A002605 -id:A002605 - cp's OEIS frontend

A160175 Expansion of 1/(1 - 2x - 2x^2 - 2x^3 - 2x^4).

1, 2, 6, 18, 54, 160, 476, 1416, 4212, 12528, 37264, 110840, 329688, 980640, 2916864, 8676064, 25806512, 76760160, 228319200, 679123872, 2020019488, 6008445440, 17871816000, 53158809600, 158118181056, 470314504192, 1398926621696, 4161036233088

Offset: 0

Geoffrey Critzer, May 03 2009, May 06 2009

nonn

a(n) is the number of ways two opposing baseball teams could score a combined total of n runs (tallying the score just prior to each "batter up!") considering the order of the scoring as important. Equivalently, a(n) is the number of 2-colored tilings of an n-board with tiles of length at most 4.

a(n) is the number of compositions (ordered partitions) of n into parts <= 4 with 2 sorts of each part. - Joerg Arndt, Aug 06 2019

Arthur Benjamin and Jennifer Quinn, Proofs that Really Count, Mathematical Association of America, 2003, p. 36.

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

Cf. A077835, A002605, A000079.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-2*x-2*x^2-2*x^3-2*x^4))); // G. C. Greubel, Sep 24 2018

RecurrenceTable[{a[n] == 2(a[n - 1] + a[n - 2] + a[n - 3] + a[n - 4]), a[0] == 1, a[1] == 2, a[2] == 6, a[3] == 18}, a, {n, 0, 20}]
LinearRecurrence[{2,2,2,2},{1,2,6,18},30] (* Harvey P. Dale, Oct 27 2013 *)
CoefficientList[Series[1/(1-2*x-2*x^2-2*x^3-2*x^4), {x, 0, 50}], x] (* G. C. Greubel, Sep 24 2018 *)

x='x+O('x^30); Vec(1/(1-2*x-2*x^2-2*x^3-2*x^4)) \\ G. C. Greubel, Sep 24 2018

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

More terms from Harvey P. Dale, Oct 27 2013

A181297 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even entries (0<=k<=n).

Original entry on oeis.org

1, 0, 2, 1, 0, 6, 0, 8, 0, 16, 3, 0, 35, 0, 44, 0, 28, 0, 132, 0, 120, 8, 0, 160, 0, 460, 0, 328, 0, 92, 0, 748, 0, 1528, 0, 896, 21, 0, 642, 0, 3117, 0, 4916, 0, 2448, 0, 290, 0, 3552, 0, 12062, 0, 15456, 0, 6688, 55, 0, 2380, 0, 17119, 0, 44318, 0, 47760, 0, 18272, 0, 888, 0

Offset: 0

Emeric Deutsch, Oct 12 2010

A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

For the statistics "number of odd entries" see A181295.

			T(2,2) = 6 because we have (0 / 2), (2 / 0), (1,0 / 0,1), (0,1 / 1,0), (1,1 / 0,0), (0,0 / 1,1) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
  1;
  0,2;
  1,0,6;
  0,8,0,16;
  3,0,35,0,44;

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Cf. A003480, A000045, A181295, A181296, A181298.

G := (1-z^2)^2/(1-3*z^2+z^4-2*s*z-2*s^2*z^2+s^2*z^4): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 11 do seq(coeff(P[n], s, k), k = 0 .. n) end do; # yields sequence in triangular form

G.f.: G(t,z) = (1-z^2)^2/(1-3*z^2+z^4-2*s*z-2*s^2*z^2+s^2*z^4).
The g.f. H(t,s,z), where z marks the size of the 2-composition and t (s) marks the number of odd (even) entries, is H=1/(1-h), where h=z(t+sz)(2s+tz-sz^2)/(1-z^2)^2.
Sum_{k=0..n} T(n,k) = A003480(n).
T(2*n-1,0) = 0.
T(2*n,0) = A000045(2*n) (Fibonacci numbers).
T(n,k) = 0 if n and k have opposite parities.
T(n,n) = A002605(n+1).
Sum_{k=0..n} k*T(n,k) = A181298(n).

