A046717 -id:A046717 - cp's OEIS frontend

A104934 Expansion of (1-x)/(1 - 3x - 2x^2).

1, 2, 8, 28, 100, 356, 1268, 4516, 16084, 57284, 204020, 726628, 2587924, 9217028, 32826932, 116914852, 416398420, 1483024964, 5281871732, 18811665124, 66998738836, 238619546756, 849856117940, 3026807447332, 10780134577876, 38394018628292, 136742325040628, 487015012378468, 1734529687216660

Offset: 0

Creighton Dement, Mar 29 2005

A floretion-generated sequence relating A007482, A007483, A007484. Inverse is A046717. Inverse of Fibonacci(3n+1), A033887. Binomial transform is A052984. Inverse binomial transform is A006131. Note: the conjectured relation 2*a(n) = A007482(n) + A007483(n-1) is a result of the FAMP identity dia[I] + dia[J] + dia[K] = jes + fam
Floretion Algebra Multiplication Program, FAMP Code: 1dia[I]tesseq[A*B] with A = - .25'i + .25'j + .25'k - .25i' + .25j' + .25k' - .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' + .25'ki' + .25'kj' - .25e and B = + 'i + i' + 'ji' + 'ki' + e
a(n) is also the number of ways to build a (2 x 2 x n)-tower using (2 X 1 X 1)-bricks (see Exercise 3.15 in Aigner's book). - Vania Mascioni (vmascioni(AT)bsu.edu), Mar 09 2009
a(n) is the number of compositions of n when there are 2 types of 1 and 4 types of other natural numbers. - Milan Janjic, Aug 13 2010
Pisano period lengths: 1, 1, 4, 1, 24, 4, 48, 1, 12, 24, 30, 4, 12, 48, 24, 1,272, 12, 18, 24, ... - R. J. Mathar, Aug 10 2012

M. Aigner, A Course in Enumeration, Springer, 2007, p.103.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
Index entries for linear recurrences with constant coefficients, signature (3,2)

Cf. A006131, A007482 (partial sums), A007483, A007484, A033887, A046717, A052984, A055099, A104935, A122542.

# Following the Pari implementation.
function a(n)
   F = BigInt[0 1; 2 3]
   Fn = F^n * [1; 2]
   Fn[1, 1]
end # Peter Luschny, Jan 06 2019

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 - x)/(1 - 3*x - 2*x^2)); // Vincenzo Librandi, Jul 13 2018

a := proc(n) option remember; `if`(n < 2, [1, 2][n+1], (3*a(n-1) + 2*a(n-2))) end:
seq(a(n), n=0..28); # Peter Luschny, Jan 06 2019

LinearRecurrence[{3, 2}, {1, 2}, 40] (* Vincenzo Librandi, Jul 13 2018 *)
CoefficientList[Series[(1-x)/(1-3x-2x^2),{x,0,40}],x] (* Harvey P. Dale, May 02 2019 *)

a(n)=([0,1; 2,3]^n*[1;2])[1,1] \\ Charles R Greathouse IV, Jun 20 2015

[(i*sqrt(2))^(n-1)*(i*sqrt(2)*chebyshev_U(n, -3*i/(2*sqrt(2))) - chebyshev_U(n-1, -3*i/(2*sqrt(2))) ) for n in (0..40)] # G. C. Greubel, Jun 27 2021

