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A052905
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a(n) = (n^2 + 7*n + 2)/2.
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20
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1, 5, 10, 16, 23, 31, 40, 50, 61, 73, 86, 100, 115, 131, 148, 166, 185, 205, 226, 248, 271, 295, 320, 346, 373, 401, 430, 460, 491, 523, 556, 590, 625, 661, 698, 736, 775, 815, 856, 898, 941, 985, 1030, 1076, 1123, 1171, 1220, 1270, 1321, 1373, 1426, 1480
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OFFSET
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0,2
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COMMENTS
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Starting 1, 5, 10, 16, 23, ... gives binomial transform of (1, 4, 1, 0, 0, 0, ...). Row sums of triangle A134199. - Gary W. Adamson, Jul 25 2007
If Y_i (i=1,2,3,4,5) are 2-blocks of an n-set X then, for n >= 10, a(n-4) is the number of (n-2)-subsets of X intersecting each Y_i (i=1,2,3,4,5). - Milan Janjic, Nov 09 2007
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LINKS
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FORMULA
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G.f.: (-2*x+2*x^2-1)/(-1+x)^3.
Recurrence: {a(0)=1, a(1)=5, a(2)=10, -2*a(n)+n^2+7*n+2}.
E.g.f.: (1/2)*(x^2 + 8*x + 2)*exp(x). - G. C. Greubel, Jul 13 2017
Sum_{n>=0} 1/a(n) = 19/20 + 2*Pi*tan(sqrt(41)*Pi/2)/sqrt(41). - Amiram Eldar, Dec 13 2022
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EXAMPLE
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Illustration of initial terms:
. o
. o o
. o o o o
. o o o o o o
. o o o o o o o o o
. o o o o o o o o o o o o
. o o o o o o o o o o . . . . . o
. o o o o o o o . . . . o o . . . . . o
. o o o o o . . . o o . . . . o o . . . . . o
. o o o . . o o . . . o o . . . . o o . . . . . o
. o o . o o . . o o . . . o o . . . . o o . . . . . o
. o o o . o o . . o o . . . o o . . . . o o . . . . . o
. o o o o o o o o o o o o o o o o o o o o o o o o o o o o
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. 1 5 10 16 23 31 40
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MAPLE
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spec := [S, {S=Prod(Sequence(Z), Sequence(Z), Union(Sequence(Z), Z, Z))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
a:=n->sum((n-4)/2, j=0..n): seq(a(n)-2, n=5..56); # Zerinvary Lajos, Apr 30 2007
with (combinat):seq((fibonacci(3, n)+n-11)/2, n=3..54); # Zerinvary Lajos, Jun 07 2008
a:=n->sum(k, k=0..n):seq(a(n)/2+sum(k, k=5..n)/2, n=3..54); # Zerinvary Lajos, Jun 10 2008
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {1, 5, 10}, 60] (* Harvey P. Dale, Sep 15 2018 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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STATUS
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approved
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