A098600 -id:A098600 - cp's OEIS frontend

A099926 Duplicate of A098600.

1, 2, 2, 5, 6, 12, 17, 30, 46, 77, 122, 200, 321, 522, 842, 1365, 2206, 3572, 5777, 9350

Offset: 0

dead

A008346 a(n) = Fibonacci(n) + (-1)^n.

1, 0, 2, 1, 4, 4, 9, 12, 22, 33, 56, 88, 145, 232, 378, 609, 988, 1596, 2585, 4180, 6766, 10945, 17712, 28656, 46369, 75024, 121394, 196417, 317812, 514228, 832041, 1346268, 2178310, 3524577, 5702888, 9227464, 14930353, 24157816, 39088170, 63245985, 102334156

Offset: 0

N. J. A. Sloane

Diagonal sums of A059260. - Paul Barry, Oct 25 2004
The absolute value of the Euler characteristic of the Boolean complex of the Coxeter group A_n. - Bridget Tenner, Jun 04 2008
a(n) is the number of compositions (ordered partitions) of n into two sorts of 2's and one sort of 3's.  Example: the a(5)=4 compositions of 5 are 2+3, 2'+3, 3+2 and 3+2'. - Bob Selcoe, Jun 21 2013
Let r = 0.70980344286129... denote the rabbit constant A014565. The sequence 2^a(n) gives the simple continued fraction expansion of the constant r/2 = 0.35490172143064565732 ... = 1/(2^1 + 1/(2^0 + 1/(2^2 + 1/(2^1 + 1/(2^4 + 1/(2^4 + 1/(2^9 + 1/(2^12 + ... )))))))). Cf. A099925. - Peter Bala, Nov 06 2013
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 1; 1, 0, 1; 1, 0, 0] or of the 3 X 3 matrix [0, 1, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014
a(n) is the number of growing self-avoiding walks with n+3 edges on the grid graph of integer points (x,y) with x >= 0 and y in {0, 1} and with a trapped endpoint. - Jay Pantone, Jul 26 2024

			The Boolean complex of Coxeter group A_4 is homotopy equivalent to the wedge of 2 spheres S^3, which has Euler characteristic 1 - 2 = -1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Bilgici, Generalized order-k Pell-Padovan-like numbers by matrix methods, Pure and Applied Mathematics Journal, 2013; 2(6): 174-178.
Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, Predators and altruists arriving on jammed Riviera, arXiv:2401.01225 [math.CO], 2024. See p. 14.
N. Gogin and A. Mylläri, Padovan-like sequences and Bell polynomials, Proceedings of Applications of Computer Algebra ACA, 2013.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 13.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 445
Jay Pantone, A. R. Klotz, and E. Sullivan, Exactly-solvable self-trapping lattice walks. II. Lattices of arbitrary height., arXiv:2407.18205 [math.CO], 2024.
K. Ragnarsson and B. E. Tenner, Homotopy type of the Boolean complex of a Coxeter system, arXiv:0806.0906 [math.CO], 2008-2009.
Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A007492, A066983, A078024, A119282, A014565, A099925, A098600, A374297.

List([0..50], n-> Fibonacci(n) + (-1)^n); # G. C. Greubel, Jul 13 2019

[Fibonacci(n) + (-1)^n: n in [0..50]]; // Vincenzo Librandi, Apr 23 2011

with(combinat): f := n->fibonacci(n)+(-1)^n; seq(f(n), n=0..40);

Table[Fibonacci[n]+(-1)^n,{n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)
CoefficientList[Series[1/(1-2x^2-x^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 10 2013 *)
LinearRecurrence[{0,2,1}, {1,0,2}, 51] (* Ray Chandler, Sep 08 2015 *)

