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Search: id:A138041
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| A138041 |
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a(1) = 1, a(2) = 10; for n>2, a(n+1) = 4*a(n) + 6*a(n-1). Also a(n) = upper left term in the 2 X 2 matrix [1,3; 3,3]. |
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| 1, 10, 46, 244, 1252, 6472, 33400, 172432, 890128, 4595104, 23721184, 122455360, 632148544, 3263326336, 16846196608, 86964744448, 448936157440, 2317533096448, 11963749330432, 61760195900416, 318823279584256
(list; graph; listen)
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OFFSET
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1,2
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FORMULA
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a(n)/a(n-1) tends to (2 + sqrt(10)) = 5.16227766... (a root of x^2 - 4*x - 6 and an eigenvalue of the matrix).
a(n) mod 9 == 1.
O.g.f.: -x*(1+6*x)/(-1+4*x+6*x^2). a(n) = A085939(n)+6*A085939(n-1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 03 2008
From the characteristic polynomial of the matrix we get g.f.: (6*x + 1)/(-6*x^2 - 4*x + 1), with roots a=-(2+sqrt(10))/6, b=-(2-sqrt(10))/6. Let A=3+3*sqrt(10)/10 and B=3-3*sqrt(10)/10. Then a(n) = (A*(1/a)^n + B*(1/b)^n)/6. - Lambert Herrgesell (zero815(AT)googlemail.com), Apr 04 2008
a(n)=(1/2)*(2+sqrt(10))^n-(13/20)*(2-sqrt(10))^n*sqrt(10)+(13/20)*(2+sqrt(10))^n*sqrt(10)+(1/2)*(2-sqrt(10))^n - Paolo P. Lava (ppl(AT)spl.at), Jun 03 2008
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EXAMPLE
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a(4) = 244 = 4*46 + 6*10 = 4*a(3) + 6*a(2).
a(4) = 244 = upper left term in [1,3; 3,3]^4
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MATHEMATICA
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a = {1, 10}; Do[AppendTo[a, 4*a[[ -1]] + 6*a[[ -2]]], {25}]; a - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com)
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CROSSREFS
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Adjacent sequences: A138038 A138039 A138040 this_sequence A138042 A138043 A138044
Sequence in context: A096045 A115712 A003765 this_sequence A000832 A143895 A034443
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 02 2008
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EXTENSIONS
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More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com) and R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 02 2008
Definition corrected by Paolo P. Lava (ppl(AT)spl.at), Jun 03 2008
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