(AT)blic.net), Aug 13 2010]
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 894
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FORMULA
| G.f.: (1-x)/(1-5*x+2*x^2).
Sum(1/17*(3+_alpha)*_alpha^(-1-n), _alpha=RootOf(1-5*_Z+2*_Z^2))
a(n) = ((17+3*sqrt(17))/34)*((5+sqrt(17))/2)^n + ((17-3*sqrt(17))/34)*((5-sqrt(17))/2)^n. - N. J. A. Sloane (njas(AT)research.att.com), Jun 03, 2002
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MAPLE
| spec := [S, {S=Sequence(Union(Prod(Sequence(Z), Union(Z, Z)), Z, Z))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
| Transpose[NestList[{Last[#], 5Last[#]-2First[#]}&, {1, 4}, 20]][[1]] (* From Harvey P. Dale, Mar 12 2011 *)
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CROSSREFS
| Sequence in context: A181610 A194460 A100192 * A129160 A187077 A143646
Adjacent sequences: A052910 A052911 A052912 * A052914 A052915 A052916
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KEYWORD
| easy,nonn
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AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
| Typo in definition corrected by Bruno Berselli, Jun 07 2010
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