Define A007483(-1) = 1. Then 2*a(n) = A007482(n) + A007483(n-1) (conjecture);
  a(n+2) = 4*A007484(n) (thus 8*A007484(n) = A007482(n+2) + A007483(n+1));
  a(n+1) = 2*A055099(n);
  a(n+2) - a(n+1) - a(n) = A007484(n+1) - A007484(n).
a(0)=1, a(1)=2, a(n) = 3*a(n-1) + 2*a(n-2) for n > 1. - Philippe Deléham, Sep 19 2006
a(n) = Sum_{k=0..n} 2^k*A122542(n,k). - Philippe Deléham, Oct 08 2006
a(n) = ((17+sqrt(17))/34)*((3+sqrt(17))/2)^n + ((17-sqrt(17))/34)*((3-sqrt(17))/2)^n. - Richard Choulet, Nov 19 2008
a(n) = 2*a(n-1) + 4*Sum_{k=0..n-2} a(k) for n > 0. - Vania Mascioni (vmascioni(AT)bsu.edu), Mar 09 2009
G.f.: (1-x)/(1-3*x-2*x^2). - M. F. Hasler, Jul 12 2018
a(n) = (i*sqrt(2))^(n-1)*( i*sqrt(2)*ChebyshevU(n, -3*i/(2*sqrt(2))) - ChebyshevU(n-1, -3*i/(2*sqrt(2))) ). - G. C. Greubel, Jun 27 2021
E.g.f.: exp(3*x/2)*(sqrt(17)*cosh(sqrt(17)*x/2) + sinh(sqrt(17)*x/2))/sqrt(17). - Stefano Spezia, May 24 2024

A102900 a(n) = 3a(n-1) + 4a(n-2), a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 7, 25, 103, 409, 1639, 6553, 26215, 104857, 419431, 1677721, 6710887, 26843545, 107374183, 429496729, 1717986919, 6871947673, 27487790695, 109951162777, 439804651111, 1759218604441, 7036874417767, 28147497671065

Offset: 0

Paul Barry, Jan 17 2005

Binomial transform of A102901.

Hankel transform is = 1,6,0,0,0,0,0,0,0,0,0,0,... - Philippe Deléham, Nov 02 2008

Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. A001045, A004171, A046717, A086901, A102901, A247666 (which appears to be the run length transform of this sequence).

a102900 n = a102900_list !! n
a102900_list = 1 : 1 : zipWith (+)
               (map (* 4) a102900_list) (map (* 3) $ tail a102900_list)
-- Reinhard Zumkeller, Feb 13 2015

[n le 2 select 1 else 3*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 28 2015

a[n_]:=(MatrixPower[{{2,2},{3,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{3, 4}, {1, 1}, 30] (* Vincenzo Librandi, Dec 28 2015 *)

a(n)=([0,1; 4,3]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Mar 28 2016

A102900=BinaryRecurrenceSequence(3,4,1,1)
[A102900(n) for n in range(51)] # G. C. Greubel, Dec 09 2022

G.f.: (1-2*x)/(1-3*x-4*x^2).
a(n) = (2*4^n + 3*(-1)^n)/5.
a(n) = ceiling(4^n/5) + floor(4^n/5) = (ceiling(4^n/5))^2 - (floor(4^n/5))^2.
a(n) + a(n+1) = 2^(2*n+1) = A004171(n).
a(n) = Sum_{k=0..n} binomial(2*n-k, 2*k)*2^k. - Paul Barry, Jan 20 2005
a(n) = upper left term in the 2 X 2 matrix [1,3; 2,2]^n. - Gary W. Adamson, Mar 14 2008
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(8*4^k-3*(-1)^k)/(x*(8*4^k-3*(-1)^k) + (2*4^k+3*(-1)^k)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013
a(n) = 2^(2*n-1) - a(n-1), a(1)=1. - Ben Paul Thurston, Dec 27 2015; corrected by Klaus Purath, Aug 02 2020
From Klaus Purath, Aug 02 2020: (Start)
a(n) = 4*a(n-1) + 3*(-1)^n.
a(n) = 6*4^(n-2) + a(n-2), n>=2. (End)

A124182 A skewed version of triangular array A081277.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 3, 4, 0, 0, 1, 8, 8, 0, 0, 0, 5, 20, 16, 0, 0, 0, 1, 18, 48, 32, 0, 0, 0, 0, 7, 56, 112, 64, 0, 0, 0, 0, 1, 32, 160, 256, 128, 0, 0, 0, 0, 0, 9, 120, 432, 576, 256, 0, 0, 0, 0, 0, 1, 50, 400, 1120, 1280, 512

Offset: 0

Philippe Deléham, Dec 05 2006

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0,...] where DELTA is the operator defined in A084938. Falling diagonal sums in A052980.

