A189315 -id:A189315 - cp's OEIS frontend

A081567 Second binomial transform of F(n+1).

1, 3, 10, 35, 125, 450, 1625, 5875, 21250, 76875, 278125, 1006250, 3640625, 13171875, 47656250, 172421875, 623828125, 2257031250, 8166015625, 29544921875, 106894531250, 386748046875, 1399267578125, 5062597656250, 18316650390625, 66270263671875, 239768066406250

Offset: 0

Paul Barry, Mar 22 2003

Binomial transform of F(2*n-1), index shifted by 1, where F is A000045. - corrected by Richard R. Forberg, Aug 12 2013
Case k=2 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=1, a(1)=k+1.
Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1, 2, ..., 2*n+1, s(0) = 3, s(2*n+1) = 4.
a(n+1) gives the number of periodic multiplex juggling sequences of length n with base state <2>. - Steve Butler, Jan 21 2008
a(n) is also the number of idempotent order-preserving partial transformations (of an n-element chain) of waist n (waist(alpha) = max(Im(alpha))). - Abdullahi Umar, Sep 14 2008
Counts all paths of length (2*n+1), n>=0, starting at the initial node on the path graph P_9, see the Maple program. - Johannes W. Meijer, May 29 2010
Given the 3 X 3 matrix M = [1,1,1; 1,1,0; 1,1,3], a(n) = term (1,1) in M^(n+1). - Gary W. Adamson, Aug 06 2010
Number of nonisomorphic graded posets with 0 and 1 of rank n+2, with exactly 2 elements of each rank level between 0 and 1. Also the number of nonisomorphic graded posets with 0 of rank n+1, with exactly 2 elements of each rank level above 0. (This is by Stanley's definition of graded, that all maximal chains have the same length.) - David Nacin, Feb 26 2012
a(n) = 3^n a(n;1/3) = Sum_{k=0..n} C(n,k) * F(k-1) * (-1)^k * 3^(n-k), which also implies the Deleham formula given below and where a(n;d), n=0,1,...,d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also the papers of Witula et al.). - Roman Witula, Jul 12 2012
The limiting ratio a(n)/a(n-1) is 1 + phi^2. - Bob Selcoe, Mar 17 2014
a(n) counts closed walks on K_2 containing 3 loops on the index vertex and 2 loops on the other. Equivalently the (1,1) entry of A^n where the adjacency matrix of digraph is A=(3,1; 1,2). - David Neil McGrath, Nov 18 2014

			a(4)=125: 35*(3 + (35 mod 10 - 10 mod 3)/(10-3)) = 35*(3 + 4/7) = 125. - _Bob Selcoe_, Mar 17 2014

R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pages 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Santiago Alzate, Oscar Correa, and Rigoberto Flórez, Fibonacci identities from Jordan Identities, arXiv:2009.02639 [math.NT], 2020.
Carolina Benedetti, Christopher R. H. Hanusa, Pamela E. Harris, Alejandro H. Morales, and Anthony Simpson, Kostant's partition function and magic multiplex juggling sequences, arXiv:2001.03219 [math.CO], 2020. See Table 1 p. 12.
S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, arXiv:0801.2597 [math.CO], 2008.
P. E. Harris, E. Insko, and L. K. Williams, The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, arXiv preprint arXiv:1401.0055 [math.RT], 2013.
Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Mathematics, 15(1) (2017), 477-485.
A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), #04.3.8.
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
D. Nacin, The Minimal Non-Koszul A(Gamma), arXiv preprint arXiv:1204.1534 [math.QA], 2012. - From _N. J. A. Sloane_, Oct 05 2012
Roman Witula and Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042.
Index entries for linear recurrences with constant coefficients, signature (5,-5).

a(n) = 5*A052936(n-1), n > 1.
Row sums of A114164.
Cf. A000045, A007051 (INVERTi transform), A007598, A028387, A030191, A039717, A049310, A081568 (binomial transform), A086351 (INVERT transform), A090041, A093129, A094441, A111776, A147748, A178381, A189315.

