A128018 -id:A128018 - cp's OEIS frontend

A157240 a(n) = A128018(n) + 1.

2, -1, -7, -7, 17, 65, 65, -127, -511, -511, 1025, 4097, 4097, -8191, -32767, -32767, 65537, 262145, 262145, -524287, -2097151, -2097151, 4194305, 16777217, 16777217, -33554431, -134217727, -134217727, 268435457, 1073741825

Offset: 0

Creighton Dement, Feb 25 2009

Generating floretion is Y = .5('i + 'j + 'k + i' + j' + k') + ee. ("tes"). Note: A current conjecture is that if X is a floretion for which 4*tes(X^n) is an integer for all n, then X+sigma(X) also has this property. "sigma" is the uniquely defined projection operator which "flips the arrows" of a floretion (i.e. sigma('i) = i', sigma('j) = j', etc.). Taking X = .5('i + 'j + 'k + ee), then tesseq(X) = [ -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, ...] is an integer sequence, thus by the conjecture 4*tes(Y^n) = 4*tes((X+sigma)^n) should also be an integer sequence for all n.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,4).

Cf. A128018, A157241.

LinearRecurrence[{3,-6,4},{2,-1,-7},40] (* Harvey P. Dale, May 30 2021 *)

Vec((2 - 7*x + 8*x^2) / ((1 - x)*(1 - 2*x + 4*x^2)) + O(x^35)) \\ Colin Barker, May 22 2019

G.f.: (2-7*x+8*x^2) / ((1-x)*(4*x^2-2*x+1)).
From Colin Barker, May 22 2019: (Start)
a(n) = (2 + (1-i*sqrt(3))^(1+n) + (1+i*sqrt(3))^(1+n)) / 2 where i=sqrt(-1).
a(n) = 3*a(n-1) - 6*a(n-2) + 4*a(n-3) for n>2.
(End)
a(n) = A138230(n+1)+1. - R. J. Mathar, Sep 11 2019

A128014 Central binomial coefficients C(2n,n) repeated.

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 20, 20, 70, 70, 252, 252, 924, 924, 3432, 3432, 12870, 12870, 48620, 48620, 184756, 184756, 705432, 705432, 2704156, 2704156, 10400600, 10400600, 40116600, 40116600, 155117520, 155117520, 601080390, 601080390

Offset: 0

Paul Barry, Feb 11 2007

Binomial transform is A097893. Hankel transform is A128017.
Hankel transform of a(n+1) is A128018. - Paul Barry, Nov 23 2009
Number of 2n-bead balanced binary necklaces that are equivalent to their reverse. - Andrew Howroyd, Sep 29 2017
Number of ballot sequences of length n in which the vote is tied or decided by 1 vote. - Nachum Dershowitz, Aug 12 2020
Number of binary strings of length n that are abelian squares. - Michael S. Branicky, Dec 21 2020

Cf. A097893, A128017, A128018.

Cf. A000984, A249095, A249308.

a128014 = a000984 . flip div 2
-- Reinhard Zumkeller, Nov 14 2014

(1+x)/Sqrt[1-4x^2] + O[x]^34 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 07 2017 *)
With[{cb=Table[Binomial[2n,n],{n,0,20}]},Riffle[cb,cb]] (* Harvey P. Dale, Feb 17 2020 *)

G.f.: (1+x)/sqrt(1-4*x^2).
a(n) = C(n,n/2)*(1+(-1)^n)/2 + C(n-1,(n-1)/2)*(1-(-1)^n)/2.
a(n) = (1/Pi)*Integral_{x=-2..2} x^n*(1+x)/(x*sqrt(4-x^2)), as moment sequence.
E.g.f. of a(n+1): Bessel_I(0,2*x)+2*Bessel_I(1,2*x). - Paul Barry, Mar 26 2010
n*a(n) +(n-2)*a(n-1) +4*(-n+1)*a(n-2) +4*(-n+3)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012
a(n) = 2^n*Product_{k=0..n-1} ((k/n+1/n)/2)^((-1)^k). - Peter Luschny, Dec 03 2013
From Reinhard Zumkeller, Nov 14 2014: (Start)
a(n) = A000984(floor(n/2)).
a(n) = A249095(n,n) = A249308(n) / 2^n. (End)

A138230 Expansion of (1-x)/(1 - 2x + 4x^2).