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 3, 4;
  0, 0, 1, 8,  8;
  0, 0, 0, 5, 20, 16;
  0, 0, 0, 1, 18, 48,  32;
  0, 0, 0, 0,  7, 56, 112,  64;
  0, 0, 0, 0,  1, 32, 160, 256,  128;
  0, 0, 0, 0,  0,  9, 120, 432,  576,  256;
  0, 0, 0, 0,  0,  1,  50, 400, 1120, 1280, 512;

Cf. A025192 (column sums). Diagonals include A011782, A001792, A001793, A001794, A006974, A006975, A006976.

T(0,0)=T(1,1)=1, T(n,k)=0 if n < k or if k < 0, T(n,k) = T(n-2,k-1) + 2*T(n-1,k-1).
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A090965(n), (-1)^n*A084120(n), (-1)^n*A006012(n), A033999(n), A000007(n), A001333(n), A084059(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively.
Sum_{k=0..floor(n/2)} T(n-k,k) = Fibonacci(n-1) = A000045(n-1).
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 respectively. - Philippe Deléham, Dec 26 2007
Sum_{k=0..n} T(n,k)*(-x)^(n-k) = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x= 0,1,2,3,4,5,6 respectively. - Philippe Deléham, Nov 14 2008
G.f.: (1-y*x)/(1-2y*x-y*x^2). - Philippe Deléham, Dec 04 2011
Sum_{k=0..n} T(n,k)^2 = A002002(n) for n > 0. - Philippe Deléham, Dec 04 2011

A212435 Expansion of e.g.f.: exp(-x) / cosh(2*x).

Original entry on oeis.org

1, -1, -3, 11, 57, -361, -2763, 24611, 250737, -2873041, -36581523, 512343611, 7828053417, -129570724921, -2309644635483, 44110959165011, 898621108880097, -19450718635716001, -445777636063460643, 10784052561125704811, 274613643571568682777

Offset: 0

Michael Somos, Jun 21 2012

sign

			G.f. = 1 - x - 3*x^2 + 11*x^3 + 57*x^4 - 361*x^5 - 2763*x^6 + 24611*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001586, A046717, A188458, A188514, A255883.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x)/Cosh(2*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 10 2018

CoefficientList[Series[2*E^x/(E^(4*x)+1), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 25 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ -x] / Cosh[ 2 x], {x, 0, n}]]; (* Michael Somos, Aug 26 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( exp(-x + A) / cosh( 2*x + A), n))};

@CachedFunction
def p(n,x) :
    if n == 0 : return 1
    w = -1 if n%2 == 0 else  0
    v =  1 if n%2 == 0 else -1
    return v*add(p(k,0)*binomial(n,k)*(x^(n-k)+w) for k in range(n)[::2])
def A212435(n) : return 2^n*p(n, 1/2)
[A212435(n) for n in (0..20)]  # Peter Luschny, Jul 19 2012

E.g.f.: 2 * exp(x) / (exp(4*x) + 1).
E.g.f. is the reciprocal of the e.g.f. of A046717.
a(n) = (-1)^n * A188458(n) = (-1)^floor((n + 1) / 2) * A001586(n).
E.g.f.: 2/E(0), where E(k) = 1 + (-1)^k/(3^k - 3*9^k*x/(3*3^k*x + (-1)^k*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013
G.f.: conjecture T(0)/(1+x), where T(k) = 1 - 4*x^2*(k+1)^2/(4*x^2*(k+1)^2 + (1+ x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013
a(n) ~ n! * (cos(Pi*n/2)-sin(Pi*n/2)) * 2^(2*n+3/2) / Pi^(n+1). - Vaclav Kotesovec, Feb 25 2014
From Peter Bala, Mar 10 2015: (Start)
a(n) = 4^n*E(n,1/4).
O.g.f.: Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - x*(4*k + 1)).
The series expansion exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 - x - x^2 + 5*x^3 + 11*x^4 - 91*x^5 - 391*x^6 + ... appears to have integer coefficients. Cf. A188514, A255883. (End)

A080928 Triangle T(n,k) read by rows: T(n,k) = Sum_{i=0..n} C(n,2i)*C(2i,k).

