A127864 -id:A127864 - cp's OEIS frontend

A002605 a(n) = 2*(a(n-1) + a(n-2)), a(0) = 0, a(1) = 1.

0, 1, 2, 6, 16, 44, 120, 328, 896, 2448, 6688, 18272, 49920, 136384, 372608, 1017984, 2781184, 7598336, 20759040, 56714752, 154947584, 423324672, 1156544512, 3159738368, 8632565760, 23584608256, 64434348032, 176037912576, 480944521216, 1313964867584

Offset: 0

Colin Mallows

Individually, both this sequence and A028859 are convergents to 1 + sqrt(3). Mutually, both sequences are convergents to 2 + sqrt(3) and 1 + sqrt(3)/2. - Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Nov 04 2001
The 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) = 2, s(n+1) = 3. - Herbert Kociemba, Jun 02 2004
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 4 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(4). - Cino Hilliard, Sep 25 2005
The Hankel transform of this sequence is [1, 2, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Nov 21 2007
[1, 3; 1, 1]^n *[1, 0] = [A026150(n), a(n)]. - Gary W. Adamson, Mar 21 2008
(1 + sqrt(3))^n = A026150(n) + a(n)*sqrt(3). - Gary W. Adamson, Mar 21 2008
a(n+1) is the number of ways to tile a board of length n using red and blue tiles of length one and two. - Geoffrey Critzer, Feb 07 2009
Starting with offset 1 = INVERT transform of the Jacobsthal sequence, A001045: (1, 1, 3, 5, 11, 21, ...). - Gary W. Adamson, May 12 2009
Starting with "1" = INVERTi transform of A007482: (1, 3, 11, 39, 139, ...). - Gary W. Adamson, Aug 06 2010
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 85, 277, 337 and 340, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A026150, without the first leading 1. - Johannes W. Meijer, Aug 15 2010
The sequence 0, 1, -2, 6, -16, 44, -120, 328, -896, ... (with alternating signs) is the Lucas U(-2,-2)-sequence. - R. J. Mathar, Jan 08 2013
a(n+1) counts n-walks (closed) on the graph G(1-vertex;1-loop,1-loop,2-loop,2-loop). - David Neil McGrath, Dec 11 2014
Number of binary strings of length 2*n - 2 in the regular language (00+11+0101+1010)*. - Jeffrey Shallit, Dec 14 2015
For n >= 1, a(n) equals the number of words of length n - 1  over {0, 1, 2, 3} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Dec 17 2015
a(n+1) is the number of compositions of n into parts 1 and 2, both of two kinds. - Gregory L. Simay, Sep 20 2017
Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1, ..., n} that have neutral elements. - J. Devillet, Sep 28 2017
(1 + sqrt(3))^n = A026150(n) + a(n)*sqrt(3), for n >= 0; integers in the real quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 10 2018
Starting with 1, 2, 6, 16, ..., number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>3, 1>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the third and fourth elements. - Sergey Kitaev, Dec 09 2020
a(n) is the number of tilings of a 2 X n board missing one corner cell, with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares). Compare to A127864. - Greg Dresden and Yilin Zhu, Jul 17 2025

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 8, 29.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA], 2017.
M. Diepenbroek, M. Maus, and A. Stoll, Pattern Avoidance in Reverse Double Lists, Preprint 2015. See Table 3.
Sergio Falcón, Binomial Transform of the Generalized k-Fibonacci Numbers, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Dale Gerdemann Bird Flock, Youtube video, 2011.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=q=2.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 476
D. Jhala, G. P. S. Rathore, and K. Sisodiya, Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124.
Tanya Khovanova, Recursive Sequences
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (39), (41) and (45), lhs, m=2.
D. H. Lehmer, On Lucas's test for the primality of Mersenne's numbers, Journal of the London Mathematical Society 1.3 (1935): 162-165. See U_n.
Toufik Mansour and Mark Shattuck, Pattern avoidance in flattened derangements, Disc. Math. Lett. (2025) Vol. 15, 67-74. See p. 74.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
Alan Prince, Counting parses, Rutgers Optimality Archive, 2010.
Index entries for linear recurrences with constant coefficients, signature (2,2).
Index entries for sequences related to Chebyshev polynomials.
Index entries for Lucas sequences.