I:=[1, 3]; [n le 2 select I[n] else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 27 2012

with(GraphTheory):G:=PathGraph(9): A:= AdjacencyMatrix(G): nmax:=23; n2:=nmax*2+2: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..9); od: seq(a(2*n+1),n=0..nmax); # Johannes W. Meijer, May 29 2010

Table[MatrixPower[{{2,1},{1,3}},n][[2]][[2]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{5,-5},{1,3},30] (* Vincenzo Librandi, Feb 27 2012 *)

Vec((1-2*x)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Mar 18 2014

def a(n, adict={0:1, 1:3}):
    if n in adict:
        return adict[n]
    adict[n]=5*a(n-1) - 5*a(n-2)
    return adict[n] # David Nacin, Mar 04 2012

a(n) = 5*a(n-1) - 5*a(n-2) for n >= 2, with a(0) = 1 and a(1) = 3.
a(n) = (1/2 - sqrt(5)/10) * (5/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2) * (sqrt(5)/2 + 5/2)^n.
G.f.: (1 - 2*x)/(1 - 5*x + 5*x^2).
a(n-1) = Sum_{k=1..n} binomial(n, k)*F(k)^2. - Benoit Cloitre, Oct 26 2003
a(n) = A090041(n)/2^n. - Paul Barry, Mar 23 2004
The sequence 0, 1, 3, 10, ... with a(n) = (5/2 - sqrt(5)/2)^n/5 + (5/2 + sqrt(5)/2)^n/5 - 2(0)^n/5 is the binomial transform of F(n)^2 (A007598). - Paul Barry, Apr 27 2004
From Paul Barry, Nov 15 2005: (Start)
a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n, j)*binomial(j+k, 2k);
a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n, k+j)*binomial(k, k-j)2^(n-k-j);
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} binomial(n+k-j, n-k-j)*binomial(k, j)(-1)^j*2^(n-k-j). (End)
a(n) = A111776(n, n). - Abdullahi Umar, Sep 14 2008
a(n) = Sum_{k=0..n} A094441(n,k)*2^k. - Philippe Deléham, Dec 14 2009
a(n+1) = Sum_{k=-floor(n/5)..floor(n/5)} ((-1)^k*binomial(2*n, n+5*k)/2). -Mircea Merca, Jan 28 2012
a(n) = A030191(n) - 2*A030191(n-1). - R. J. Mathar, Jul 19 2012
G.f.: Q(0,u)/x - 1/x, where u=x/(1-2*x), Q(k,u) = 1 + u^2 + (k+2)*u - u*(k+1 + u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
For n>=3: a(n) = a(n-1)*(3+(a(n-1) mod a(n-2) - a(n-2) mod a(n-3))/(a(n-2) - a(n-3))). - Bob Selcoe, Mar 17 2014
a(n) = sqrt(5)^(n-1)*(3*S(n-1, sqrt(5)) - sqrt(5)*S(n-2, sqrt(5))) with Chebyshev's S-polynomials (see A049310), where S(-1, x) = 0 and S(-2, x) = -1. This is the (1,1) entry of A^n with the matrix A=(3,1;1,2). See the comment by David Neil McGrath, Nov 18 2014. - Wolfdieter Lang, Dec 04 2014
Conjecture: a(n) = 2*a(n-1) + A039717(n). - Benito van der Zander, Nov 20 2015
a(n) = A189315(n+1) / 10. - Tom Copeland, Dec 08 2015
a(n) = A093129(n) + A030191(n-1). - Gary W. Adamson, Apr 24 2023
E.g.f.: exp(5*x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Jun 03 2024

A147748 Row sums of Riordan array ((1-3x+x^2)/(1-4x+3x^2), x(1-2x)/(1-4x+3x^2)).

