A073371 -id:A073371 - cp's OEIS frontend

A095977 Expansion of g.f. 2x / ((1+x)^2(1-2*x)^2).

2, 4, 14, 32, 82, 188, 438, 984, 2202, 4852, 10622, 23056, 49762, 106796, 228166, 485448, 1029162, 2174820, 4582670, 9631360, 20194802, 42253724, 88235734, 183927992, 382769082, 795364308, 1650380958, 3420066544, 7078742402, 14634703372, 30223843942, 62356562216

Offset: 1

Ralf Stephan, Jul 16 2004

Number of 2 X 2 tiles in all tilings of a 3 X (n+1) rectangle with 1 X 1 and 2 X 2 square tiles. - Emeric Deutsch, Feb 18 2007

The terms of this sequence have a primitive divisor for all terms beyond the 4th if and only if n is not of the form 4k+2, for some nonnegative integer k. - Anthony Flatters (Anthony.Flatters(AT)uea.ac.uk), Aug 17 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and J. Int. Seq. 17 (2014) #14.1.5
A. Flatters, Prime divisors of some Lehmer-Pierce sequences, arXiv:0708.2190 [math.NT], 2007.
R. P. Grimaldi, Tilings, Compositions, and Generalizations, J. Int. Seq. 13 (2010), 10.6.5, page 7.
Luka Podrug, Horadam cubes, arXiv:2410.03193 [math.CO], 2024. See p. 11.
Helmut Prodinger, On binary representations of integers with digits -1,0,1, Integers 0 (2000), #A08.
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).

Cf. A128099, A073371.

a:=n->n/9*2^(n+2)+1/27*2^(n+3)-2*n/9*(-1)^n-8/27*(-1)^n: seq(a(n),n=1..30); # Emeric Deutsch, Feb 18 2007

Table[(1/27)*((3*n + 2)*2^(n + 2) - (6*n + 8)*(-1)^n) , {n,1,50}] (* G. C. Greubel, Dec 28 2016 *)

Vec(2*x / ((1+x)^2 * (1-2*x)^2) + O(x^50)) \\ Michel Marcus, Nov 07 2015

a(n) = (1/27)*((3*n + 2)*2^(n + 2) - (6*n + 8)*(-1)^n).
a(n) = 2 * A073371(n-1).
a(n) = Sum_{k=0..floor((n+1)/2)} k*2^k*binomial(n+1-k,k). - Emeric Deutsch, Feb 18 2007
E.g.f.: 2*(cosh(x/2) + sinh(x/2))*(15*x*cosh(3*x/2) + (8 + 9*x)*sinh(3*x/2))/27. - Stefano Spezia, Oct 12 2024

A102713 Total sum of odd parts in all compositions of n.

Original entry on oeis.org

1, 2, 8, 18, 48, 110, 260, 586, 1320, 2918, 6412, 13954, 30192, 64926, 138964, 296122, 628664, 1330134, 2805916, 5903090, 12388736, 25942542, 54215268, 113090858, 235502408, 489646150, 1016575020, 2107715426, 4364561680, 9027384958, 18651293172, 38495632794

Offset: 1

Vladeta Jovovic, Feb 06 2005

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).

Cf. A066967, A073371.

a(n)={((15*n+4)*2^(n-1) - 2*(3*n+1)*(-1)^n)/27} \\ Andrew Howroyd, Jan 08 2020

Vec(x*(1 + x^2) / ((1 + x)^2*(1 - 2*x)^2) + O(x^35)) \\ Colin Barker, Jan 08 2020

a(n) = ((15*n+4)*2^(n-1)-2*(3*n+1)*(-1)^n)/27.
From Colin Barker, Jan 08 2020: (Start)
G.f.: x*(1 + x^2) / ((1 + x)^2*(1 - 2*x)^2).
a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) - 4*a(n-4) for n>4.
(End)

Terms a(26) and beyond from Andrew Howroyd, Jan 08 2020

A128100 Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 5, 1, 8, 10, 3, 13, 20, 9, 1, 21, 38, 22, 4, 34, 71, 51, 14, 1, 55, 130, 111, 40, 5, 89, 235, 233, 105, 20, 1, 144, 420, 474, 256, 65, 6, 233, 744, 942, 594, 190, 27, 1, 377, 1308, 1836, 1324, 511, 98, 7, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 987, 3970