Original entry on oeis.org

1, 1, -2, -8, -8, 16, 64, 64, -128, -512, -512, 1024, 4096, 4096, -8192, -32768, -32768, 65536, 262144, 262144, -524288, -2097152, -2097152, 4194304, 16777216, 16777216, -33554432, -134217728, -134217728, 268435456, 1073741824, 1073741824, -2147483648, -8589934592

Offset: 0

Paul Barry, Mar 06 2008

In general, the expansion of (1-x)/(1 - 2*x + (m+1)*x^2) has general term given by a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*(-m)^k = ((1+sqrt(-m))^n + (1-sqrt(-m))^n)/2.

Binomial transform of [1, 0, -3, 0, 9, 0, -27, 0, 81, 0, ...] = powers of -3 with interpolated zeros. - Philippe Deléham, Dec 02 2008

Michael De Vlieger, Table of n, a(n) for n = 0..3322
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras, Vol. 29 (2019), Article 54.
Index entries for linear recurrences with constant coefficients, signature (2,-4).

Cf. A087204, A088138, A098158, A104537, A124182, A128018.

[2^n*Evaluate(ChebyshevFirst(n), 1/2): n in [0..30]]; // G. C. Greubel, Feb 11 2023

CoefficientList[Series[(1-x)/(1-2x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,-4},{1,1},30] (* Harvey P. Dale, Nov 11 2014 *)

[2^n*chebyshev_T(n,1/2) for n in range(31)] # G. C. Greubel, Feb 11 2023

From Philippe Deléham, Nov 14 2008: (Start)
a(n) = 2*a(n-1) - 4*a(n-2), a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A098158(n,k)*(-3)^(n-k). (End)
a(n) = Sum_{k=0..n} A124182(n,k)*(-4)^(n-k). - Philippe Deléham, Nov 15 2008
a(n) = 2^n*cos(Pi*n/3). - Richard Choulet, Nov 19 2008
a(n) = -8*a(n-3). - Paul Curtz, Apr 22 2011
From Sergei N. Gladkovskii, Jul 27 2012: (Start)
G.f.: G(0) where G(k) = 1 + x/(1 + 2*x/(1 - 2*x - 4*x/(4*x + 1/G(k+1)))); (continued fraction).
E.g.f.: exp(x)*cos(sqrt(3)*x) = G(0) where G(k) = 1 + x/(3*k+1 + 2*x*(3*k+1)/(3*k+2 - 2*x - 4*x*(3*k+2)/(4*x + 3*(k+1)/G(k+1)))); (continued fraction). (End)
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = A088138(n+1) - A088138(n). - R. J. Mathar, Mar 04 2018
a(n) = (-1)^n*A104537(n). - R. J. Mathar, May 21 2019
a(n) = 2^(n-1)*A087204(n). - G. C. Greubel, Feb 11 2023
Sum_{n>=0} 1/a(n) = 4/3. - Amiram Eldar, Feb 14 2023

A168175 Expansion of 1/(1 - 4x + 7x^2).

Original entry on oeis.org

1, 4, 9, 8, -31, -180, -503, -752, 513, 7316, 25673, 51480, 26209, -255524, -1205559, -3033568, -3695359, 6453540, 51681673, 161551912, 284435937, 6880364, -1963530103, -7902282960, -17864421119, -16141703756, 60484132809

Offset: 0

Roger L. Bagula, Nov 19 2009

Also the coefficient of i of Q^(n+1), Q being the quaternion 2+i+j+k. The real part of the quaternion power is A213421, see also A087455, A088138, A128018. - Stanislav Sykora, Jun 11 2012

a(n)*(-1)^n gives the coefficient c(7^n) of (eta(z^6))^4, a modular cusp form of weight 2, when expanded in powers of q = exp(2*Pi*i*z), Im(z) > 0, assuming alpha-multiplicativity (but not for primes 2 and 3) with alpha(x) = x (weight 2) and input c(7) = -4. Eta is the Dedekind function. See the Apostol reference, p. 138, eq. (54) for alpha-multiplicativity and p. 130, eq. (39) with k=2. See also A000727(n) = b(n) where c(7^n) = b((7^n-1)/6) = b(A023000(n)), n >= 0. Proof: The alpha-multiplicity with alpha(1) = 1 and c(1) = 1 leads from p^n = p^(n-1)*p to the recurrence c_n = c*c_(n-1) - a*c(n-2), with c_n = c(p^n), c = c(p) and a = alpha(p). Inputs are c_{-1} = 0 and c_0 = c(1) = 1. This gives the polynomial c_n = sqrt(a)^n * S(n,c/sqrt(a)), with Chebyshev's S-polynomials (A049310). See the Apostol reference, Exercise 6., p. 139. Here p = 7, c = -4. - Wolfdieter Lang, Apr 27 2016