Original entry on oeis.org

1, 2, 6, 20, 70, 250, 900, 3250, 11750, 42500, 153750, 556250, 2012500, 7281250, 26343750, 95312500, 344843750, 1247656250, 4514062500, 16332031250, 59089843750, 213789062500, 773496093750, 2798535156250, 10125195312500

Offset: 0

Paul Barry, Nov 11 2008

Row sums of A147747. Binomial transform of A061646.
Counts all paths of length (2*n), n>=0, starting at the initial node on the path graph P_9, see the Maple program. - Johannes W. Meijer, May 29 2010
From L. Edson Jeffery, Apr 19 2011: (Start)
For the 5 X 5 unit-primitive matrix (see [Jeffery])
A_(10,1) = [0,1,0,0,0; 1,0,1,0,0; 0,1,0,1,0; 0,0,1,0,1; 0,0,0,2,0],
a(n) = (Trace([A_(10,1)]^(2*n)))/5. (See also A189315.) (End)

S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
L. Edson Jeffery, Unit-primitive matrices.
Index entries for linear recurrences with constant coefficients, signature (5,-5).

Cf. A000984, A030191, A061646, A081567, A147746, A147747, A178381, A189315.

with(GraphTheory): G:=PathGraph(9): A:= AdjacencyMatrix(G): nmax:=24; n2:=nmax*2: for n from 0 to n2 do B(n):=A^n; a(n):= add(B(n)[1,k], k=1..9); od: seq(a(2*n), n=0..nmax); # Johannes W. Meijer, May 29 2010

(1 - 3x + x^2)/(1 - 5x + 5x^2) + O[x]^25 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 05 2016 *)

G.f.: (1-3*x+x^2)/(1-5*x+5*x^2).
a(n) = 5*a(n-1) - 5*a(n-2) for n > 2, a(0)=1, a(1)=2, a(2)=6. - Philippe Deléham, Nov 13 2008
For n >= 1: a(n) = (2/5)*((5-sqrt(5))/2)^n + (2/5)*((5+sqrt(5))/2)^n. - Richard Choulet, Nov 14 2008
G.f.: 1/(1-2x/(1-x/(1-x/(1-x)))) (hence sequence approximates A000984 in first few terms). - Paul Barry, Aug 05 2009
a(n) = (1/5)*Sum_{k=1..5} (x_k)^(2*n), x_k=2*cos((2*k-1)*Pi/10). - L. Edson Jeffery, Apr 19 2011
From R. J. Mathar, Apr 20 2011: (Start)
a(n) = A030191(n) - 3*A030191(n-1) + A030191(n-2).
a(n) = 2*A081567(n-1), n > 0. (End)
a(n) = Sum_{k=0..n} A147746(n,k)*2^k. - Philippe Deléham, Oct 30 2011
E.g.f.: (1 + 4*exp(5*x/2)*cosh(sqrt(5)*x/2))/5. - Stefano Spezia, Jul 09 2024

A189316 Expansion of g.f. 5(1-x-x^2)/((1+x)(1-3*x+x^2)).

Original entry on oeis.org

5, 5, 15, 35, 95, 245, 645, 1685, 4415, 11555, 30255, 79205, 207365, 542885, 1421295, 3720995, 9741695, 25504085, 66770565, 174807605, 457652255, 1198149155, 3136795215, 8212236485, 21499914245, 56287506245, 147362604495, 385800307235, 1010038317215

Offset: 0

L. Edson Jeffery, Apr 20 2011

(Start) Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,2)=
(0 0 1 0 0)
(0 1 0 1 0)
(1 0 1 0 1)
(0 1 0 2 0)
(0 0 2 0 1).
Then a(n)=Trace(A^n). For m=1,2,..., A^(m) can also be written
A^(m)=
[   F(m-1)^2     0     F(m)^2      0      F(m-1)*F(m)      ]
[       0     F(2*m-1)    0      F(2*m)        0           ]
[    F(m)^2      0    F(m+1)^2     0      F(m)*F(m+1)      ]
[       0      F(2*m)     0     F(2*m+1)       0           ]
[ 2*F(m-1)*F(m)  0  2*F(m)*F(m+1)  0  F(2*m+1)-F(m)*F(m+1) ],
where F(m-1)=A000045(n) are the Fibonacci numbers and m=n+1. Hence also a(n+1)=Trace(A^(n+1))=F(m-1)^2+F(2*m-1)+F(m+1)^2+2*F(2*m+1)-F(m)*F(m+1). (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix A_(N,r), 0

L. E. Jeffery, Unit-primitive matrices.
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000045, A061646, A189315, A189317, A189318.