First differences are given by A026150.
a(n) = A073387(n, 0), n>=0 (first column of triangle).
Equals (1/3) A083337. First differences of A077846. Pairwise sums of A028860 and abs(A077917).
a(n) = A028860(n)/2 apart from the initial terms.
Row sums of A081577 and row sums of triangle A156710.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Cf. A080953, A052948, A080040, A028859, A030195, A106435, A108898, A125145, A265106, A265107, A265278, A270810, A293005, A293006, A293007.
Cf. A175289 (Pisano periods).
Cf. A002530.
Cf. A127864.

a002605 n = a002605_list !! n
a002605_list =
   0 : 1 : map (* 2) (zipWith (+) a002605_list (tail a002605_list))
-- Reinhard Zumkeller, Oct 15 2011

[Floor(((1 + Sqrt(3))^n - (1 - Sqrt(3))^n)/(2*Sqrt(3))): n in [0..30]]; // Vincenzo Librandi, Aug 18 2011

[n le 2 select n-1 else 2*Self(n-1) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 07 2018

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+2*a[n-2]od: seq(a[n], n=0..33); # Zerinvary Lajos, Dec 15 2008
a := n -> `if`(n<3, n, 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -2));
seq(simplify(a(n)), n=0..29); # Peter Luschny, Dec 16 2015

Expand[Table[((1 + Sqrt[3])^n - (1 - Sqrt[3])^n)/(2Sqrt[3]), {n, 0, 30}]] (* Artur Jasinski, Dec 10 2006 *)
a[n_]:=(MatrixPower[{{1,3},{1,1}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{2, 2}, {0, 1}, 30] (* Robert G. Wilson v, Apr 13 2013 *)
Round@Table[Fibonacci[n, Sqrt[2]] 2^((n - 1)/2), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2016 *)
nxt[{a_,b_}]:={b,2(a+b)}; NestList[nxt,{0,1},30][[All,1]] (* Harvey P. Dale, Sep 17 2022 *)

Vec(x/(1-2*x-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 10 2011

A002605(n)=([2,2;1,0]^n)[2,1] \\ M. F. Hasler, Aug 06 2018

[lucas_number1(n,2,-2) for n in range(0, 30)] # Zerinvary Lajos, Apr 22 2009

a = BinaryRecurrenceSequence(2,2)
print([a(n) for n in (0..29)])  # Peter Luschny, Aug 29 2016

a(n) = (-I*sqrt(2))^(n-1)*U(n-1, I/sqrt(2)) where U(n, x) is the Chebyshev U-polynomial. - Wolfdieter Lang
G.f.: x/(1 - 2*x - 2*x^2).
From Paul Barry, Sep 17 2003: (Start)
E.g.f.: x*exp(x)*(sinh(sqrt(3)*x)/sqrt(3) + cosh(sqrt(3)*x)).
a(n) = (1 + sqrt(3))^(n-1)*(1/2 + sqrt(3)/6) + (1 - sqrt(3))^(n-1)*(1/2 - sqrt(3)/6), for n>0.
Binomial transform of 1, 1, 3, 3, 9, 9, ... Binomial transform is A079935. (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(n - k, k)*2^(n - k). - Paul Barry, Jul 13 2004
a(n) = A080040(n) - A028860(n+1). - Creighton Dement, Jan 19 2005
a(n) = Sum_{k=0..n} A112899(n,k). - Philippe Deléham, Nov 21 2007
a(n) = Sum_{k=0..n} A063967(n,k). - Philippe Deléham, Nov 03 2006
a(n) = ((1 + sqrt(3))^n - (1 - sqrt(3))^n)/(2*sqrt(3)).
a(n) = Sum_{k=0..n} binomial(n, 2*k + 1) * 3^k.
Binomial transform of expansion of sinh(sqrt(3)x)/sqrt(3) (0, 1, 0, 3, 0, 9, ...). E.g.f.: exp(x)*sinh(sqrt(3)*x)/sqrt(3). - Paul Barry, May 09 2003
a(n) = (1/3)*Sum_{k=1..5} sin(Pi*k/2)*sin(2*Pi*k/3)*(1 + 2*cos(Pi*k/6))^n, n >= 1. - Herbert Kociemba, Jun 02 2004
a(n+1) = ((3 + sqrt(3))*(1 + sqrt(3))^n + (3 - sqrt(3))*(1 - sqrt(3))^n)/6. - Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009
Antidiagonals sums of A081577. - J. M. Bergot, Dec 15 2012
G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k + 2 + 2*x)/(x*(4*k + 4 + 2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013
a(n) = 2^(n - 1)*hypergeom([1 - n/2, (1 - n)/2], [1 - n], -2) for n >= 3. - Peter Luschny, Dec 16 2015
Sum_{k=0..n} a(k)*2^(n-k) = a(n+2)/2 - 2^n. - Greg Dresden, Feb 11 2022
a(n) = 2^floor(n/2) * A002530(n). - Gregory L. Simay, Sep 22 2022
From Peter Bala, May 08 2024: (Start)
G.f.: x/(1 - 2*x - 2*x^2) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (k + 2*x + 1)/(1 + k*x) )
Also x/(1 - 2*x - 2*x^2) = Sum_{n >= 0} (2*x)^n *( x*Product_{k = 1..n} (m*k + 2 - m + x)/(1 + 2*m*k*x) ) for arbitrary m (both series are telescoping). (End)
a(n) = A127864(n-1) + A127864(n-2). - Greg Dresden and Yilin Zhu, Jul 17 2025