a(n)=fibonacci(n)+(-1)^n \\ Charles R Greathouse IV, Feb 03 2014

[fibonacci(n)+(-1)^n for n in (0..50)] # G. C. Greubel, Jul 13 2019

G.f.: 1/(1 - 2*x^2 - x^3).
a(n) = 2*a(n-2) + a(n-3).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} (-1)^(n-k-j)binomial(j, k). Diagonal sums of A059260. - Paul Barry, Sep 23 2004
From Paul Barry, Oct 04 2004: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)2^(3k-n).
a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)2^k(1/2)^(n-2k). (End)
From Paul Barry, Oct 25 2004: (Start)
G.f.: 1/((1+x)*(1-x-x^2)).
a(n) = Sum_{k=0..n} binomial(n-k-1, k). (End)
a(n) = |1 + (-1)^(n-1)*Fibonacci(n-1)|. - Bridget Tenner, Jun 04 2008
a(n) = A000045(n) + A033999(n). - Michel Marcus, Nov 14 2013
a(n) = Fibonacci(n+1) - a(n-1), with a(0) = 1. - Franklin T. Adams-Watters, Mar 26 2014
a(n) = b(n+1) where b(n) = b(n-1) + b(n-2) + (-1)^(n+1), b(0) = 0, b(1) = 1. See also A098600.  - Richard R. Forberg, Aug 30 2014
a(n) = b(n+2) where b(n) = Sum_{k=1..n} b(n-k)*A000931(k+1), b(0) = 1. - J. Conrad, Apr 19 2017
a(n) = Sum_{j=n+1..2*n+1} F(j) mod Sum_{j=0..n} F(j) for n > 2 and F(j)=A000045(j). - Art Baker, Jan 20 2019

A098599 Riordan array ((1+2*x)/(1+x), (1+x)).

Original entry on oeis.org

1, 1, 1, -1, 2, 1, 1, 0, 3, 1, -1, 0, 2, 4, 1, 1, 0, 0, 5, 5, 1, -1, 0, 0, 2, 9, 6, 1, 1, 0, 0, 0, 7, 14, 7, 1, -1, 0, 0, 0, 2, 16, 20, 8, 1, 1, 0, 0, 0, 0, 9, 30, 27, 9, 1, -1, 0, 0, 0, 0, 2, 25, 50, 35, 10, 1, 1, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, 1, -1, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12, 1, 1, 0, 0, 0, 0, 0, 0, 13, 91, 182, 156, 65, 13, 1

Offset: 0

Paul Barry, Sep 17 2004

			Triangle begins as:
   1;
   1, 1;
  -1, 2, 1;
   1, 0, 3, 1;
  -1, 0, 2, 4, 1;
   1, 0, 0, 5, 5,  1;
  -1, 0, 0, 2, 9,  6,  1;
   1, 0, 0, 0, 7, 14,  7,  1;
  -1, 0, 0, 0, 2, 16, 20,  8, 1;
   1, 0, 0, 0, 0,  9, 30, 27, 9, 1;

G. C. Greubel, Table of n, a(n) for n = 0..1325

Cf. A000007, A005408, A029635, A040000, A079757, A098601.
Row sums are A098600.
Diagonal sums are A098601.
Apart from signs, same as A100218.
Very similar to triangle A111125.

A098599:= func< n,k | n eq 0 select 1 else Binomial(k, n-k) + Binomial(k-1, n-k-1) >;
[A098599(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 27 2024

T[n_, k_]:= If[n==0, 1, Binomial[k,n-k] +Binomial[k-1,n-k-1]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 27 2024 *)

def A098599(n,k): return 1 if n==0 else binomial(k, n-k) + binomial(k-1, n-k-1)
flatten([[A098599(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 27 2024

Triangle: T(n, k) = binomial(k, n-k) + binomial(k-1, n-k-1), with T(0, 0) = 1.
Sum_{k=0..n} T(n, k) = A098600(n) (row sums).
T(n,k) = T(n-1,k-1) - T(n-1,k) + 2*T(n-2,k-1) + T(n-3,k-1), T(0,0)=1, T(1,0)=1, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 09 2014
From G. C. Greubel, Mar 27 2024: (Start)
T(2*n, n) = A040000(n).
T(2*n+1, n) = A000007(n).
T(2*n-1, n) = A005408(n-1), n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = A079757(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A098601(n). (End)

A355021 a(n) = (-1)^n * A000032(n) - 1.

Original entry on oeis.org

1, -2, 2, -5, 6, -12, 17, -30, 46, -77, 122, -200, 321, -522, 842, -1365, 2206, -3572, 5777, -9350, 15126, -24477, 39602, -64080, 103681, -167762, 271442, -439205, 710646, -1149852, 1860497, -3010350, 4870846, -7881197, 12752042, -20633240, 33385281

Offset: 0

Clark Kimberling, Jun 21 2022

There are the partial sums of L(1) - L(2) + L(3) - L(4) + L(5) - ... .
Closely related (Fibonacci, A000045) partial sums of F(1) - F(2) + F(3) - F(4) + F(5) - ... are given by A355020.
Apart from signs, same as A098600 and A181716.

			a(0) = 1;
a(1) = 1 - 3 = -2;
a(2) = 1 - 3 + 4 = 2;
a(3) = 1 - 3 + 4 - 7 = -5.

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

Cf. A000032, A000045, A098600, A181716, A355018, A355019, A355020.