Offset: 0

Emeric Deutsch, Feb 18 2007

Row sums are the Jacobsthal numbers (A001045). Column 0 yields the Fibonacci numbers (A000045); the other columns yield convolved Fibonacci numbers (A001629, A001628, A001872, A001873, etc.). Sum_{k=0..floor(n/2)} k*T(n,k) = A073371(n-2).
Triangle T(n,k), with zeros omitted, given by (1, 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. - Philippe Deléham, Jan 24 2012
Riordan array (1/(1-x-x^2), x^2/(1-x-x^2)), with zeros omitted. - Philippe Deléham, Feb 06 2012
Diagonal sums are A000073(n+2) (tribonacci numbers). - Philippe Deléham, Feb 16 2014
Number of induced subgraphs of the Fibonacci cube Gamma(n-1) that are isomorphic to the hypercube Q_k. Example: row n=4 is 5, 5, 1; indeed, the Fibonacci cube Gamma(3) is a square with an additional pendant edge attached to one of its vertices; it has 5 vertices (i.e., Q_0's), 5 edges (i.e., Q_1's) and 1 square (i.e., Q_2). - Emeric Deutsch, Aug 12 2014
Row n gives the coefficients of the polynomial p(n,x) defined as the numerator of the rational function given by f(n,x) = 1 + (x + 1)/f(n-1,x), where f(x,0) = 1. Conjecture: for n > 2, p(n,x) is irreducible if and only if n is a (prime - 2). - Clark Kimberling, Oct 22 2014

			Triangle starts:
   1;
   1;
   2,  1;
   3,  2;
   5,  5,  1;
   8, 10,  3;
  13, 20,  9,  1;
  21, 38, 22,  4;
From _Philippe Deléham_, Jan 24 2012: (Start)
Triangle (1, 1, -1, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,  0;
   3,  2,  0,  0;
   5,  5,  1,  0,  0;
   8, 10,  3,  0,  0,  0;
  13, 20,  9,  1,  0,  0,  0;
  21, 38, 22,  4,  0,  0,  0,  0; (End)
From _Clark Kimberling_, Oct 22 2014: (Start)
Here are the first 4 polynomials p(n,x) as in Comment and generated by Mathematica program:
  1
  2 +  x
  3 + 2x
  5 + 5x + x^2. (End)

C.-P. Chou and H. A. Witek, An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids, MATCH: Commun. Math. Comput. Chem, 68 (2012) 3-30. See Eq. (9). - From _N. J. A. Sloane_, Dec 23 2012
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, preprint.
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, Acta Appl. Math. 117, 2012, 93-105. - _Emeric Deutsch_, Aug 12 2014

Cf. A001045, A000045, A001629, A001628, A001872, A001873, A073371.

Cf. A109466, A119473.

G:=1/(1-z-(1+t)*z^2): Gser:=simplify(series(G,z=0,19)): for n from 0 to 16 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 16 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

p[x_, n_] := 1 + (x + 1)/p[x, n - 1]; p[x_, 1] = 1;
Numerator[Table[Factor[p[x, n]], {n, 1, 20}]]  (* Clark Kimberling, Oct 22 2014 *)

G.f.: 1/(1-z-(1+t)z^2).
Sum_{k=0..n} T(n,k)*x^k = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, and -13, respectively. - Philippe Deléham, Jan 24 2012
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1). - Philippe Deléham, Jan 24 2012
G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x/((2*k+2+ x*(1+y))*x + 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
T(n,k) = Sum_{i=k..floor(n/2)} binomial(n-i,i)*binomial(i,k). See Corollary 3.3 in the Klavzar et al. link. - Emeric Deutsch, Aug 12 2014

A127985 a(n) = floor(2^n*(n/3 + 4/9)).

Original entry on oeis.org

0, 1, 4, 11, 28, 67, 156, 355, 796, 1763, 3868, 8419, 18204, 39139, 83740, 178403, 378652, 800995, 1689372, 3553507, 7456540, 15612131, 32622364, 68040931, 141674268, 294533347, 611436316, 1267611875, 2624702236, 5428361443

Offset: 0

Artur Jasinski, Feb 09 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (corrected by Ray Chandler, Jan 19 2019)
Wieb Bosma, Signed bits and fast exponentiation, J. Th. Nombres de Bordeaux, 13 no. 1 (2001), p. 27-41.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-4,4).

