A096980 -id:A096980 - cp's OEIS frontend

A108051 a(n+1) = 4*(a(n)+a(n-1)) for n>1, a(1)=1, a(2)=6.

0, 1, 6, 28, 136, 656, 3168, 15296, 73856, 356608, 1721856, 8313856, 40142848, 193826816, 935878656, 4518821888, 21818802176, 105350496256, 508677193728, 2456110759936, 11859151814656, 57261050298368, 276480808452096

Offset: 0

Creighton Dement, Jun 01 2005

Let (a_n) be the sequence and (a_(n+1)) the sequence beginning at 1. Let B and iB be the binomial and inverse binomial transforms, respectively. Then B((a_n)) = A001108(n) (a(n)-th triangular number is a square); B((a_(n+1))) = A002315(n) (NSW Numbers); iB((a_(n+1))) = A096980(n). Note: a 2nd sequence generated by the same floretion is A057087 (Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence.). As is often the case with two sequences corresponding to a single floretion, both satisfy the same recurrence relation.

Floretion Algebra Multiplication Program, FAMP Code: (a_n) = 2ibasekseq[A*B] (with initial term zero), (a_(n+1)) = 1tesseq[A*B], A = + .5'i - .5'j + .5'k + .5i' - .5j' + .5k' - .5'ij' - .5'ik' - .5'ji' - .5'ki'; B = - .5'i + .5'j + .5'k - .5i' + .5j' + .5k' - .5'ik' - .5'jk' - .5'ki' - .5'kj'

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (4,4).

Cf. A057087, A001108, A002315, A096980.

I:=[0, 1, 6]; [n le 3 select I[n] else 4*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 26 2012

CoefficientList[Series[x*(1+2*x)/(1-4*x-4*x^2),{x,0,40}],x] (* Vincenzo Librandi, Jun 26 2012 *)

a(n+1) = -(1/2)*(2-2*2^(1/2))^n*(-1+2^(1/2))-(1/2)*(2+2*2^(1/2))^n(-1-2^(1/2)); G.f.: x*(1+2*x)/(1-4*x-4*x^2).
a(n) = sum{k=0..n, (-1)^k*C(n-1, k)*(Pell(2n-2k)-Pell(2n-2k-1))}, n>0, where Pell(n) = A000129(n). - Paul Barry, Jun 07 2005
a(n+1) = ((3+sqrt18)(2+sqrt8)^n+(3-sqrt18)(2-sqrt8)^n)/6. - Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009, index corrected Jul 11 2012
a(n) = 2^(n-1) * A001333(n), n>0. - Ralf Stephan, Dec 02 2010
a(n) = A057087(n-1) + 2*A057087(n-2). - R. J. Mathar, Jul 11 2012

A164737 a(n) = 8*a(n-2) for n > 2; a(1) = 5, a(2) = 12.

Original entry on oeis.org

5, 12, 40, 96, 320, 768, 2560, 6144, 20480, 49152, 163840, 393216, 1310720, 3145728, 10485760, 25165824, 83886080, 201326592, 671088640, 1610612736, 5368709120, 12884901888, 42949672960, 103079215104, 343597383680, 824633720832

Offset: 1

Klaus Brockhaus, Aug 24 2009

nonn

Interleaving of 5*A001018 and 12*A001018.

Binomial transform is A096980 without initial terms 1. Second binomial transform is A164593. Third binomial transform is A101386.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,8).

Cf. A001018 (powers of 8), A067412, A096980, A101386, A164593.

[ n le 2 select 7*n-2 else 8*Self(n-2): n in [1..26] ];

seq(coeff(series( x*(5+12*x)/(1-8*x^2) , x, n+1), x, n), n=1..30); # G. C. Greubel, Apr 16 2020

LinearRecurrence[{0,8}, {5,12}, 30] (* G. C. Greubel, Apr 16 2020 *)

[(13 -7*(-1)^n)*2^((6*n -11 +3*(-1)^n)/4) for n in (1..30)] # G. C. Greubel, Apr 16 2020

a(n) = (13 - 7*(-1)^n)*2^(1/4*(6*n - 11 + 3*(-1)^n)).

G.f.: x*(5 + 12*x)/(1 - 8*x^2).