Edited by N. J. A. Sloane, Apr 15 2009

A167762 a(n) = 2a(n-1)+3a(n-2)-6*a(n-3) starting a(0)=a(1)=0, a(2)=1.

Original entry on oeis.org

0, 0, 1, 2, 7, 14, 37, 74, 175, 350, 781, 1562, 3367, 6734, 14197, 28394, 58975, 117950, 242461, 484922, 989527, 1979054, 4017157, 8034314, 16245775, 32491550, 65514541, 131029082, 263652487, 527304974, 1059392917, 2118785834, 4251920575, 8503841150

Offset: 0

Paul Curtz, Nov 11 2009

Inverse binomial transform yields two zeros followed by A077917 (a signed variant of A127864).
a(n) mod 10 is zero followed by a sequence with period length 8: 0, 1, 2, 7, 4, 7, 4, 5 (repeat).
a(n) is the number of length n+1 binary words with some prefix w such that w contains three more 1's than 0's and no prefix of w contains three more 0's than 1's. - Geoffrey Critzer, Dec 13 2013
From Gus Wiseman, Oct 06 2023: (Start)
Also the number of subsets of {1..n} with two distinct elements summing to n + 1. For example, the a(2) = 1 through a(5) = 14 subsets are:
  {1,2}  {1,3}    {1,4}      {1,5}
         {1,2,3}  {2,3}      {2,4}
                  {1,2,3}    {1,2,4}
                  {1,2,4}    {1,2,5}
                  {1,3,4}    {1,3,5}
                  {2,3,4}    {1,4,5}
                  {1,2,3,4}  {2,3,4}
                             {2,4,5}
                             {1,2,3,4}
                             {1,2,3,5}
                             {1,2,4,5}
                             {1,3,4,5}
                             {2,3,4,5}
                             {1,2,3,4,5}
The complement is counted by A038754.
Allowing twins gives A167936, complement A108411.
For n instead of n + 1 we have A365544, complement A068911.
The version for all subsets (not just pairs) is A366130.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

Cf. A024495, A167710.
First differences are A167936, complement A108411.
Cf. A004526, A004737, A008967, A038754, A046663, A068911, A088809, A093971, A365376, A365544, A366130.

LinearRecurrence[{2,3,-6},{0,0,1},40] (* Harvey P. Dale, Sep 17 2013 *)
CoefficientList[Series[x^2/((2 x - 1) (3 x^2 - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 17 2013 *)
Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#,{2}],n+1]&]],{n,0,10}] (* Gus Wiseman, Oct 06 2023 *)

a(n) mod 9 = A153130(n), n>3 (essentially the same as A154529, A146501 and A029898).
a(n+1)-2*a(n) = 0 if n even, = A000244((1+n)/2) if n odd.
a(2*n) = A005061(n). a(2*n+1) = 2*A005061(n).
G.f.: x^2/((2*x-1)*(3*x^2-1)). a(n) = 2^n - A038754(n). - R. J. Mathar, Nov 12 2009
G.f.: x^2/(1-2*x-3*x^2+6*x^3). - Philippe Deléham, Nov 11 2009

