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A136252
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a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).
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5
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1, 3, 5, 9, 13, 21, 29, 45, 61, 93, 125, 189, 253, 381, 509, 765, 1021, 1533, 2045, 3069, 4093, 6141, 8189, 12285, 16381, 24573, 32765, 49149, 65533, 98301, 131069, 196605, 262141, 393213, 524285, 786429, 1048573, 1572861, 2097149, 3145725, 4194301
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OFFSET
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0,2
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COMMENTS
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A060482 without the term 2.
For n >= 2, number of n X n arrays with values that are squares of integers, having all 2 X 2 subblocks summing to 4. - R. H. Hardin, Apr 03 2009
Number of moves required in 4-peg Tower of Hanoi solution using a (suboptimal) recursive algorithm: Move (n-2) disks, move bottom 2 disks, move (n-2) disks. Cf. A007664. - Toby Gottfried, Nov 29 2010
Partial sums of A163403.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).
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FORMULA
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a(n) = 2^((1/2)*n-1)*(4 + 4(-1)^n + 3*sqrt(2)*(1-(-1)^n)) - 3. - Emeric Deutsch, Mar 31 2008
a(n) = -2*(1 + sqrt(2))^(-1)*a(0)*(1 - sqrt(2))^(-1) - (1/4)*sqrt(2)*sqrt(2)^n*(1 - sqrt(2))^(-1)*a(2) + (1/2)*sqrt(2)^n*a(0)*(1 - sqrt(2))^(-1) + (-4 - 2*sqrt(2))^(-1)*(-sqrt(2))^n*a(1) + (-4 - 2*sqrt(2))^(-1)*sqrt(2)*(-sqrt(2))^n*a(1) - (-4 - 2*sqrt(2))^(-1)*sqrt(2)*(-sqrt(2))^n*a(0) - (1/2)*sqrt(2)^n*a(1)*(1 - sqrt(2))^(-1) + (1 + sqrt(2))^(-1)*(1 - sqrt(2))^(-1)*a(2) - (-4 - 2*sqrt(2))^(-1)*(-sqrt(2))^n*a(2) + (1/4)*sqrt(2)*sqrt(2)^n*a(1)*(1 - sqrt(2))^(-1). - Alexander R. Povolotsky, Mar 31 2008
G.f.: (1+2*x)/((1-x)*(1-2*x^2)). - Jaume Oliver Lafont, Aug 30 2009
a(n) = 2*a(n-2) + 3; first differences are powers of 2, occurring in pairs. - Toby Gottfried, Nov 29 2010
a(n) = A027383(n+1) - 1. - Jason Kimberley, Nov 01 2011
a(2n+1) = (a(2n) + a(2n+2))/2. - Richard R. Forberg, Nov 30 2013
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MAPLE
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a:=proc(n) options operator, arrow: 2^((1/2)*n-1)*(4+4*(-1)^n+3*sqrt(2)*(1-(-1)^n))-3 end proc: seq(a(n), n=0..40); # Emeric Deutsch, Mar 31 2008
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MATHEMATICA
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LinearRecurrence[{1, 2, -2}, {1, 3, 5}, 100] (* G. C. Greubel, Feb 18 2017 *)
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PROG
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(PARI) x='x+O('x^50); Vec((1+2*x)/((1-x)*(1-2*x^2))) \\ G. C. Greubel, Feb 18 2017
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CROSSREFS
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Same recurrence as in A135530.
Cf. A007664 (Optimal 4-peg Tower of Hanoi).
Sequence in context: A206297 A320596 A227565 * A187212 A141325 A248604
Adjacent sequences: A136249 A136250 A136251 * A136253 A136254 A136255
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KEYWORD
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nonn
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AUTHOR
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Paul Curtz, Mar 17 2008
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EXTENSIONS
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Edited by N. J. A. Sloane, Apr 18 2008
More terms from Emeric Deutsch, Mar 31 2008
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STATUS
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approved
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