[Lucas(-n) -1: n in [0..50]]; // G. C. Greubel, Mar 17 2024

f[n_] := Fibonacci[n]; g[n_] := LucasL[n];
f1 = Table[(-1)^n f[n] + 1, {n, 0, 40}]   (* A355020 *)
g1 = Table[(-1)^n g[n] - 1, {n, 0, 40}]   (* this sequence *)
LucasL[-Range[0, 50]] - 1 (* G. C. Greubel, Mar 17 2024 *)
LinearRecurrence[{0,2,-1},{1,-2,2},40] (* Harvey P. Dale, Sep 06 2024 *)

[lucas_number2(-n,1,-1) -1 for n in range(51)] # G. C. Greubel, Mar 17 2024

a(n) = 2*a(n-2) - a(n-3) for n >= 3. [Corrected by Georg Fischer, Sep 30 2022]

G.f.: (1 - 2*x)/(1 - 2*x^2 + x^3).

A181716 a(n) = a(n-1) + a(n-2) + (-1)^n, with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 2, 2, 5, 6, 12, 17, 30, 46, 77, 122, 200, 321, 522, 842, 1365, 2206, 3572, 5777, 9350, 15126, 24477, 39602, 64080, 103681, 167762, 271442, 439205, 710646, 1149852, 1860497, 3010350, 4870846, 7881197, 12752042, 20633240, 33385281, 54018522, 87403802

Offset: 0

Robert G. Wilson v, Nov 07 2010

Aside from the first term, duplicate of A098600.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A000032, A000045, A008346, A119282.

First differences of A014217.

I:=[0, 1, 2]; [n le 3 select I[n] else 2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jan 09 2012

[Lucas(n-1)+(-1)^n: n in [0..40]]; // G. C. Greubel, Mar 25 2024

a[0]= 0; a[1]= 1; a[n_]:= a[n]= a[n-1] +a[n-2] +(-1)^n; Array[a,38,0]
LinearRecurrence[{0,2,1},{0,1,2},40] (* Vincenzo Librandi, Jan 09 2012 *)

[lucas_number2(n-1,1,-1)+(-1)^n for n in range(41)] # G. C. Greubel, Mar 25 2024

a(n) = a(n-1) + a(n-2) + (-1)^n.
a(n) = 2*a(n-2) + a(n-3).
a(n) - A000045(n) = A008346(n-2).
G.f.: x*(1+2*x)/(1-2*x^2-x^3). - Colin Barker, Jan 09 2012
a(n) = A000032(n-1) + (-1)^n. - G. C. Greubel, Mar 25 2024
E.g.f.: exp(x/2)*(sqrt(5)*sinh(sqrt(5)*x/2) - cosh(sqrt(5)*x/2)) + exp(-x). - Stefano Spezia, Jun 18 2024

A182143 Number of independent vertex sets in the Moebius ladder graph with 2n nodes (n >= 0).

Original entry on oeis.org

1, 3, 5, 15, 33, 83, 197, 479, 1153, 2787, 6725, 16239, 39201, 94643, 228485, 551615, 1331713, 3215043, 7761797, 18738639, 45239073, 109216787, 263672645, 636562079, 1536796801, 3710155683, 8957108165, 21624372015, 52205852193, 126036076403, 304278004997

Offset: 0

Cesar Bautista, Apr 14 2012

Also the number of vertex covers. - Eric W. Weisstein, Jan 04 2014

Cesar Bautista, Table of n, a(n) for n = 0..1000
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Moebius Ladder
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Cf. A000129, A001333, A002203, A098600, A100227.

I:=[1,3,5]; [n le 3 select I[n] else Self(n-1)+3*Self(n-2)+Self(n-3): n in [1..31]]; // Bruno Berselli, Apr 14 2012

Table[(1 + Sqrt[2])^n + (1 - Sqrt[2])^n - (-1)^n, {n, 0, 30}] (* Bruno Berselli, Apr 14 2012 *)
Table[LucasL[n, 2] - (-1)^n, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
LinearRecurrence[{1, 3, 1}, {1, 3, 5}, 20] (* Eric W. Weisstein, Mar 31 2017 *)
CoefficientList[Series[(-1 - 2 x + x^2)/(-1 + x + 3 x^2 + x^3), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)

Vec((x^2-2*x-1)/((x+1)*(x^2+2*x-1))+O(x^31)) \\ Bruno Berselli, Apr 14 2012

G.f.: (x^2-2*x-1)/((x+1)*(x^2+2*x-1)).
a(n) = (1+sqrt(2))^n + (1-sqrt(2))^n - (-1)^n = A002203(n) - (-1)^n.
a(n) = a(n-1) + 3*a(n-2) + a(n-3) with a(0)=1, a(1)=3, a(2)=5.
From Peter Bala, Jun 29 2015: (Start)
a(n) = Pell(n-1) + Pell(n+1) - (-1)^n.
a(n) = [x^n] ( (1 + 2*x + sqrt(1 + 8*x + 8*x^2))/2 )^n.
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 7*x^3 + 17*x^4 + 41*x^5 + ... = Sum_{n >= 0} A001333*x^n. Cf. A098600. (End)

A099925 a(n) = Lucas(n) + (-1)^n.