A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, -1, 3, -2, -1, 1, 0, -2, 5, -3, -1, 1, 1, -2, -2, 7, -4, -1, 1, -1, 5, -7, -1, 9, -5, -1, 1, 0, -3, 12, -15, 1, 11, -6, -1, 1, 1, -3, -3, 21, -26, 4, 13, -7, -1, 1, -1, 7, -15, 3, 31, -40, 8, 15, -8, -1, 1, 0, -4, 22, -42

Offset: 0

Philippe Deléham, Nov 12 2011

			Triangle begins :
1
-1, 1
0, -1, 1
1, -1, -1, 1
-1, 3, -2, -1, 1
0, -2, 5, -3, -1, 1
1, -2, -2, 7, -4, -1, 1
-1, 5, -7, -1, 9, -5, -1, 1

Cf. A026729, A063967, A129267, A176971 (diagonals sums).

T(n,k)=T(n-1,k-1)+T(n-2,k-1)-T(n-1,k)-T(n-2,k), T(0,0)=1.
G.f.: 1/(1-(y-1)*x-(y-1)*x^2).
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000748(n), A108520(n), A049347(n), A000007(n), A000045(n+1), A002605(n+1), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for x = -2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.

A368155 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - 2*x^2.

Original entry on oeis.org

1, 1, 3, 2, 3, 7, 3, 9, 5, 15, 5, 15, 26, 3, 31, 8, 30, 43, 63, -15, 63, 13, 54, 104, 87, 144, -81, 127, 21, 99, 203, 273, 115, 333, -275, 255, 34, 177, 416, 549, 609, -9, 806, -789, 511, 55, 315, 811, 1263, 1146, 1260, -725, 2043, -2071, 1023, 89, 555, 1573

Offset: 1

Clark Kimberling, Jan 20 2024

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1    3
   2    3     7
   3    9     5    15
   5   15    26     3    31
   8   30    43    63   -15    63
  13   54   104    87   144   -81    127
  21   99   203   273   115   333   -275   255
Row 4 represents the polynomial p(4,x) = 3 + 9*x + 5*x^2 + 15*x^3, so (T(4,k)) = (3,9,5,15), k=0..3.

Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers 18 (2018) 1-28.

Cf. A000045 (column 1); A000225, (p(n,n-1)); A001787 (row sums), (p(n,1)); A002605 (alternating row sums), (p(n,-1)); A004254, (p(n,-2)); A057084, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368156.

p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 3x - 2x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 3*x, u = p(2,x), and v = 1 - 3*x - 2*x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 - 6*x + x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A038508 Expansion of (1-2x-x^2)/((1-2x)(1-2x+2*x^2)).

Original entry on oeis.org

1, 2, 5, 12, 30, 76, 196, 512, 1352, 3600, 9648, 25984, 70240, 190400, 517184, 1406976, 3831936, 10445056, 28488448, 77735936, 212186624, 579320832, 1581966336, 4320477184, 11800692736, 32233951232

Offset: 0

N. J. A. Sloane

Number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+1, s(0) = 1, s(n+1) = 2. - Herbert Kociemba, Jun 17 2004

Index entries for linear recurrences with constant coefficients, signature (4,-2,-4).

First differences are in A094297.

Cf. A002605.

Vec((1-2*x-x^2)/(1-4*x+2*x^2+4*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = (3*2^(n+1) - sqrt(3)*(1-sqrt(3))^(n+1) + sqrt(3)*(1+sqrt(3))^(n+1))/12. - Herbert Kociemba, Jun 17 2004
2*a(n) = 2^n + A002605(n+1). - R. J. Mathar, Sep 11 2019
a(n) = 4*a(n-1)-2*a(n-2)-4*a(n-3). - Wesley Ivan Hurt, May 14 2021

A052611 Expansion of e.g.f. 1/(1-2x-2x^2).

Original entry on oeis.org

1, 2, 12, 96, 1056, 14400, 236160, 4515840, 98703360, 2426941440, 66305433600, 1992646656000, 65328154214400, 2320237766246400, 88746105588940800, 3636883029491712000, 158978387626426368000, 7383729547341987840000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..380
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 556

Cf. A002605, A080599.

