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A111566
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a(n) = ((1+sqrt(8))*(2+sqrt(2))^n + (1-sqrt(8))*(2-sqrt(2))^n)/2.
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5
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1, 6, 22, 76, 260, 888, 3032, 10352, 35344, 120672, 412000, 1406656, 4802624, 16397184, 55983488, 191139584, 652591360, 2228086272, 7607162368, 25972476928, 88675582976, 302757378048, 1033678346240, 3529198628864
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OFFSET
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0,2
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COMMENTS
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Binomial transform of A048655: generalized Pellian with second term equal to 5.
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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 (4,-2).
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FORMULA
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a(n) = 4*a(n-1) - 2*a(n-2), a(0) = 1, a(1) = 6.
Program "FAMP" returns: a(n) = A007052(n) - A006012(n) + A111567(n).
From R. J. Mathar, Apr 02 2008: (Start)
O.g.f.: (1+2*x)/(1-4*x+2*x^2).
a(n) = A007070(n) + 2*A007070(n-1). (End)
a(n) = Sum_{k=0..n} A207543(n,k)*2^k. - Philippe Deléham, Feb 25 2012
a(n) = 4*A007070(n) - A007052(n+1). - Yuriy Sibirmovsky, Sep 13 2016
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MATHEMATICA
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LinearRecurrence[{4, -2}, {1, 6}, 30] (* Harvey P. Dale, Jan 31 2015 *)
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PROG
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Floretion Algebra Multiplication Program, FAMP Code: 1vesseq[K*J] with K = + .5'i + .5'j + .5k' + .5'kk' and J = + .5i' + .5j' + 2'kk' + .5'ki' + .5'kj'.
(MAGMA) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+2*r)*(2+r)^n+(1-2*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 27 2009
(PARI) x='x+O('x^30); Vec((1+2*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jan 27 2018
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CROSSREFS
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Cf. A007052, A006012, A111567, A007070.
Sequence in context: A159555 A032195 A217530 * A200052 A051945 A253070
Adjacent sequences: A111563 A111564 A111565 * A111567 A111568 A111569
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KEYWORD
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easy,nonn
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AUTHOR
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Creighton Dement, Aug 06 2005
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EXTENSIONS
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Edited by N. J. A. Sloane, Jul 27 2009 using new definition from Al Hakanson (hawkuu(AT)gmail.com)
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STATUS
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approved
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