Original entry on oeis.org

3, 0, 4, 3, 8, 10, 19, 28, 48, 75, 124, 198, 323, 520, 844, 1363, 2208, 3570, 5779, 9348, 15128, 24475, 39604, 64078, 103683, 167760, 271444, 439203, 710648, 1149850, 1860499, 3010348, 4870848, 7881195, 12752044, 20633238, 33385283, 54018520, 87403804

Offset: 0

Ralf Stephan, Nov 02 2004

Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio and put c = sum {n = 1..inf} 1/2^floor(n*(phi + 2)). The bicimal expansion of the constant c begins 0.001000100100010001001.... The binary digits are the generalized Fibonacci word A221150.

The sequence 2^a(n) for n >= 1 gives the partial quotients, apart from the first, in the simple continued fraction expansion of the constant 1/2*c = 0.06692 72114 83804 90296 ... = 1/(14 + 1/(2^0 + 1/(2^4 + 1/(2^3 + 1/(2^8 + 1/(2^10 + 1/(2^19 + ...))))))). Cf. A008346. - Peter Bala, Nov 06 2013

Colin Barker, Table of n, a(n) for n = 0..1000
Yüksel Soykan, Generalized Pell-Padovan Numbers, Asian Journal of Advanced Research and Reports (2020) Vol. 11, No. 2, 8-28, Article No. 57839.
Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A000032, A008346, A098600, A068397.

CoefficientList[Series[(3 - 2 x^2)/((1 + x) (1 - x - x^2)), {x, 0, 38}], x] (* Michael De Vlieger, Sep 16 2020 *)

Vec((3-2*x^2)/((1+x)*(1-x-x^2)) + O(x^40)) \\ Colin Barker, Jun 03 2016

G.f.: (3-2*x^2)/((1+x)*(1-x-x^2)).
a(0) = 3, a(1) = 0, a(2) = 4 and a(n) = 2*a(n-2) + a(n-3) for n >= 3. - Peter Bala, Nov 06 2013
a(n) = A068397(n) - 1 for n>2.
a(n) = ((-1)^n+(1/2*(1-sqrt(5)))^n+(1/2*(1+sqrt(5)))^n). - Colin Barker, Jun 03 2016

A138123 Antidiagonal sums of a triangle of coefficients of recurrences of the Fibonacci sequence.

Original entry on oeis.org

1, 1, 3, 0, 3, 0, 7, 1, 11, 0, 17, 0, 29, 1, 47, 0, 75, 0, 123, 1, 199, 0, 321, 0, 521, 1, 843, 0, 1363, 0, 2207, 1, 3571, 0, 5777, 0, 9349, 1, 15127, 0, 24475, 0, 39603, 1, 64079, 0, 103681, 0, 167761, 1, 271443, 0, 439203, 0, 710647, 1, 1149851, 0, 1860497, 0

Offset: 1

Paul Curtz, May 04 2008

nonn

Consider the irregular sparse triangle T(p,p) = A000204(p), T(p,2p)= -A033999(p)=(-1)^(p+1), T(p,m) =0 else; 1<=m<=2p, p>=1. Then a(n)=sum_{m=1..[2(n+1)/3]} T(1+n-m,m).
The T are coefficients in recurrences f(n)=sum_{m=1..2p} T(p,m)*f(n-m).
The recurrence for p=1, f(n)=f(n-1)+f(n-2), is satisfied by the Fibonacci sequence A000045. The recurrence for p=2, f(n)=3f(n-2)-f(n-4), is satisfied by A005013, A005247, A075091, A075270, A108362 and A135992.
Conjecture: The Fibonacci sequence F obeys all the recurrences: A000045(n)=F(n)= L(p)*F(n-p)-(-1)^p*F(n-2p), any p>0, L=A000204.
[Proof: conjecture is equivalent to the existence of a g.f. of F with denominator 1-L(p)x^p+(-1)^p*x^(2p). Since 1-x-x^2 is known to be a denominator of such a g.f. of A000045, the conjecture is that 1-L(p)*x^p+(-1)^p*x^(2p) can be reduced to 1-x-x^2. One finds: {1-L(p)*x^p+(-1)^p*x^(2p)}/(1-x-x^2) = sum{n=0..p-1}F(n+1)x^n-sum{n=0..p-2} (-1)^(n+p)F(n+1)x^(2p-n-2) is a polynomial with integer coefficients, which is proved by multiplication with 1-x-x^2 and via F(n)+F(n+1)=F(n+2) and L(n)=F(n-1)+F(n+1). - R. J. Mathar, Jul 10 2008].
Conjecture: The Lucas sequence L also obeys all the recurrences: L(n)= L(p)*L(n-p)-(-1)^p*L(n-2p), any p>0, L=A000204.