Edited and extended by R. J. Mathar, Nov 12 2009

A220054 Number A(n,k) of tilings of a k X n rectangle using right trominoes and 1 X 1 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 11, 11, 1, 1, 1, 1, 33, 39, 33, 1, 1, 1, 1, 87, 195, 195, 87, 1, 1, 1, 1, 241, 849, 2023, 849, 241, 1, 1, 1, 1, 655, 3895, 16839, 16839, 3895, 655, 1, 1, 1, 1, 1793, 17511, 151817, 249651, 151817, 17511, 1793, 1, 1

Offset: 0

Alois P. Heinz, Dec 03 2012

			A(2,2) = 5, because there are 5 tilings of a 2 X 2 rectangle using right trominoes and 1 X 1 tiles:
  ._._.   ._._.   .___.   .___.   ._._.
  |_|_|   | |_|   | ._|   |_. |   |_| |
  |_|_|   |___|   |_|_|   |_|_|   |___|
Square array A(n,k) begins:
  1,  1,   1,     1,       1,        1,          1,            1, ...
  1,  1,   1,     1,       1,        1,          1,            1, ...
  1,  1,   5,    11,      33,       87,        241,          655, ...
  1,  1,  11,    39,     195,      849,       3895,        17511, ...
  1,  1,  33,   195,    2023,    16839,     151817,      1328849, ...
  1,  1,  87,   849,   16839,   249651,    4134881,     65564239, ...
  1,  1, 241,  3895,  151817,  4134881,  128938297,   3814023955, ...
  1,  1, 655, 17511, 1328849, 65564239, 3814023955, 207866584389, ...

Liang Kai, Antidiagonals n = 0..35, flattened (antidiagonals n = 0..29 from Alois P. Heinz)
Kai Liang, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025. See p. 25, Table 4.
Wikipedia, Tromino

Columns (or rows) k=0+1, 2-10 give: A000012, A127864, A127867, A127870, A220055, A220056, A220057, A220058, A220059, A220060.

Main diagonal gives: A220061.

b:= proc(n, l) option remember; local k, t;
      if max(l[])>n then 0 elif n=0 or l=[] then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; b(n, subsop(k=1, l))+
         `if`(k>1 and l[k-1]=1, b(n, subsop(k=2, k-1=2, l)), 0)+
         `if`(k `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, l_] := b[n, l] = Module[{k, t}, Which[ Max[l] > n , 0, n == 0 || l == {} , 1 , Min[l] > 0 , t := Min[l]; b[n - t, l - t] , True, For[k = 1, True, k++, If[ l[[k]] == 0 , Break[] ] ]; b[n, ReplacePart[l, k -> 1]] + If[k > 1 && l[[k - 1]] == 1, b[n, ReplacePart[l, {k -> 2, k - 1 -> 2}]], 0] + If[k < Length[l] && l[[k + 1]] == 1, b[n, ReplacePart[l, {k -> 2, k + 1 -> 2}]], 0] + If[k < Length[l] && l[[k + 1]] == 0, b[n, ReplacePart[l, {k -> 1, k + 1 -> 2}]] + b[n, ReplacePart[l, {k -> 2, k + 1 -> 1}]] + b[n, ReplacePart[l, {k -> 2, k + 1 -> 2}]], 0] + If[k + 1 < Length[l] && l[[k + 1]] == 0 && l[[k + 2]] == 0, b[n, ReplacePart[l, {k -> 2, k + 1 -> 2, k + 2 -> 2}]], 0] ] ]; a[n_, k_] := If[n >= k, b[n, Array[0 &, k]], b[k, Array[0 &, n]]]; Table [Table [a[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

A127867 Number of tilings of a 3 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

1, 1, 11, 39, 195, 849, 3895, 17511, 79339, 358397, 1620843, 7326991, 33127155, 149766353, 677103839, 3061202815, 13839823275, 62570318397, 282882722979, 1278922980071, 5782057329219, 26140890761969, 118183916056327, 534313772133687, 2415651952691819

Offset: 0

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

nonn

			a(2) = 11 because the 3 X 2 board can be tiled in one way with only square tiles, in 8 ways using one L-tile and 3 square tiles and in 2 ways with 2 L-tiles.

Alois P. Heinz, Table of n, a(n) for n = 0..500
P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8
Index entries for linear recurrences with constant coefficients, signature (3, 7, -1, 2).

Cf. A127864, A127865, A127866, A127868, A127869, A127870.