			G.f. = 1 + 4*x + 9*x^2 + 8*x^3 - 31*x^4 - 180*x^5 - 503*x^6 - 752*x^7 + ... - _Michael Somos_, Feb 23 2020

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 130, 138 - 139.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-7).
Index entries for sequences related to Chebyshev polynomials.

Cf. A023000, A049310.

I:=[1,4]; [n le 2 select I[n] else 4*Self(n-1)-7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 25 2012

CoefficientList[Series[1/(1-4x+7x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {4,-7},{1,4},30] (* Harvey P. Dale, Nov 28 2014 *)

{a(n) = my(s=1, t=1); if( n<0, n=-2-n; s=-1; t=1/7); s * t^(n+1) * polcoeff(1 / (1 - 4*x + 7*x^2) + x * O(x^n), n)}; /* Michael Somos, Feb 23 2020 */

a(n) = (1/2 - i/sqrt(3))*(2 + i*sqrt(3))^n + (1/2 + i/sqrt(3))*(2 - i*sqrt(3))^n (Binet formula), where i is the imaginary unit.
a(n) = 4*a(n-1) - 7*a(n-2).
a(n) = sqrt(7)^n * S(n, 4/sqrt(7)), n >= 0, with Chebyshev's S polynomials (A049310). - Wolfdieter Lang, Apr 27 2016
E.g.f.: (2*sqrt(3)*sin(sqrt(3)*x) + 3*cos(sqrt(3)*x))*exp(2*x)/3. - Ilya Gutkovskiy, Apr 27 2016
a(n) = (-1) * 7^(n+1) * a(-2-n) for all n in Z. - Michael Somos, Feb 23 2020

A213421 Real part of Q^n, Q being the quaternion 2+i+j+k.

Original entry on oeis.org

1, 2, 1, -10, -47, -118, -143, 254, 2017, 6290, 11041, 134, -76751, -307942, -694511, -622450, 2371777, 13844258, 38774593, 58188566, -38667887, -561991510, -1977290831, -3975222754, -2059855199, 19587138482

Offset: 0

Stanislav Sykora, Jun 11 2012

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras Vol. 29, No. 3 (2019), Article 54.
Wikipedia, Lucas sequence

Cf. A168175, A087455, A088138, A128018, A146559.

#A213421
seq(simplify(1/2*((2+I*sqrt(3))^n+(2-I*sqrt(3))^n)), n = 0 .. 25); # Peter Bala, Mar 29 2015

QuaternionToN(a,b,c,d,nmax) = {local (C);C = matrix(nmax+1,4);C[1,1]=1;for(n=2,nmax+1,C[n,1]=a*C[n-1,1]-b*C[n-1,2]-c*C[n-1,3]-d*C[n-1,4];C[n,2]=b*C[n-1,1]+a*C[n-1,2]+d*C[n-1,3]-c*C[n-1,4];C[n,3]=c*C[n-1,1]-d*C[n-1,2]+a*C[n-1,3]+b*C[n-1,4];C[n,4]=d*C[n-1,1]+c*C[n-1,2]-b*C[n-1,3]+a*C[n-1,4];);return (C);}
Q=QuaternionToN(2,1,1,1,1000);
for(n=1,#Q[,1],write("A213421.txt",n-1," ",Q[n,1]));

Conjecture: G.f. (1-2x)/(1-4x+7x^2). a(n) = A168175(n)-2*A168175(n-1). - R. J. Mathar, Jun 25 2012
From Peter Bala, Mar 29 2015: (Start)
The above o.g.f. is correct; this is the Lucas sequence V_n(4,7).
a(n) = Re( (2 + sqrt(3)*i)^n )= 1/2*( (2 + sqrt(3)*i)^n + (2 - sqrt(3)*i)^n ).
a(n) = 1/2 * trace( [ 2 + i, 1 + i; -1 + i, 2 - i ]^n ) = 1/2 * trace( [ 2 , sqrt(3)*i ; sqrt(3)*i, 2 ]^n ).
a(n) = 4*a(n-1) - 7*a(n-2) with a(0) = 1, a(1) = 2. (End)

A128019 Expansion of (1 - 3x)/(1 + 3*x^2).

