A090131 -id:A090131 - cp's OEIS frontend

A099456 Expansion of 1/(1 - 4x + 5x^2).

1, 4, 11, 24, 41, 44, -29, -336, -1199, -3116, -6469, -10296, -8839, 16124, 108691, 354144, 873121, 1721764, 2521451, 1476984, -6699319, -34182196, -103232189, -242017776, -451910159, -597551756, -130656229, 2465133864

Offset: 0

Paul Barry, Oct 16 2004

Associated to the knot 9_44 by the modified Chebyshev transform A(x) -> (1/(1+x^2)^2)A(x/(1+x^2)). See A099457 and A099458.

Imaginary part of (2+i)^n. - Gary W. Adamson, Apr 05 2008; Franklin T. Adams-Watters, Jan 06 2009

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Mei Bai, Wenchang Chu, and Dongwei Guo, Reciprocal Formulae among Pell and Lucas Polynomials, Mathematics, 10, 2691, (2022). See p. 5.
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Dror Bar-Natan, The Rolfsen Knot Table
Index entries for linear recurrences with constant coefficients, signature (4,-5).

Cf. A139011, A090131 (inv. bin. transf.)

seq(((2+I)^(n+1) - (2-I)^(n+1))/(2*I),n=0..30);  # James R. Buddenhagen, Dec 29 2017

CoefficientList[Series[1/(1-4*x+5*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 09 2013 *)
Table[((1+2*I)*(2-I)^n + (1-2*I)*(2+I)^n)/2,{n,0,20}] (* Vaclav Kotesovec, Oct 09 2013 *)

[lucas_number1(n,4,5) for n in range(1, 29)] # Zerinvary Lajos, Apr 22 2009

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-5)^k*4^(n-2k).
E.g.f. (with offset 1): exp(2*x)*sin(x). - Zerinvary Lajos, Apr 06 2009 [corrected by Joerg Arndt, Apr 24 2011]
a(n) = 4*a(n-1) - 5*a(n-2), a(0)=1, a(1)=4. - Vincenzo Librandi, Mar 22 2011
From Paul Curtz, Apr 24 2011: (Start)
a(n) - a(n-4) = 40 * A118444(n);
a(n) - a(n-2) = 10 * A139011(n). (End)
a(n) = ((1+2*i)*(2-i)^n + (1-2*i)*(2+i)^n)/2. - Vaclav Kotesovec, Oct 09 2013
a(n) = ((2+i)^(n+1) - (2-i)^(n+1))/(2*i).
Lim sup n->infinity |a(n)|/5^((n+1)/2) = 1. - Vaclav Kotesovec, Oct 09 2013
a(n) = Sum_{k=0..n} (-1)^k*2^(n-2*k)*binomial(n+1,2*k+1). - Gerry Martens, Sep 18 2022
E.g.f.: exp(2*x)*(cos(x) + 2*sin(x)). - Stefano Spezia, Jul 24 2025

A078069 Expansion of (1-x)/(1+2x+2x^2).

Original entry on oeis.org

1, -3, 4, -2, -4, 12, -16, 8, 16, -48, 64, -32, -64, 192, -256, 128, 256, -768, 1024, -512, -1024, 3072, -4096, 2048, 4096, -12288, 16384, -8192, -16384, 49152, -65536, 32768, 65536, -196608, 262144, -131072, -262144, 786432, -1048576, 524288, 1048576, -3145728, 4194304, -2097152, -4194304

Offset: 0

N. J. A. Sloane, Nov 17 2002

Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4,... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (-2,-2).

Cf. A090131.

CoefficientList[Series[(1-x)/(1+2x+2x^2),{x,0,50}],x] (* or *) LinearRecurrence[{-2,-2},{1,-3},50] (* Harvey P. Dale, Jan 19 2012 *)

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

a(n) = (-2)*(a(n-1)+a(n-2)), n>1 ; a(0)=1, a(1)=-3. - Philippe Deléham, Nov 19 2008
a(n) = A108520(n)-A108520(n-1). - R. J. Mathar, Aug 11 2012
G.f.: G(0)*(1 - x)/(2*(1 + x)), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
a(n) = (1/2 - I)*(-1 - I)^n + (1/2 + I)*(-1 + I)^n, n>=0. Taras Goy, Apr 20 2019

A208509 Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.

Original entry on oeis.org

1, 3, 5, 1, 7, 5, 9, 14, 1, 11, 30, 7, 13, 55, 27, 1, 15, 91, 77, 9, 17, 140, 182, 44, 1, 19, 204, 378, 156, 11, 21, 285, 714, 450, 65, 1, 23, 385, 1254, 1122, 275, 13, 25, 506, 2079, 2508, 935, 90, 1, 27, 650, 3289, 5148, 2717, 442, 15, 29, 819, 5005, 9867

Offset: 1

Clark Kimberling, Feb 27 2012

			First five rows:
  1
  3
  5    1
  7    5
  9   14   1
First five polynomials v(n,x):
  1
  3
  5 +   x
  7 +  5x
  9 + 14x + x^2

Columns 0 to 5: A005408, A000330, A005585, A050486, A054333, A057788.
Row sums, v(n,1): A003948.
Alternating row sums, v(n,-1): A090131.
Cf. A208508.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208508 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208509 *)

u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x) + 1, with u(1,x)=1, v(1,x)=1.

Conjecture: T(n,k) = binomial(n-1,2*k+1) + binomial(n,2*k+1). - Knud Werner, Jan 11 2022

A207543 Triangle read by rows, expansion of (1+yx)/(1-2y*x+y(y-1)x^2).