Cf. A073371, A127976, A127978, A127979, A127980, A127981, A127982, A127983, A127984, A073371, A000337.

[(n/3 + 4/9)*2^n - 1/2 + (-1)^n/18: n in [1..40]]; // Vincenzo Librandi, May 26 2011

Table[(n/3 + 4/9) 2^n - 1/2 + (-1)^n/18, {n, 1, 50}]
LinearRecurrence[{4,-3,-4,4},{1,4,11,28},50] (* Harvey P. Dale, May 15 2011 *)

PARI
```
a(n)=(n*3+4)<M. F. Hasler, Oct 07 2014
```

a(n) = (n/3 + 4/9)*2^n - 1/2 + (-1)^n/18.
a(1)=1, a(2)=4, a(3)=11, a(4)=28, a(n) = 4*a(n-1)-3*a(n-2)-4*a(n-3)+4*a(n-4). - Harvey P. Dale, May 15 2011
G.f.: x*(1-2*x^2)/((1-2*x)^2*(1-x^2)). - Harvey P. Dale, May 15 2011
E.g.f.: ((4 + 6*x)*cosh(2*x) - 5*sinh(x) + 4*cosh(x)*((2 + 3*x)*sinh(x) - 1))/9. - Stefano Spezia, May 25 2023

Definition simplified by M. F. Hasler, Oct 07 2014

Sequence extended to a(0)=0 by M. F. Hasler, Oct 08 2014

A301699 Generating function = g(g(x)), where g(x) = g.f. of Jacobsthal numbers A001045.

Original entry on oeis.org

0, 1, 2, 8, 26, 94, 330, 1178, 4186, 14914, 53098, 189122, 673530, 2398834, 8543498, 30428162, 108371354, 385970386, 1374653610, 4895901602, 17437011514, 62102837746, 221182535242, 787753281218, 2805624912090, 9992381298706, 35588393716202

Offset: 0

N. J. A. Sloane, Mar 29 2018

The Dira (2017) article describes this as the self-convolution of A001045, but it is really the self-composition. - N. J. A. Sloane, Apr 07 2019, following a suggestion from Ilya Gutkovskiy. Note that A073371 is the convolution of A001045(n+1) with itself, with g.f.: g(x)^2/x^2, where g(x) = g.f. of A001045.

The Dira (2017) article contains on pages 851 and 852 several other sequences that could be added to the OEIS.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Index entries for linear recurrences with constant coefficients, signature (3,4,-6,-4).

Cf. A001045, A073371, A270863.

I:=[0,1,2,8]; [n le 4 select I[n] else 3*Self(n-1)+4*Self(n-2)-6*Self(n-3)-4*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Mar 30 2018

f:=proc(a,b) local t1;
t1:=(x-a*x^2-b*x^3)/(1-3*a*x+(2*a^2-3*b)*x^2+3*a*b*x^3 + b^2*x^4);
lprint(t1);
series(t1,x,50);
seriestolist(%);
end;
f(1,2);

CoefficientList[Series[(-2 x^3 - x^2 + x) / (4 x^4 + 6 x^3 - 4 x^2 - 3 x + 1), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 30 2018 *)

G.f.: (-2*x^3-x^2+x)/(4*x^4+6*x^3-4*x^2-3*x+1).

a(n) = 3*a(n-1) + 4*a(n-2) - 6*a(n-3) - 4*a(n-4). - Vincenzo Librandi, Mar 30 2018

cp's OEIS Frontend

A095977 Expansion of g.f. 2*x / ((1+x)^2*(1-2*x)^2).

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A102713 Total sum of odd parts in all compositions of n.

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A128100 Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).

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A127985 a(n) = floor(2^n*(n/3 + 4/9)).

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A301699 Generating function = g(g(x)), where g(x) = g.f. of Jacobsthal numbers A001045.

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A095977 Expansion of g.f. 2x / ((1+x)^2(1-2*x)^2).