[n le 2 select n else 2*(n-1)*Self(n-1) + 2*(n-1)*(n-2)*Self(n-2): n in [1..41]]; // G. C. Greubel, Jan 31 2023

spec := [S,{S=Sequence(Union(Z,Z,Prod(Z,Union(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[1/(1-2x-2x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 14 2015 *)
Table[n!*(-I*Sqrt[2])^(n)*ChebyshevU[n,I/Sqrt[2]], {n,0,40}] (* G. C. Greubel, Jan 31 2023 *)

A002605=BinaryRecurrenceSequence(2,2,0,1)
def A052611(n): return factorial(n)*A002605(n+1)
[A052611(n) for n in range(41)] # G. C. Greubel, Jan 31 2023

E.g.f.: 1/(1 - 2*x - 2*x^2).
a(n) = 2^n * A080599(n).
a(n) = 2*n*a(n+1) + 2*n*(n-1)*a(n), a(0) = 1, a(1) = 2.
a(n) = (n!/6) * Sum_{p = RootOf(2*z^2+2*z-1)} (1+2*p)*p^(-n-1).
a(n) = n!*A002605(n+1). - R. J. Mathar, Nov 27 2011

A072946 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=bnu(n-1)+lambda(n-1)_q*nu(n-2) with (b,lambda)=(2,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

Original entry on oeis.org

1, 2, 6, 4, 12, 8, 24, 16, 48, 32, 96, 64, 192, 128, 384, 256, 768, 512, 1536, 1024, 3072, 2048, 6144, 4096, 12288, 8192, 24576, 16384, 49152, 32768, 98304, 65536, 196608, 131072, 393216, 262144, 786432, 524288, 1572864, 1048576, 3145728, 2097152

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Instead of listing the coefficients of the highest power of q in each nu(n), if we listed the coefficients of the smallest power of q (i.e., constant terms), we get a weighted Fibonacci numbers described by f(0)=1, f(1)=1, for n>=2, f(n)=2f(n-1)+2f(n-2).

			nu(0)=1,
nu(1)=2,
nu(2)=6,
nu(3)=16+4q,
nu(4)=44+20q+12q^2,
nu(5)=120+80q+64q^2+40q^3+8q^4,
nu(6)=328+288q+280q^2+232q^3+168q^4+64q^5+24q^6.
By listing the coefficients of the highest power in each nu(n) we get 1,2,6,4,12,8,24,...

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Index entries for linear recurrences with constant coefficients, signature (0, 2).

Essentially the same as A162255 and A164073.

Cf. A002605.

LinearRecurrence[{0,2},{1,2,6},50] (* Harvey P. Dale, Dec 31 2015 *)

For given b and lambda, the recurrence relation is given by; t(0)=1, t(1)=b, t(2)=b^2+lambda and for n>=3, t(n)=lambda*T(n-2).

O.g.f.: (1+2*x+4*x^2)/(1-2*x^2). - R. J. Mathar, Dec 05 2007

More terms from R. J. Mathar, Dec 05 2007

A084157 a(n) = 8a(n-1) - 16a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.

Original entry on oeis.org

0, 1, 4, 22, 112, 556, 2704, 13000, 62080, 295312, 1401664, 6644320, 31472896, 149017792, 705395968, 3338614912, 15800258560, 74772443392, 353840161792, 1674425579008, 7923565146112, 37494981225472, 177428889407488

Offset: 0

Paul Barry, May 16 2003

Binomial transform of A084156.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-16,0,12).

Cf. A026150, A083881, A084155, A084156.

Cf. A002605, A003462, A080040.

I:=[0,1,4,22]; [n le 4 select I[n] else 8*Self(n-1) -16*Self(n-2) +12*Self(n-4): n in [1..41]]; // G. C. Greubel, Oct 11 2022

LinearRecurrence[{8,-16,0,12},{0,1,4,22},30] (* Harvey P. Dale, Feb 19 2017 *)

A083881 = BinaryRecurrenceSequence(6,-6,1,3)
A026150 = BinaryRecurrenceSequence(2,2,1,1)
def A084157(n): return (A083881(n) - A026150(n))/2
[A084157(n) for n in range(41)] # G. C. Greubel, Oct 11 2022

a(n) = (A083881(n) - A026150(n))/2.
a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4).
a(n) = ((3+sqrt(3))^n + (3-sqrt(3))^n - (1+sqrt(3))^n - (1-sqrt(3))^n)/4.
G.f.: x*(1-4*x+6*x^2)/((1-2*x-2*x^2)*(1-6*x+6*x^2)).
E.g.f.: exp(2*x)*sinh(x)*cosh(sqrt(3)*x).
From G. C. Greubel, Oct 11 2022: (Start)
a(2*n) = A003462(n)*A026150(2*n) = A003462(n)*A080040(2*n)/2.
a(2*n+1) = (1/2)*(3^(n+1)*A002605(2*n+1) - A026150(2*n+1)). (End)