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 4, 6, 3, 0, 8, 16, 12, 4, 1, 16, 40, 40, 20, 5, 0, 32, 96, 120, 80, 30, 6, 1, 64, 224, 336, 280, 140, 42, 7, 0, 128, 512, 896, 896, 560, 224, 56, 8, 1, 256, 1152, 2304, 2688, 2016, 1008, 336, 72, 9, 0, 512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 10

Offset: 0

Paul Barry, Feb 26 2003

Gives the general solution to a(n) = 2*a(n-1) + k(k+2)*a(n-2), a(0) = a(1) = 1. The value k=1 gives the row sums of the triangle, or 1,1,5,13,... This is A046717, the solution to a(n) = 2*a(n-1) + 3*a(n-2), a(0)=a(1)=1.

Product of A007318 and A007318 with every odd-indexed row set to zero. - Paul Barry, Nov 08 2005

			Triangle begins:
    1;
    1,    0;
    2,    2,    1;
    4,    6,    3,    0;
    8,   16,   12,    4,    1;
   16,   40,   40,   20,    5,    0;
   32,   96,  120,   80,   30,    6,   1;
   64,  224,  336,  280,  140,   42,   7,  0;
  128,  512,  896,  896,  560,  224,  56,  8, 1;
  256, 1152, 2304, 2688, 2016, 1008, 336, 72, 9, 0; etc.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 156.
J-L. Kim, Relation between weight distribution and combinatorial identities, Bulletin of the Institute of Combinatorics and its Applications, Canada, 31, 2001, pp. 69-79.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150).
Saul Schleimer, Bert Wiest, On the conjugacy problem in braid groups: Garside theory and subsurfaces, arXiv:1807.01500 [math.GT], 2018.

Apart from k=n, T(n, k) equals (1/2)*A038207(n, k).

Columns include A011782, 2*A001792, A080929, 4*A080930. Row sums are in A046717.

Table[Sum[Binomial[n, 2 i] Binomial[2 i, k], {i, 0, n}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 11 2018 *)

T(n, n) = (n+1) mod 2, T(n, k) = C(n, k)*2^(n-k-1).
T(n, 0) = A011782(n), T(n, k)=0, k>n, T(2n, 2n)=1, T(2n-1, 2n-1)=0, T(n+1, n)=n+1. Otherwise T(n, k) = T(n-1, k-1) + 2T(n-1, k). Rows are the coefficients of the polynomials in the expansion of (1-x)/((1+k*x)*(1-(k+2)*x)). The main diagonal is 1, 0, 1, 0, 1, 0, ... with g.f. 1/(1-x^2). Subsequent subdiagonals are given by A011782(k)*C(n+k, k) with g.f. A011782(k)/(1-x)^k.
T(n, k) = Sum_{j=0..n} C(n, j)*C(j, k)*(1+(-1)^j)/2; T(n, k) = 2^(n-k-1)*(C(n, k) + (-1)^n*C(0, n-k)). - Paul Barry, Nov 08 2005

Edited by Ralf Stephan, Feb 04 2005

A122951 Number of walks from (0,0) to (n,n) in the region x >= y with the steps (1,0), (0,1), (2,0) and (0,2).

Original entry on oeis.org

1, 1, 5, 22, 117, 654, 3843, 23323, 145172, 921508, 5942737, 38825546, 256431172, 1709356836, 11485249995, 77703736926, 528893901963, 3619228605738, 24884558358426, 171828674445330, 1191050708958096, 8284698825305832

Offset: 0

Arvind Ayyer, Oct 25 2006

When this walk is further restricted to the subset of the plane x-y <= 2, this gives the sequence A046717. Similarly, the sequence for such a walk restricted to x-y <= w (w > 2) is not present in the OEIS. The reference provided proves recurrences for generating functions in w.

			a(2) = 5 because we can reach (2,2) in the following ways:
  (0,0),(1,0),(1,1),(2,1),(2,2)
  (0,0),(2,0),(2,2)
  (0,0),(1,0),(2,0),(2,2)
  (0,0),(2,0),(2,1),(2,2)
  (0,0),(1,0),(2,0),(2,1),(2,2).