Column k=3 of A220054. - Alois P. Heinz, Dec 03 2012

Table[Coefficient[Normal[Series[(1 - x)^2/(1 - 3x - 7x^2 + x^3 - 2x^4), {x, 0, 30}]], x, n], {n, 0, 30}]

G.f.: (1-x)^2/(1-3x-7x^2+x^3-2x^4).

A127870 Number of tilings of a 4 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

1, 1, 33, 195, 2023, 16839, 151817, 1328849, 11758369, 103628653, 914646205, 8068452381, 71189251649, 628067760289, 5541284098945, 48888866203241, 431331449340441, 3805499681885145, 33574725778806817, 296219181642118401, 2613448287490035073

Offset: 0

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

nonn

			a(2) = 33 because the 4x2 board can be tiled in one way with only square tiles, in 12 ways using one L-tile and 5 square tiles and in 20 ways with 2 L-tiles and 2 square tiles.

Alois P. Heinz, Table of n, a(n) for n = 0..500
P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8
Index entries for linear recurrences with constant coefficients, signature (5, 34, 6, -72, -28, 74, -10, -4, -4).

Cf. A127864, A127865, A127866, A127867, A127868, A127869.

Column k=4 of A220054. - Alois P. Heinz, Dec 03 2012

Table[Coefficient[Normal[Series[(1 - 4 z - 6 z^2 - 10 z^3 - 8 z^4 - 4 z^5)/(1 - 5z - 34 z^2 - 6 z^3 + 72 z^4 + 28 z^5 - 74 z^6 + 10 z^7 + 4 z^8 + 4 z^9), {x, 0, 30}]], x, n], {n, 0, 30}]

G.f.: (1 - 4 z - 6 z^2 - 10 z^3 - 8 z^4 - 4 z^5) / (1 - 5z - 34 z^2 - 6 z^3 + 72 z^4 + 28 z^5 - 74 z^6 + 10 z^7 + 4 z^8 + 4 z^9).

A127865 Number of square tiles in all tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

2, 8, 30, 108, 354, 1152, 3614, 11204, 34170, 103176, 308598, 916236, 2702834, 7929872, 23155182, 67333140, 195082218, 563367960, 1622185958, 4658753564, 13347741666, 38160007200, 108881256414, 310108078116, 881761288154

Offset: 1

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

			a(2) = 8 because the 2 X 2 board can be tiled either with 4 squares or with a single L-shaped tile (in four orientations) together with a single square tile and thus all the tilings of the 2 X 2 board contain 8 square tiles.

P. Z. Chinn, R. Grimaldi and S. Heubach, Tiling with Ls and Squares, J. Int. Sequences 10 (2007) #07.2.8.
S. Heubach, Tiling with Ls and Squares, 2005.
Index entries for linear recurrences with constant coefficients, signature (2, 7, -4, -20, -16, -4).

Cf. A127864, A127866, A127867, A127868, A127869, A127870, A127871, A127872.

Table[(2n - 12)(-1)^n + (2/3)((9 - 5Sqrt[3])(1 + Sqrt[3])^n + (9 + 5Sqrt[3])(1 - Sqrt[3])^n) + (n/Sqrt[3])((Sqrt[3] - 1)( 1 + Sqrt[3])^n + (Sqrt[3] + 1)(1 - Sqrt[3])^n), {n, 1, 30}]

a(n) = (2*n - 12)*(-1)^n + (2/3)*((9-5*sqrt(3))*(1+sqrt(3))^n + (9+5*sqrt(3))*(1-sqrt(3))^n) + (n/sqrt(3))*((sqrt(3)-1)*(1+sqrt(3))^n+ (sqrt(3)+1)*(1-sqrt(3))^n).

G.f.: 2*x*(1+2*x)/((1+x)^2*(1-2*x-2*x^2)^2). - Colin Barker, Apr 30 2012

A127866 Number of L-shaped tiles in all tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

4, 12, 52, 172, 580, 1852, 5828, 17980, 54788, 165116, 493316, 1463036, 4312068, 12641276, 36887556, 107201532, 310427652, 896045052, 2579017732, 7403843580, 21205303300, 60604891132, 172872744964, 492233179132, 1399272374276

Offset: 2

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

			a(2) = 4 because the 2 X 2 board can be tiled either with 4 squares or with a single L-shaped tile (in four orientations) together with a single square tile and thus all the tilings of the 2 X 2 board contain 4 L-shaped tiles.