A086404 Square array of numbers T(n,k) = ((1+sqrt(3))(k+sqrt(3))^n-(1-sqrt(3))(k-sqrt(3))^n)/(2*sqrt(3)), read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 3, 1, 4, 11, 16, 9, 1, 5, 18, 41, 44, 9, 1, 6, 27, 84, 153, 120, 27, 1, 7, 38, 151, 396, 571, 328, 27, 1, 8, 51, 248, 857, 1872, 2131, 896, 81, 1, 9, 66, 381, 1644, 4893, 8856, 7953, 2448, 81, 1, 10, 83, 556, 2889, 10984, 28003, 41904, 29681

Offset: 0

Paul Barry, Jul 19 2003

			Rows begin
  1, 1,  3,   3,   9, ...
  1, 2,  6,  16,  44, ...
  1, 3, 11,  41, 153, ...
  1, 4, 18,  84, 396, ...
  1, 5, 27, 151, 857, ...

Rows include A002605, A079935, A086405. Main diagonal is A086406. Rows are successive binomial transforms of (1, 1, 3, 3, 9, 9, ...).

Cf. A086350.

A099040 Riordan array (1, 2+2x).

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 0, 8, 8, 0, 0, 4, 24, 16, 0, 0, 0, 24, 64, 32, 0, 0, 0, 8, 96, 160, 64, 0, 0, 0, 0, 64, 320, 384, 128, 0, 0, 0, 0, 16, 320, 960, 896, 256, 0, 0, 0, 0, 0, 160, 1280, 2688, 2048, 512, 0, 0, 0, 0, 0, 32, 960, 4480, 7168, 4608, 1024, 0, 0, 0, 0, 0, 0, 384, 4480, 14336, 18432, 10240, 2048

Offset: 0

Paul Barry, Sep 23 2004

Row sums give A002605. Diagonal sums give A052907.
The Riordan array (1,s+t*x) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).
T(n,k) is the number of compositions of n into two types of parts of size 1 and 2 that have exactly k parts. - Geoffrey Critzer, Aug 18 2012.
Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 22 2020

			Rows begin {1}, {0,2}, {0,2,4}, {0,0,8,8}, {0,0,4,24,16}, {0,0,0,24,64,32},...
T(3,2)=8 because we have: 1+2,1+2',1'+2,1'+2',2+1,2+1',2'+1,2'+1' where a part of the second type is designated by '. - _Geoffrey Critzer_, Aug 18 2012

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A002605, A026729, A038208, A052907.

nn = 8; CoefficientList[Series[1/(1 - 2 y x - 2 y x^2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 18 2012 *)

Number triangle T(n, k) = 2^k*binomial(k, n-k).
Columns have g.f. (2x+2x^2)^k.
T(n,k) = A026729(n,k)*2^k. - Philippe Deléham, Jul 28 2006
O.g.f.: 1/(1-2*y*x-2*y*x^2). - Geoffrey Critzer, Aug 18 2012.

cp's OEIS Frontend

A160175 Expansion of 1/(1 - 2*x - 2*x^2 - 2*x^3 - 2*x^4).

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A181297 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k even entries (0<=k<=n).

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A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A368155 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3*x - 2*x^2.

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A038508 Expansion of (1-2*x-x^2)/((1-2*x)*(1-2*x+2*x^2)).

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A052611 Expansion of e.g.f. 1/(1-2*x-2*x^2).

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A072946 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(2,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

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A084157 a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.

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A160175 Expansion of 1/(1 - 2x - 2x^2 - 2x^3 - 2x^4).

A368155 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - 2*x^2.

A038508 Expansion of (1-2x-x^2)/((1-2x)(1-2x+2*x^2)).

A052611 Expansion of e.g.f. 1/(1-2x-2x^2).

A072946 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=bnu(n-1)+lambda(n-1)_q*nu(n-2) with (b,lambda)=(2,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

A084157 a(n) = 8a(n-1) - 16a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.

A086404 Square array of numbers T(n,k) = ((1+sqrt(3))(k+sqrt(3))^n-(1-sqrt(3))(k-sqrt(3))^n)/(2*sqrt(3)), read by antidiagonals.