Robert Israel, Table of n, a(n) for n = 0..1000
Arvind Ayyer and Doron Zeilberger, The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points, arXiv:math/0610734 [math.CO], 2006-2007.

Cf. A000108, A046717.

N:= 100: # to get a[0] to a[N]
S:= series(RootOf(z^4*F^4-2*z^3*F^3-z^2*F^3+2*z^2*F^2+3*z*F^2-2*z*F-F+1,F), z, N+1):
seq(coeff(S,z,j),j=0..N); # Robert Israel, Feb 18 2013

f[x_] = (2x+Sqrt[4(x-2)x+1] - Sqrt[2]Sqrt[2x(-2x + Sqrt[4(x-2)x+1]-1) + Sqrt[4(x-2)x+1]+1]+1)/(4x^2);
CoefficientList[Series[f[x],{x,0,21}],x]
(* Jean-François Alcover, May 19 2011, after g.f. *)

In Maple, GF is given by solve(z^4*F^4 -2*z^3*F^3 -z^2*F^3 +2*z^2*F^2 +3*z*F^2 -2*z*F-F+1, F);

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 3, 0, 0, 8, 8, 1, 0, 0, 16, 20, 5, 0, 0, 0, 32, 48, 18, 1, 0, 0, 0, 64, 112, 56, 7, 0, 0, 0, 0, 128, 256, 160, 32, 1, 0, 0, 0, 0, 256, 576, 432, 120, 9, 0, 0, 0, 0, 0, 512, 1280, 1120, 400, 50, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 03 2011

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
Skewed version of triangle in A200139.
Triangle without zeros: A207537.
For the version with negative odd numbered columns, which is Riordan ((1-x)/(1-2*x), -x^2/(1-2*x)) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014
This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015

			The triangle T(n,k) begins:
  n\k      0     1     2     3     4    5   6  7 8 9 10 11 ...
  0:       1
  1:       1     0
  2:       2     1     0
  3:       4     3     0     0
  4:       8     8     1     0     0
  5:      16    20     5     0     0    0
  6:      32    48    18     1     0    0   0
  7:      64   112    56     7     0    0   0  0
  8:     128   256   160    32     1    0   0  0 0
  9:     256   576   432   120     9    0   0  0 0 0
  10:    512  1280  1120   400    50    1   0  0 0 0  0
  11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0
  ...  reformatted and extended. - _Wolfdieter Lang_, Aug 06 2014

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Columns include A011782, A001792, A001793, A001794, A006974, A006975, A006976.
Diagonals sums are in A052980.
Cf. A028297, A081265, A124182, A131577, A039991 (zero-columns deleted, unsigned and zeros appended).
Cf. A098158, A200139, A207537.
Cf. A028297 (signed version, zeros deleted). Cf. A034839.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - #)/(1 - 2 #)&, #^2/(1 - 2 #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 for k<0 or for n

Sum_{k=0..n} T(n,k)^2 = A002002(n) for n>0.

Sum_{k=0..n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n), A087455(n), A146559(n), A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 respectively.

G.f.: (1-x)/(1-2*x-y*x^2). - Philippe Deléham, Mar 03 2012

From Peter Bala, Jul 14 2015: (Start)

Factorizes as A034839 * A007318 = (1/(1 - x), x^2/(1 - x)^2) * (1/(1 - x), x/(1 - x)) as a product of Riordan arrays.

T(n,k) = Sum_{i = k..floor(n/2)} binomial(n,2*i) *binomial(i,k). (End)

Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by Wolfdieter Lang, Aug 06 2014

A102345 a(n) = 3^n + (-1)^n.