A164588 a(n) = ((3 + sqrt(18))(5 + sqrt(8))^n + (3 - sqrt(18))(5 - sqrt(8))^n)/6.

Original entry on oeis.org

1, 9, 73, 577, 4529, 35481, 277817, 2174993, 17027041, 133295529, 1043495593, 8168931937, 63949894289, 500627099961, 3919122796697, 30680567267633, 240180585132481, 1880236207775049, 14719292130498313, 115228905772807297, 902061091509601649

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

nonn

Binomial transform of A057084. Second binomial transform of A002315. Third binomial transform of A108051 without initial 0. Fourth binomial transform of A096980. Fifth binomial transform of A094015.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-17).

Cf. A057084, A002315, A108051, A096980, A094015.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+3*r)*(5+2*r)^n+(3-3*r)*(5-2*r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

LinearRecurrence[{10,-17},{1,9},30] (* Harvey P. Dale, Sep 11 2016 *)

x='x+O('x^50); Vec((1-x)/(1-10*x+17*x^2)) \\ G. C. Greubel, Aug 12 2017

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 9.
G.f.: (1-x)/(1-10*x+17*x^2).
E.g.f.: (1/3)*exp(5*x)*(3*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017

Extended by Klaus Brockhaus and R. J. Mathar Aug 24 2009

A164593 a(n) = ((5 + sqrt(18))(2 + sqrt(8))^n + (5 - sqrt(18))(2 - sqrt(8))^n)/2.

Original entry on oeis.org

5, 22, 108, 520, 2512, 12128, 58560, 282752, 1365248, 6592000, 31828992, 153683968, 742051840, 3582943232, 17299980288, 83531694080, 403326697472, 1947433566208, 9403041054720, 45401898483712, 219219758153728, 1058486626549760, 5110825538813952, 24677248661454848

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Binomial transform of A096980 without initial 1. Second binomial transform of A164737. Inverse binomial transform of A101386.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (4,4).

Cf. A000129, A096980, A101386, A164737.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(2+2*r)^n+(5-3*r)*(2-2*r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

seq(coeff(series( (5+2*x)/(1-4*x-4*x^2) , x, n+1), x, n), n = 0..25); # G. C. Greubel, Apr 16 2020

LinearRecurrence[{4, 4}, {5, 22}, 25] (* G. C. Greubel, Aug 12 2017 *)
Table[2^n*(Fibonacci[n+2, 2] + 3*Fibonacci[n+1, 2]), {n,0,25}] (* G. C. Greubel, Apr 16 2020 *)

my(x='x+O('x^25)); Vec((5+2*x)/(1-4*x-4*x^2)) \\ G. C. Greubel, Aug 12 2017

[2^n*(lucas_number1(n+2, 2, -1) + 3*lucas_number1(n+1, 2, -1)) for n in range(25)] # G. C. Greubel, Apr 16 2020

a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 5, a(1) = 22.
G.f.: (5 + 2*x)/(1-4*x-4*x^2).
E.g.f.: exp(2*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017
a(n) = 2^n*(Pell(n+2) + 3*Pell(n+1)), where Pell(n) = A000129(n). - G. C. Greubel, Apr 16 2020

cp's OEIS Frontend

A108051 a(n+1) = 4*(a(n)+a(n-1)) for n>1, a(1)=1, a(2)=6.

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A164737 a(n) = 8*a(n-2) for n > 2; a(1) = 5, a(2) = 12.

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A164588 a(n) = ((3 + sqrt(18))*(5 + sqrt(8))^n + (3 - sqrt(18))*(5 - sqrt(8))^n)/6.

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A164593 a(n) = ((5 + sqrt(18))*(2 + sqrt(8))^n + (5 - sqrt(18))*(2 - sqrt(8))^n)/2.

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Magma

Maple

Mathematica

PARI

Sage

Formula

A164588 a(n) = ((3 + sqrt(18))(5 + sqrt(8))^n + (3 - sqrt(18))(5 - sqrt(8))^n)/6.

A164593 a(n) = ((5 + sqrt(18))(2 + sqrt(8))^n + (5 - sqrt(18))(2 - sqrt(8))^n)/2.