			The triangle T(p,m) with Lucas numbers on the diagonal starts
  1, 1;
  0, 3, 0,-1;
  0, 0, 4, 0, 0, 1;
  0, 0, 0, 7, 0, 0, 0,-1;
  0, 0, 0, 0,11, 0, 0, 0, 0, 1;
The antidiagonal sums are a(1)=1. a(2)=0+1=1. a(3)=0+3=3. a(4)=0+0+0=0. a(5)=0+0+4-1=3.

Row sums: Sum_{m=1..2p} T(p,m) = A098600(p).
Conjectures from Chai Wah Wu, Apr 15 2024: (Start)
a(n) = a(n-2) - a(n-3) + a(n-4) + a(n-5) + a(n-7) for n > 7.
G.f.: x*(-x^5 - 2*x^2 - x - 1)/((x + 1)*(x^2 - x + 1)*(x^4 + x^2 - 1)). (End)

Edited and extended by R. J. Mathar, Jul 10 2008

A237498 Riordan array (1/(1-x-x^2), x/(1+2*x)).

Original entry on oeis.org

1, 1, 1, 2, -1, 1, 3, 4, -3, 1, 5, -5, 10, -5, 1, 8, 15, -25, 20, -7, 1, 13, -22, 65, -65, 34, -9, 1, 21, 57, -152, 195, -133, 52, -11, 1, 34, -93, 361, -542, 461, -237, 74, -13, 1, 55, 220, -815, 1445, -1464, 935, -385, 100, -15, 1, 89, -385, 1850, -3705

Offset: 0

Philippe Deléham, Feb 08 2014

First column: Fibonacci numbers A000045(n+1).

			Triangle begins:
   1;
   1,    1;
   2,   -1,    1;
   3,    4,   -3,    1;
   5,   -5,   10,   -5,   1;
   8,   15,  -25,   20,  -7,   1;
  13,  -22,   65,  -65,  34,  -9,  1;
  ...
Production matrix is:
   1,  1;
   1, -2,  1;
   2,  0, -2,  1;
   4,  0,  0, -2,  1;
   8,  0,  0,  0, -2,  1;
  16,  0,  0,  0,  0, -2,  1;
  32,  0,  0,  0,  0,  0, -2,  1;
  64,  0,  0,  0,  0,  0,  0, -2,  1;
  ...

Indranil Ghosh, Rows 0..100, flattened

Columns: A000045, A084179.

nmax=10;Flatten[CoefficientList[Series[CoefficientList[Series[(1 + 2*x) / ((1 + 2*x - y*x) * (1 - x - x^2)), {x, 0, nmax }], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 15 2017 *)

Sum_{k=0..n} T(n,k)*x^k = A000045(n+1), A098600(n), A000032(n+1), A027961(n+1), A027974(n) for x = 0, 1, 2, 3, 4 respectively.
T(n,k) = T(n-1,k-1) - T(n-1,k) + 3*T(n-2,k) - T(n-2,k-1) + 2*T(n-3,k) - T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 2, T(2,1) = -1, T(n,k) = 0 if k<0 or if k>n.
T(n,0) = T(n-1,0) + T(n-2,0) with T(0,0) = T(1,0) = 1, T(n,k) = T(n-1,k-1) - 2*T(n-1,k) for k>=1.
G.f.: (1+2*x)/((1+2*x-y*x)*(1-x-x^2)).

cp's OEIS Frontend

A099926 Duplicate of A098600.

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A008346 a(n) = Fibonacci(n) + (-1)^n.

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A098599 Riordan array ((1+2*x)/(1+x), (1+x)).

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A355021 a(n) = (-1)^n * A000032(n) - 1.

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A181716 a(n) = a(n-1) + a(n-2) + (-1)^n, with a(0)=0 and a(1)=1.

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A182143 Number of independent vertex sets in the Moebius ladder graph with 2n nodes (n >= 0).

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PARI

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A099925 a(n) = Lucas(n) + (-1)^n.

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