Original entry on oeis.org

1, -3, -3, 9, 9, -27, -27, 81, 81, -243, -243, 729, 729, -2187, -2187, 6561, 6561, -19683, -19683, 59049, 59049, -177147, -177147, 531441, 531441, -1594323, -1594323, 4782969, 4782969, -14348907, -14348907, 43046721, 43046721, -129140163, -129140163, 387420489, 387420489

Offset: 0

Paul Barry, Feb 11 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,-3)

Cf. A108411, A128018.

CoefficientList[Series[(1 - 3x)/(1 + 3*x^2),{x,0,40}],x] (* Stefano Spezia, Dec 31 2022 *)
LinearRecurrence[{0,-3},{1,-3},40] (* Harvey P. Dale, Jun 09 2025 *)

a(n) = 3^floor((n+1)/2)*(-1)^C(n+1,2).
Binomial transform is A128018.
E.g.f.: cos(sqrt(3)*x) - sqrt(3)*sin(sqrt(3)*x). - Stefano Spezia, Dec 31 2022

A157241 Expansion of x / ((1-x)(4x^2-2*x+1)).

Original entry on oeis.org

0, 1, 3, 3, -5, -21, -21, 43, 171, 171, -341, -1365, -1365, 2731, 10923, 10923, -21845, -87381, -87381, 174763, 699051, 699051, -1398101, -5592405, -5592405, 11184811, 44739243, 44739243, -89478485, -357913941

Offset: 0

Creighton Dement, Feb 25 2009

Generating floretion is Y = .5('i + 'j + 'k + i' + j' + k') + ee. ("ibasek"). This is the same floretion which generates A157240.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,4).

Cf. A128018, A157240.

CoefficientList[Series[x/((1-x)(4x^2-2x+1)),{x,0,40}],x] (* or *) LinearRecurrence[{3,-6,4},{0,1,3},40] (* Harvey P. Dale, Oct 27 2013 *)
Table[1/9 (3 + (-1)^Floor[1/3 (-2 + n)] 2^(4 + 3 Floor[1/3 (-2 + n)]) + (-1)^Floor[1/3 (-1 + n)] 2^(3 + 3 Floor[1/3 (-1 + n)])), {n, 0, 500}] (* John M. Campbell, Dec 23 2016 *)

concat(0, Vec(x / ((1 - x)*(1 - 2*x + 4*x^2)) + O(x^40))) \\ Colin Barker, May 22 2019

a(n+1) - a(n) = A088138(n+1).
a(n+1) = Sum_{k=0..n} A120987(n,k)*(-1)^(n-k). - Philippe Deléham, Oct 25 2011
G.f.: 2*x-2*x/(G(0) + 1) where G(k)= 1 + 2*(2*k+3)*x/(2*k+1 - 2*x*(k+2)*(2*k+1)/(2*x*(k+2) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2012
a(n) = 1/9*(3 + (-1)^floor((n-2)/3)*2^(4+3*floor((n-2)/3)) + (-1)^floor((n-1)/3)*2^(3+3*floor((n-1)/3))). - John M. Campbell, Dec 23 2016
From Colin Barker, May 22 2019: (Start)
a(n) = (2 - (1+i*sqrt(3))^(1+n) + i*(1-i*sqrt(3))^n*(i+sqrt(3))) / 6 where i=sqrt(-1).
a(n) = 3*a(n-1) - 6*a(n-2) + 4*a(n-3) for n>2.
(End)

A103970 Expansion of (1 - sqrt(1 - 4x - 12x^2))/(2*x).