CoefficientList[Series[5 (1-x-x^2)/((1+x)(1-3x+x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,2,-1},{5,5,15},40] (* Harvey P. Dale, Nov 26 2016 *)

G.f.: 5*(1-x-x^2)/((1+x)*(1-3*x+x^2)).
a(n) = 2*a(n-1)+2*a(n-2)-a(n-3), n>2, a(0)=5, a(1)=5, a(2)=15.
a(n) = Sum_{k=1..5} ((w_k)^2-1)^n, w_k = 2*cos((2*k-1)*Pi/10).
a(n) = (-1)^n+2*(1/tau^(2*n)+tau^(2*n)), tau = (1+sqrt(5))/2=1.618033....
a(n) = 5*A061646(n), n>=0 (offset for A061646 is -1).
E.g.f.: cosh(x) + 4*exp(3*x/2)*cosh(sqrt(5)*x/2) - sinh(x). - Stefano Spezia, Jul 09 2024

A189318 Expansion of 5(1-2x)/(1-3x-2x^2+4*x^3).

Original entry on oeis.org

5, 5, 25, 65, 225, 705, 2305, 7425, 24065, 77825, 251905, 815105, 2637825, 8536065, 27623425, 89391105, 289275905, 936116225, 3029336065, 9803137025, 31723618305, 102659784705, 332214042625, 1075067224065, 3478990618625, 11258250133505

Offset: 0

L. Edson Jeffery, Apr 20 2011

(Start) Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,4)=
(0 0 0 0 1)
(0 0 0 2 0)
(0 0 2 0 1)
(0 2 0 2 0)
(2 0 2 0 1).
Then a(n)=Trace(A^n). (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix A_(N,r) (0

L. E. Jeffery, Unit-primitive matrices.
Index entries for linear recurrences with constant coefficients, signature (3, 2, -4).

Cf. A052899.

A189315, A189316, A189317.

CoefficientList[Series[5(1-2x)/(1-3x-2x^2+4x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,2,-4},{5,5,25},30] (* Harvey P. Dale, Jun 02 2014 *)

Vec(5*(1-2*x)/(1-3*x-2*x^2+4*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

G.f.: 5*(1-2*x)/(1-3*x-2*x^2+4*x^3).
a(n)=3*a(n-1)+2*a(n-2)-4*a(n-3), n>3, a(0)=5, a(1)=5, a(2)=25, a(3)=65.
a(n)=Sum_{k=1..5} ((w_k)^4-3*(w_k)^2+1)^n, w_k=2*cos((2*k-1)*Pi/10).
a(n)=1+2*(1-Sqrt(5))^n+2*(1+Sqrt(5))^n.
a(n)=5*A052899(n).

A265185 Non-vanishing traces of the powers of the adjacency matrix for the simple Lie algebra B_4: 2 * ((2 + sqrt(2))^n + (2 - sqrt(2))^n).

Original entry on oeis.org

4, 8, 24, 80, 272, 928, 3168, 10816, 36928, 126080, 430464, 1469696, 5017856, 17132032, 58492416, 199705600, 681837568, 2327939072, 7948081152, 27136446464, 92649623552, 316325601280, 1080003158016, 3687361429504, 12589439401984, 42983034748928