P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8
Index entries for linear recurrences with constant coefficients, signature (3, 4, -8, -12, -4).

Cf. A127864, A127865, A127867, A127868, A127869, A127870, A127871, A127872.

Table[Coefficient[Normal[Series[4x^2/((1 + x)(1 - 2x - 2x^2)^2), {x, 0, 20}]], x, n], {n, 0, 20}]

a(n) = 4 (-1)^n - (2/9)[(9-5*Sqrt(3))(1+Sqrt(3))^n + (9+5*Sqrt(3))(1-Sqrt(3))^n] - (n/3)[(1-Sqrt(3))(1+Sqrt(3))^n+ (1+Sqrt(3))(1-Sqrt(3))^n].

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

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.

A127868 Number of square tiles in all tilings of a 3 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

3, 30, 171, 1044, 5691, 30678, 159891, 821100, 4151511, 20764590, 102880755, 505866804, 2471159019, 12004723878, 58037429739, 279405305676, 1340130574407, 6406579480446, 30536794325547, 145166910196116, 688444702671291, 3257788855054518, 15385512460164963

Offset: 1

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

nonn

			a(2) = 30 because the 3 X 2 board can be tiled in one way with only square tiles, in 8 ways using one L-tile and 3 square tiles and in 2 ways with 2 L-tiles, so there are altogether 6 + 8*3 = 30 square tiles in all of the 3x2 tilings.

Robert Israel, Table of n, a(n) for n = 1..1510
P. Z. Chinn, R. Grimaldi and S. Heubach, Tiling with Ls and Squares, J. Int. Sequences 10 (2007) #07.2.8.
S. Heubach, Tiling with Ls and Squares, 2005.
Index entries for linear recurrences with constant coefficients, signature (6,5,-44,-39,2,-29,4,-4).

Cf. A127864, A127865, A127866, A127867, A127869, A127870.

I:=[3,30,171,1044,5691,30678,159891,821100]; [n le 8 select I[n] else 6*Self(n-1)+5*Self(n-2)-44*Self(n-3)-39*Self(n-4)+2*Self(n-5)-29*Self(n-6)+4*Self(n-7)-4*Self(n-8): n in [1..30]]; // Vincenzo Librandi, Dec 23 2015

f:= gfun:-rectoproc({a(n) - 6*a(n-1)-5*a(n-2)+44*a(n-3)+39*a(n-4)-2*a(n-5)+29*a(n-6)-4*a(n-7)+4*a(n-8), a(0) = 0, a(1) = 3, a(2) = 30, a(3) = 171, a(4) = 1044, a(5) = 5691, a(6) = 30678, a(7) = 159891},a(n), remember):
seq(f(n), n=1..40); # Robert Israel, Dec 22 2015

Table[Coefficient[Normal[Series[3x(1-x)^2(1+6x+3x^2)/(1-3x-7x^2+x^3-2x^4)^2, {x, 0, 30}]], x, n], {n, 0, 30}]
LinearRecurrence[{6, 5, -44, -39, 2, -29, 4, -4}, {3, 30, 171, 1044, 5691, 30678, 159891, 821100}, 25] (* Vincenzo Librandi, Dec 23 2015 *)

my(x='x+O('x^100)); Vec(3*x*(1-x)^2*(1+6*x+3*x^2)/(1-3*x-7*x^2+x^3-2*x^4)^2) \\ Altug Alkan, Dec 22 2015

G.f.: 3x(1-x)^2(1+6x+3x^2)/(1-3x-7x^2+x^3-2x^4)^2.

cp's OEIS Frontend

A002605 a(n) = 2*(a(n-1) + a(n-2)), a(0) = 0, a(1) = 1.

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A167762 a(n) = 2*a(n-1)+3*a(n-2)-6*a(n-3) starting a(0)=a(1)=0, a(2)=1.

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A220054 Number A(n,k) of tilings of a k X n rectangle using right trominoes and 1 X 1 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A127867 Number of tilings of a 3 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

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A127870 Number of tilings of a 4 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

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A127865 Number of square tiles in all tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

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A127866 Number of L-shaped tiles in all tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

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A167762 a(n) = 2a(n-1)+3a(n-2)-6*a(n-3) starting a(0)=a(1)=0, a(2)=1.