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<a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10, -22).

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(MAGMAMagma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((3+r)*(5+r)^n+(3-r)*(5-r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 13 2009

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a(n) = ((3+sqrt(3))*(5+sqrt(3))^n + (3-sqrt(3))*(5-sqrt(3))^n)/6.

2nd binomial transform of A086405. [From _- _R. J. Mathar_, Jul 17 2009]

a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0) = 1, a(1) = 6.

From R. J. Mathar, Jul 17 2009: (Start)

a(n) = 10*a(n-2) - 22*a(n-2).

a(n)=10*a(n-2)-22*a(n-2). G.f.: (1-4*x)/(1-10*x+22*x^2). [From _R. J. Mathar_, Jul 17 2009](End)

(MAGMA) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((3+r)*(5+r)^n+(3-r)*(5-r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; [From _// _Klaus Brockhaus_, Jul 13 2009]

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2nd binomial transform of A086405. [From _R. J. Mathar (mathar(AT)strw.leidenuniv.nl), _, Jul 17 2009]

a(n)=10*a(n-2)-22*a(n-2). G.f.: (1-4*x)/(1-10*x+22*x^2). [From _R. J. Mathar (mathar(AT)strw.leidenuniv.nl), _, Jul 17 2009]

More terms from _R. J. Mathar (mathar(AT)strw.leidenuniv.nl), _, Jul 17 2009

(MAGMA) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((3+r)*(5+r)^n+(3-r)*(5-r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; [From _Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), _, Jul 13 2009]

Edited and extended beyond a(5) by _Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), _, Jul 13 2009

a(n) = ((3+sqrt(3))*(5+sqrt(3))^n+(3-sqrt(3))*(5-sqrt(3))^n)/6.

1, 6, 38, 248, 1644, 10984, 73672, 495072, 3329936, 22407776, 150819168, 1015220608, 6834184384, 46006990464, 309717848192, 2085024691712, 14036454256896, 94493999351296, 636137999861248, 4282512012883968

0,2

Fifth binomial transform of A108411. Binomial transform of A162557. Inverse binomial transform of A162757.

2nd binomial transform of A086405. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 17 2009]

a(n) = 10*a(n-1)-22*a(n-2) for n > 1; a(0) = 1, a(1) = 6.

G.f.: (1-4*x)/(1-10*x+22*x^2).

a(n)=10*a(n-2)-22*a(n-2). G.f.: (1-4*x)/(1-10*x+22*x^2). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 17 2009]

(MAGMA) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((3+r)*(5+r)^n+(3-r)*(5-r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 13 2009]

Cf. A108411 (powers of 3 repeated), A162557, A162757.

nonn

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

Edited and extended beyond a(5) by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 13 2009

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 17 2009

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