Offset: 0

Tom Copeland, Dec 04 2015

a(n) is the trace of the 2*n-th power of the adjacency matrix M for the simple Lie algebra B_4, given in the Damianou link. M = Matrix[row 1; row 2; row 3; row 4] = Matrix[0,1,0,0; 1,0,1,0; 0,1,0,2; 0,0,1,0]. Equivalently, the trace tr(M^(2*k)) is the sum of the 2*n-th powers of the eigenvalues of M. The eigenvalues are the zeros of the characteristic polynomial of M, which is det(x*I - M) = x^4 - 4*x^2 + 2 = A127672(4,x), and are (+-) sqrt(2 + sqrt(2)) and (+-) sqrt(2 - sqrt(2)), or the four unique values generated by 2*cos((2*n+1)*Pi/8). Compare with A025192 for B_3. The odd power traces vanish.
-log(1 - 4*x^2 + 2*x^4) = 8*x^2/2 + 24*x^4/4 + 80*x^6/6 + ... = Sum_{n>0} tr(M^k) x^k / k = Sum_{n>0} a(n) x^(2k) / 2k gives an aerated version of the sequence a(n), excluding a(0), and exp(-log(1 - 4*x + 2*x^2)) = 1 / (1 - 4*x + 2*x^2) is the e.g.f. for A007070.
As in A025192, the cycle index partition polynomials P_k(x[1],...,x[k]) of A036039 evaluated with the negated power sums, the aerated a(n), are P_2(0,-a(1)) = P_2(0,-8) = -8, P_4(0,-a(1),0,-a(2)) = P_4(0,-8,0,-24) = 48, and all other P_k(0,-a(1),0,-a(2),0,...) = 0 since 1 - 4*x^2 + 2*x^4 = 1 - 8*x^2/2! + 48*x^4/4! = det(I - x M) = exp(-Sum_{k>0} tr(M^k) x^k / k) = exp[P.(-tr(M),-tr(M^2),...)x] = exp[P.(0,-a(1),0,-a(2),...)x].
Because of the inverse relation between the Faber polynomials F_n(b1,b2,...,bn) of A263916 and the cycle index polynomials, F_n(0,-4,0,2,0,0,0,...) = tr(M^n) gives aerated a(n), excluding a(0). E.g., F_2(0,-4) = -2 * -4 = 8, F_4(0,-4,0,2) = -4 * 2 + 2 * (-4)^2 = 24, and F_6(0,-4,0,2,0,0) = -2*(-4)^3 + 6*(-4)*2 = 80.

G. C. Greubel, Table of n, a(n) for n = 0..1000
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2011-2014.
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A006012, A007052, A007070, A025192, A036039, A056236, A127672, A189315, A263916.

[Floor(2 * ((2 + Sqrt(2))^n + (2 - Sqrt(2))^n)): n in [0..30]]; // Vincenzo Librandi, Dec 06 2015

4 LinearRecurrence[{4, -2}, {1, 2}, 30] (* Vincenzo Librandi, Dec 06 2015 and slightly modified by Robert G. Wilson v, Feb 13 2018 *)

my(x='x+O('x^30)); Vec((4-8*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Feb 12 2018

a(n) = 2 * ((2 + sqrt(2))^n + (2 - sqrt(2))^n) = Sum_{k=0..3} 2^(2n) (cos((2k+1)*Pi/8))^(2n) = 2*2^(2n) (cos(Pi/8)^(2n) + cos(3*Pi/8)^(2n)) = 2 Sum_{k=0..1} (exp(i(2k+1)*Pi/8) + exp(-i*(2k+1)*Pi/8))^(2n).
E.g.f.: 2 * e^(2*x) * (e^(sqrt(2)*x) + e^(-sqrt(2)*x)) = 4*e^(2*x)*cosh(sqrt(2)*x) = 2*(exp(4*x*cos(Pi/8)^2) + exp(4*x cos(3*Pi/8)^2) ).
a(n) = 4*A006012(n) = 8*A007052(n-1) = 2*A056236(n).
G.f.: (4-8*x)/(1-4*x+2*x^2). - Robert Israel, Dec 07 2015
Note the preceding o.g.f. is four times that of A006012 and the denominator is y^4 * A127672(4,1/y) with y = sqrt(x). Compare this with those of A025192 and A189315. - Tom Copeland, Dec 08 2015

More terms from Vincenzo Librandi, Dec 06 2015

A189317 Expansion of 5(1-6x+x^2)/(1-10x+5x^2).

Original entry on oeis.org

5, 20, 180, 1700, 16100, 152500, 1444500, 13682500, 129602500, 1227612500, 11628112500, 110143062500, 1043290062500, 9882185312500, 93605402812500, 886643101562500, 8398404001562500, 79550824507812500, 753516225070312500, 7137408128164062500

Offset: 0

L. Edson Jeffery, Apr 20 2011

(Start) Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,3)=
(0 0 0 1 0)
(0 0 1 0 1)
(0 1 0 2 0)
(1 0 2 0 1)
(0 2 0 2 0).
Then a(n)=Trace(A^(2*n)). (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers (here they are A^(2*n)) of a unit-primitive matrix A_(N,r) (0

L. E. Jeffery, Unit-primitive matrices.
Index entries for linear recurrences with constant coefficients, signature (10, -5).