Original entry on oeis.org

1, 4, 8, 32, 128, 576, 2688, 13056, 65024, 330752, 1710080, 8962048, 47497216, 254132224, 1370849280, 7447117824, 40707293184, 223731253248, 1235630948352, 6853893292032, 38166664839168, 213288826699776, 1195775593807872, 6723691157127168, 37908469021409280, 214260335517892608, 1213784937073737728, 6890689428042285056

Offset: 0

Paul Barry, Feb 23 2005

Image of c(x), the g.f. of the Catalan numbers A000108 under the mapping g(x) -> (1+3x)g(x(1+3x)). In general, the image of the Catalan numbers under the mapping g(x) -> (1+i*x)g(x(1+i*x)) is given by a(n) = Sum_{k=0..n} i^(n-k)*C(k)*C(k+1,n-k).
Hankel transform is 4^C(n+1,2)*A128018(n). [Paul Barry, Nov 20 2009]
By following L. Comtet [Analyse Combinatoire Tomes 1 et 2, PUF, Paris 1970], we also obtain (n+1)*C(n) - 2*a*(2*n-1)*C(n-1) + 4*(n-2)*(a^2-b)*C(n-2) = 0. In the present case, we also have the asymptotic result: a(n) ~ sqrt(4/3)*2^(n-1)*3^(n+1)/sqrt(Pi*n^3) for large n. - Richard Choulet, Dec 17 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A025227, A025229, A103971, A103972.

Cf. A000108, A025228, A025230, A025231, A025232. [Richard Choulet, Dec 17 2009]

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( (1-Sqrt(1-4*x-12*x^2))/(2*x) )); // G. C. Greubel, Mar 16 2019

n:=30:a(0):=1:a(1):=4: k:=1: for k from 1 to n do a(k+1):=sum('a(p)*a(k-p)','p'=0..k):od:seq(a(k),k=0..n); # Richard Choulet, Dec 17 2009
taylor(((1-(1-4*z-12*z^2)^0.5)/(2*z)),z=0,32); # Richard Choulet, Dec 17 2009

CoefficientList[Series[(1 - Sqrt[1-4x-12x^2])/(2x), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 18 2017 *)

my(x='x+O('x^35)); Vec((1-sqrt(1-4*x-12*x^2))/(2*x)) \\ G. C. Greubel, Mar 16 2019

((1-sqrt(1-4*x-12*x^2))/(2*x)).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, Mar 16 2019

G.f.: (1 - sqrt(1-4*x*(1+3*x)))/(2*x).
a(n) = Sum_{k=0..n} 3^(n-k)*C(k)*C(k+1, n-k).
D-finite with recurrence: (n+1)*a(n) = 2*(2*n-1)*a(n-1) + 12*(n-2)*a(n-2). - Richard Choulet, Dec 17 2009

A138340 Expansion of (1-8x)/(1-4x+16x^2).

Original entry on oeis.org

1, -4, -32, -64, 256, 2048, 4096, -16384, -131072, -262144, 1048576, 8388608, 16777216, -67108864, -536870912, -1073741824, 4294967296, 34359738368, 68719476736, -274877906944, -2199023255552, -4398046511104, 17592186044416

Offset: 0

Paul Barry, Mar 15 2008

abs(a(n)) = 2^A047267(n).

Index entries for linear recurrences with constant coefficients, signature (4, -16).

CoefficientList[Series[(1-8x)/(1-4x+16x^2),{x,0,30}],x] (* or *) LinearRecurrence[{4,-16},{1,-4},30] (* Harvey P. Dale, Sep 30 2014 *)

a(n) = 2*4^n(cos(Pi*(n+1)/3) - sqrt(3)*sin(Pi*(n+1))/3).
a(n) = 4^n*Sum_{k=0..n} A121314(n,k)*(-1)^k*3^(n-k). - Philippe Deléham, Nov 01 2008
a(n) = A128018(n)*2^n. - Philippe Deléham, Nov 14 2008
a(n) = 4*a(n-1) - 16*a(n-2); a(0)=1, a(1)=-4. - Harvey P. Dale, Sep 30 2014

A266046 Real part of Q^n, where Q is the quaternion 2 + j + k.

Original entry on oeis.org

1, 2, 2, -4, -28, -88, -184, -208, 272, 2336, 7712, 16832, 21056, -16768, -193408, -673024, -1531648, -2088448, 836096, 15875072, 58483712, 138684416, 203835392, -16764928, -1290072064, -5059698688, -12498362368, -19635257344, -3550855168, 103608123392