Original entry on oeis.org

1, 0, 3, 0, 1, 5, 0, 0, 5, 7, 0, 0, 1, 14, 9, 0, 0, 0, 7, 30, 11, 0, 0, 0, 1, 27, 55, 13, 0, 0, 0, 0, 9, 77, 91, 15, 0, 0, 0, 0, 1, 44, 182, 140, 17, 0, 0, 0, 0, 0, 11, 156, 378, 204, 19, 0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21, 0

Offset: 0

Philippe Deléham, Feb 24 2012

Previous name was: "A scaled version of triangle A082985."

Triangle, read by rows, given by (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

			Triangle begins :
1
0, 3
0, 1, 5
0, 0, 5, 7
0, 0, 1, 14, 9
0, 0, 0, 7, 30, 11
0, 0, 0, 1, 27, 55, 13
0, 0, 0, 0, 9, 77, 91, 15
0, 0, 0, 0, 1, 44, 182, 140, 17
0, 0, 0, 0, 0, 11, 156, 378, 204, 19
0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21
0, 0, 0, 0, 0, 0, 13, 275, 1122, 1254, 385, 23

R. Zielinski, Induction and Analogy in a Problem of Finite Sums arXiv:1608.04006 [math.GM], 2016.

Cf. A082985 which is another version of this triangle.

Cf. Diagonals : A005408, A000330, A005585, A050486, A054333, A057788. Cf. A119900.

s := (1+y*x)/(1-2*y*x+y*(y-1)*x^2): t := series(s,x,12):
seq(print(seq(coeff(coeff(t,x,n),y,m),m=0..n)),n=0..11); # Peter Luschny, Aug 17 2016

T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 3.
G.f.: (1+y*x)/(1-2*y*x+y*(y-1)*x^2).
Sum_{i, i>=0} T(n+i,n) = A000204(2*n+1).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A078069(n), A000007(n), A003945(n), A111566(n) for x = -1, 0, 1, 2 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A090131(n), A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively.
From Peter Bala, Aug 17 2016: (Start)
Let S(k,n) = Sum_{i = 1..n} i^k. Calculations in Zielinski 2016 suggest the following identity holds involving the p-th row elements of this triangle:
Sum_{k = 0..p} T(p,k)*S(2*k,n) = 1/2*(2*n + 1)*(n*(n + 1))^p.
For example, for row 6 we find S(6,n) + 27*S(8,n) + 55*S(10,n) + 13*S(12,n) = 1/2*(2*n + 1)*(n*(n + 1))^6.
There appears to be a similar result for the odd power sums S(2*k + 1,n) involving A119900. (End)

New name using a formula of the author from Peter Luschny, Aug 17 2016

A106664 Expansion of g.f.: (1-3x+x^2)/((1-x)(1+x)(1-2x+2*x^2)).

Original entry on oeis.org

-1, 1, 2, 5, 4, 1, -8, -15, -16, 1, 32, 65, 64, 1, -128, -255, -256, 1, 512, 1025, 1024, 1, -2048, -4095, -4096, 1, 8192, 16385, 16384, 1, -32768, -65535, -65536, 1, 131072, 262145, 262144, 1, -524288, -1048575, -1048576, 1, 2097152, 4194305, 4194304, 1, -8388608, -16777215, -16777216, 1, 33554432

Offset: 0

Creighton Dement, May 13 2005

Superseeker finds that a(n+2) - a(n) = A090131(n+1) (or with different signs, see A078069).

Floretion Algebra Multiplication Program, FAMP Code: 2ibaseiseq[ + .5'i + .5i' - .5'ii' + .5'jj' + .5'kk' + .5e]

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,-2,2).

Cf. A000035, A010673, A078069, A090131, A099087, A146559.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(  (1-3*x+x^2)/((1-x^2)*(1-2*x+2*x^2)) )); // G. C. Greubel, Sep 08 2021

CoefficientList[Series[(1-3x+x^2)/((1-x)(1+x)(1-2x+2x^2)),{x,0,60}],x] (* Harvey P. Dale, Mar 20 2013 *)

def A106664_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( sinh(x) -exp(x)*(cos(x)-sin(x)) ).egf_to_ogf().list()
A106664_list(50) # G. C. Greubel, Sep 08 2021

a(n) = (1/2)*(A010673(n) - A099087(n+2)).
a(n) = (1/2)*(1 - (-1)^n - (1-i)^(n+1) - (1+i)^(n+1)), with i=sqrt(-1). - Ralf Stephan, Nov 16 2010
From G. C. Greubel, Sep 08 2021: (Start)
a(n) = (1-(-1)^n)/2 - 2^((n+1)/2)*cos((n+1)*Pi/4).
a(n) = A000035(n) - A146559(n).
E.g.f.: sinh(x) - exp(x)*(cos(x) - sin(x)). (End)

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A099456 Expansion of 1/(1 - 4*x + 5*x^2).

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A078069 Expansion of (1-x)/(1+2*x+2*x^2).

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A208509 Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.

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A207543 Triangle read by rows, expansion of (1+y*x)/(1-2*y*x+y*(y-1)*x^2).

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A106664 Expansion of g.f.: (1-3*x+x^2)/((1-x)*(1+x)*(1-2*x+2*x^2)).

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A099456 Expansion of 1/(1 - 4x + 5x^2).

A078069 Expansion of (1-x)/(1+2x+2x^2).

A207543 Triangle read by rows, expansion of (1+yx)/(1-2y*x+y(y-1)x^2).

A106664 Expansion of g.f.: (1-3x+x^2)/((1-x)(1+x)(1-2x+2*x^2)).