A189315, A189316, A189318.

CoefficientList[Series[5*(1-6x+x^2)/(1-10x+5x^2),{x,0,30}],x] (* or *) Join[ {5},LinearRecurrence[{10,-5},{20,180},30]] (* Harvey P. Dale, Apr 02 2013 *)

Vec(5*(1-6*x+x^2)/(1-10*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

G.f.: 5*(1-6*x+x^2)/(1-10*x+5*x^2).
a(n)=10*a(n-1)-5*a(n-2), n>2, a(0)=5, a(1)=20, a(2)=180.
a(n)=Sum_{k=1..5} ((w_k)^3-2*w_k)^(2*n), w_k=2*cos((2*k-1)*Pi/10).
a(n)=2*((5-2*Sqrt(5))^n+(5+2*Sqrt(5))^n), for n>0, with a(0)=5.

A189334 Expansion of g.f. (1-6x+x^2)/(1-10x+5*x^2).

Original entry on oeis.org

1, 4, 36, 340, 3220, 30500, 288900, 2736500, 25920500, 245522500, 2325622500, 22028612500, 208658012500, 1976437062500, 18721080562500, 177328620312500, 1679680800312500, 15910164901562500, 150703245014062500, 1427481625632812500, 13521300031257812500, 128075592184414062500

Offset: 0

L. Edson Jeffery, Apr 20 2011

(Start) Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,3)=
(0 0 0 1 0)
(0 0 1 0 1)
(0 1 0 2 0)
(1 0 2 0 1)
(0 2 0 2 0).
Then a(n)=(1/5)*Trace(A^(2*n)). (See also A189317.) (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers (here they are A^(2*n)) of a unit-primitive matrix A_(N,r) (0

L. E. Jeffery, Unit-primitive matrices.
Index entries for linear recurrences with constant coefficients, signature (10,-5).

Cf. A189315, A189316, A189317, A189318.

LinearRecurrence[{10,-5},{1,4,36},22] (* Stefano Spezia, Jul 09 2024 *)

a(n) = 10*a(n-1) - 5*a(n-2), n>2, a(0)=1, a(1)=4, a(2)=36.
a(n) = (1/5)*Sum_{k=1..5} ((w_k)^3-2*w_k)^(2*n), w_k = 2*cos((2*k-1)*Pi/10).
From Stefano Spezia, Jul 09 2024: (Start)
a(n) = 2*((5 - 2*sqrt(5))^n + (5 + 2*sqrt(5))^n)/5 for n > 0.
E.g.f.: (1 + 4*exp(5*x)*cosh(2*sqrt(5)*x))/5. (End)

a(20)-a(21) from Stefano Spezia, Jul 09 2024

Showing 1-7 of 7 results.

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A081567 Second binomial transform of F(n+1).

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A147748 Row sums of Riordan array ((1-3x+x^2)/(1-4x+3x^2), x(1-2x)/(1-4x+3x^2)).

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A189316 Expansion of g.f. 5*(1-x-x^2)/((1+x)*(1-3*x+x^2)).

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A189318 Expansion of 5*(1-2*x)/(1-3*x-2*x^2+4*x^3).

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A265185 Non-vanishing traces of the powers of the adjacency matrix for the simple Lie algebra B_4: 2 * ((2 + sqrt(2))^n + (2 - sqrt(2))^n).

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A189317 Expansion of 5*(1-6*x+x^2)/(1-10*x+5*x^2).

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A189334 Expansion of g.f. (1-6*x+x^2)/(1-10*x+5*x^2).

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A189316 Expansion of g.f. 5(1-x-x^2)/((1+x)(1-3*x+x^2)).

A189318 Expansion of 5(1-2x)/(1-3x-2x^2+4*x^3).

A189317 Expansion of 5(1-6x+x^2)/(1-10x+5x^2).

A189334 Expansion of g.f. (1-6x+x^2)/(1-10x+5*x^2).