Offset: 0

Stanislav Sykora, Dec 20 2015

In general, given a quaternion Q = r+u*i+v*j+w*k with integer coefficients [r,u,v,w], its powers Q^n = R(n)+U(n)*i+V(n)*j+W(n)*k define four integer sequences R(n),U(n),V(n),W(n). The process can be also transcribed as a four-term, first order recurrence for the elements of the four sequences. Since |Q^n| = |Q|^n, we have, for any n, R(n)^2+U(n)^2+V(n)^2+W(n)^2 = (L^2)^n, where L^2 = r^2+u^2+v^2+w^2 is a constant. The normalized sequence Q^n/L^n describes a unitary quaternion undergoing stepwise rotations by the angle phi = arctan(sqrt(u^2+v^2+w^2)/r). Consequently, the four sequences exhibit sign changes with the mean period of P = 2*Pi/phi steps.
When Q has a symmetry with respect to permutations and/or inversions of the imaginary axes, the four sequences become even more interdependent.
In this particular case Q = 2+j+k, and Q^n = a(n)+b(n)*(j+k), where b(n) is the sequence A190965. The first-order recurrence reduces to two-terms, namely a(n+1)=2*a(n)-2*b(n), b(n+1)=2*b(n)+a(n). This implies further a single-term, second order recurrence a(n+2)=4*a(n+1)-6*a(n), shared by both a(n) and b(n), but with different starting terms. The mean period of sign changes is P = 10.208598624... steps.
The following OEIS sequences can be also cast as quaternion powers:
Q = 1+i+j+k: Q^n = A128018(n)+A088138(n)*(i+j+k), P = 6.000,
Q = 1+j+k  : Q^n = A087455(n)+A088137(n)*(j+k),   P = 6.577071086...,
Q = 2+i+j+k: Q^n = A213421(n)+A168175(n)*(i+j+k), P = 8.803377735...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras Vol. 29, No. 3 (2019), Article 54.
Index entries for linear recurrences with constant coefficients, signature (4,-6).

Cf. A087455 (Inv. Bin. Transf.), A088137, A088138, A128018, A168175, A190965, A213421.

[n le 2 select n else  4*Self(n-1)-6*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Dec 22 2015

LinearRecurrence[{4, -6}, {1, 2}, 30] (* Bruno Berselli, Dec 22 2015 *)

\\ A simple function to generate quaternion powers:
QuaternionToN(r, u, v, w, nmax) = {local (M); M = matrix(nmax+1, 4); M[1, 1]=1; for(n=2, nmax+1, M[n, 1]=r*M[n-1, 1]-u*M[n-1, 2]-v*M[n-1, 3]-w*M[n-1, 4]; M[n, 2]=u*M[n-1, 1]+r*M[n-1, 2]+w*M[n-1, 3]-v*M[n-1, 4]; M[n, 3]=v*M[n-1, 1]-w*M[n-1, 2]+r*M[n-1, 3]+u*M[n-1, 4]; M[n, 4]=w*M[n-1, 1]+v*M[n-1, 2]-u*M[n-1, 3]+r*M[n-1, 4]; ); return (M); }
a=QuaternionToN(2, 0, 1, 1, 1000)[,1]; \\ Select the real parts

Vec((1-2*x)/(1-4*x+6*x^2) + O(x^40)) \\ Colin Barker, Dec 21 2015

a(n)^2 + 2*A190965(n)^2 = 6^n.
From Colin Barker, Dec 21 2015: (Start)
a(n) = ((2-i*sqrt(2))^n+(2+i*sqrt(2))^n)/2, where i=sqrt(-1).
a(n) = 4*a(n-1) - 6*a(n-2) for n>1.
G.f.: (1-2*x) / (1-4*x+6*x^2). (End)

cp's OEIS Frontend

A157240 a(n) = A128018(n) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A128014 Central binomial coefficients C(2n,n) repeated.

Views

Author

Keywords

Comments

Crossrefs

Programs

Haskell

Mathematica

Formula

A138230 Expansion of (1-x)/(1 - 2*x + 4*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A168175 Expansion of 1/(1 - 4*x + 7*x^2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A213421 Real part of Q^n, Q being the quaternion 2+i+j+k.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A128019 Expansion of (1 - 3x)/(1 + 3*x^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A157241 Expansion of x / ((1-x)*(4*x^2-2*x+1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A103970 Expansion of (1 - sqrt(1 - 4*x - 12*x^2))/(2*x).

Views

A138230 Expansion of (1-x)/(1 - 2x + 4x^2).

A168175 Expansion of 1/(1 - 4x + 7x^2).

A157241 Expansion of x / ((1-x)(4x^2-2*x+1)).

A103970 Expansion of (1 - sqrt(1 - 4x - 12x^2))/(2*x).