Original entry on oeis.org

2, 2, 10, 26, 82, 242, 730, 2186, 6562, 19682, 59050, 177146, 531442, 1594322, 4782970, 14348906, 43046722, 129140162, 387420490, 1162261466, 3486784402, 10460353202, 31381059610, 94143178826, 282429536482, 847288609442

Offset: 0

Graeme McRae, Feb 16 2005

a(n) = A105723(n) + 2*(-1)^n; (a(n) + A105723(n))/2 = A000244(n). - Reinhard Zumkeller, Apr 18 2005

Michael De Vlieger, Table of n, a(n) for n = 0..2095
Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,3).

Apart from leading term, same as A084182.

Cf. A000244, A046717, A105723.

Table[3^n+(-1)^n,{n,0,30}] (* or *) LinearRecurrence[{2,3},{2,2},30] (* Harvey P. Dale, Jun 19 2016 *)

[lucas_number2(n,2,-3) for n in range(0, 26)] # Zerinvary Lajos, Apr 30 2009

a(n) = 2*a(n-1) + 3*a(n-2).
From Elmo R. Oliveira, Dec 18 2023: (Start)
G.f.: 2*(1-x)/((1+x)*(1-3*x)).
E.g.f.: exp(-x) + exp(3*x).
a(n) = 2*A046717(n). (End)

A080925 Binomial transform of Jacobsthal gap sequence (A080924).

Original entry on oeis.org

0, 1, 5, 13, 41, 121, 365, 1093, 3281, 9841, 29525, 88573, 265721, 797161, 2391485, 7174453, 21523361, 64570081, 193710245, 581130733, 1743392201, 5230176601, 15690529805, 47071589413, 141214768241, 423644304721, 1270932914165

Offset: 0

Paul Barry, Feb 26 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (2,3).

Cf. A080926. Essentially the same as A046717.

CoefficientList[Series[x (1 + 3 x) / ((1 + x) (1 - 3 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 05 2013 *)

a(n)=Sum{k=1..n, Binomial(n, 2k-2)2^(2k-2)}

a(n)=(3^n-2*0^n+(-1)^n)/2; G.f.: x(1+3x)/((1+x)(1-3x)); E.g.f.: (exp(3x)-2exp(0)+exp(-x))/2. - Paul Barry, May 19 2003

A092437 Triangle read by rows, arising from enumeration of domino tilings of Aztec Pillow-like regions.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 6, 6, 1, 1, 5, 13, 26, 30, 20, 1, 1, 5, 13, 41, 90, 140, 140, 70, 1, 1, 5, 13, 41, 121, 302, 560, 742, 630, 252

Offset: 0

Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004

The rows are of lengths 1, 3, 5, 7, ...
Call the first row row 0 and entries starting from 0. Then entries i=0 through k in row k are A046717(i).
In row k, entry k+1 is sequence A092438 and entry k+2 is sequence A092439.
In row k, entry 2k-1 is A002457(k-1) and entry 2k is A000984(k).

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 5, 6, 6;
  1, 1, 5, 13, 26, 30, 20;
  ...

James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 13).

James Propp, Publications and Preprints.
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics.

Cf. A092438-A092443.

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cp's OEIS Frontend

A104934 Expansion of (1-x)/(1 - 3*x - 2*x^2).

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A102900 a(n) = 3*a(n-1) + 4*a(n-2), a(0)=a(1)=1.

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A124182 A skewed version of triangular array A081277.

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A212435 Expansion of e.g.f.: exp(-x) / cosh(2*x).

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A080928 Triangle T(n,k) read by rows: T(n,k) = Sum_{i=0..n} C(n,2i)*C(2i,k).

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A122951 Number of walks from (0,0) to (n,n) in the region x >= y with the steps (1,0), (0,1), (2,0) and (0,2).

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A201701 Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).

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A104934 Expansion of (1-x)/(1 - 3x - 2x^2).

A102900 a(n) = 3a(n-1) + 4a(n-2), a(0)=a(1)=1.

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).