A052947 -id:A052947 - cp's OEIS frontend

A128099 Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).

1, 1, 1, 2, 1, 4, 1, 6, 4, 1, 8, 12, 1, 10, 24, 8, 1, 12, 40, 32, 1, 14, 60, 80, 16, 1, 16, 84, 160, 80, 1, 18, 112, 280, 240, 32, 1, 20, 144, 448, 560, 192, 1, 22, 180, 672, 1120, 672, 64, 1, 24, 220, 960, 2016, 1792, 448, 1, 26, 264, 1320, 3360, 4032, 1792, 128, 1, 28

Offset: 0

Emeric Deutsch, Feb 18 2007

Row sums are the Jacobsthal numbers (A001045).
Apparently, T(n,k)/2^n equals the probability P that n will occur as a partial sum in a randomly-generated infinite sequence of 1s and 2s with n compositions (ordered partitions) into (n-2k) 1s and k 2s. Example: T(6,2)=24; P = 3/8 (24/2^6) that 6 will occur as a partial sum in the sequence with 2 (6-2*2) 1s and 2 2s. - Bob Selcoe, Jul 06 2013
From Johannes W. Meijer, Aug 28 2013: (Start)
The antidiagonal sums are A077949 and the backwards antidiagonal sums are A052947.
Moving the terms in each column of this triangle, see the example, upwards to row 0 gives the Pell-Jacobsthal triangle A013609 as a square array. (End)
The numbers in rows of the triangle are along "first layer" skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along (first layer) skew diagonals pointing top-left in center-justified triangle given in A038207 ((2+x)^n), see links. - Zagros Lalo, Jul 31 2018
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.000..., when n approaches infinity. - Zagros Lalo, Jul 31 2018
It appears that the rows of this array are the coefficients of the Jacobsthal polynomials (see MathWorld link). - Michel Marcus, Jun 15 2019

			Triangle starts:
  1;
  1;
  1,  2;
  1,  4;
  1,  6,  4;
  1,  8, 12;
  1, 10, 24,  8;
  1, 12, 40, 32;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 80-83, 357-358

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Intrinsic Properties of a Non-Symmetric Number Triangle, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
Richard Fors, Independence Complexes of Certain Families of Graphs, Master's thesis in Mathematics at KTH, presented Aug 19 2011.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares arXiv:1609.03964 [math.CO] (2016).
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n
Eric Weisstein's World of Mathematics, Jacobsthal Polynomial

Cf. (Triangle sums) A001045, A095977, A077949, A052947, A113726, A052942, A077909.
Cf. (Similar triangles) A008315, A011973, A102541.
Cf. A013609, A038207
Cf. A207538

T := proc(n,k) if k<=n/2 then 2^k*binomial(n-k,k) else 0 fi end: for n from 0 to 16 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form
T := proc(n, k) option remember: if k<0 or k > floor(n/2) then return(0) fi: if k = 0 then return(1) fi: 2*procname(n-2, k-1) + procname(n-1, k): end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..13); # Johannes W. Meijer, Aug 28 2013

Table[2^k*Binomial[n - k, k] , {n,0,25}, {k,0,Floor[n/2]}] // Flatten  (* G. C. Greubel, Dec 28 2016 *)
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Zagros Lalo, Jul 31 2018 *)

T(n, k) = 2^k*binomial(n-k,k) = 2^k*A011973(n,k).
G.f.: 1/(1-z-2*t*z^2).
Sum_{k=0..floor(n/2)} k*T(n,k) = A095977(n-1).
From Johannes W. Meijer, Aug 28 2013: (Start)
T(n, k) = 2*T(n-2, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, k) = 0 for k < 0 and k > floor(n/2).
T(n, k) = A013609(n-k, k), n >= 0 and 0 <= k <= floor(n/2). (End)

A159288 Expansion of (1 + x + x^2)/(1 - x^2 - 2*x^3).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 15, 24, 35, 54, 83, 124, 191, 290, 439, 672, 1019, 1550, 2363, 3588, 5463, 8314, 12639, 19240, 29267, 44518, 67747, 103052, 156783, 238546, 362887, 552112, 839979, 1277886, 1944203, 2957844, 4499975, 6846250, 10415663

Offset: 0

Creighton Dement, Apr 08 2009

A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj')

Starting with offset 1 the sequence appears to be the INVERT transform of (1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, ...). - Gary W. Adamson, Aug 27 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier [broken link]
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A159284, A159285, A159286, A159287.

I:=[1, 1, 2]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 27 2018

CoefficientList[Series[(1+x+x^2)/(1-x^2-2x^3),{x,0,50}],x]  (* Harvey P. Dale, Mar 09 2011 *)
LinearRecurrence[{0, 1, 2}, {1, 1, 2}, 50] (* G. C. Greubel, Jun 27 2018 *)

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[1;1;2])[1,1] \\ Charles R Greathouse IV, Aug 27 2016

Vec((1 + x + x^2) / (1 - x^2 - 2*x^3) + O(x^40)) \\ Colin Barker, Apr 29 2019

a(n) = A159287(n) + A159287(n+1) + A159287(n+2). - R. J. Mathar, Apr 10 2009
a(n) = a(n-2) + 2*a(n-3) for n>2. - Colin Barker, Apr 29 2019
a(n)= A052947(n) + A052947(n-1) +A052947(n-2). - R. J. Mathar, Mar 23 2023

A183111 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [RED ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; BLUE] pre-colored puzzle.

Original entry on oeis.org

0, 1, 3, 9, 25, 75, 223, 665, 1993, 5971, 17903, 53697, 161065, 483163, 1449439, 4348233, 13044585, 39133571, 117400431, 352200881, 1056601993, 3169805003, 9509413535, 28528238329, 85584711561, 256754129459, 770262380399, 2310787129121, 6932361368937

Offset: 0

Uri Levy, Dec 25 2010

A. The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [RED ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; BLUE], given in [Source ; Intermediate ; Destination] order. The solution algorithm producing the sequence is optimal (the sequence presented gives the minimum number of moves to solve the puzzle for the given pre-coloring configurations). Optimal solutions are discussed and their optimality is proved in link 2 listed below.
B. Disk numbering is from largest disk (k = 1) to smallest disk (k = N)
C. The above-listed "original" sequence generates a "partial-sums" sequence - describing the total number of moves required to solve the puzzle.
D. The number of moves of disk k, for large k, is close to (10/11)*3^(k-1) ~ 0.909*3^(k-1). Series designation: P909(k).

Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173; arXiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
Web applet to play The Magnetic Tower of Hanoi
Index entries for linear recurrences with constant coefficients, signature (3,1,-1,-6).

A000244 "Powers of 3" is the sequence (also) describing the number of moves of the k-th disk solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi.

Cf. A183111 - A183125.

LinearRecurrence[{3,1,-1,-6},{0,1,3,9,25},30] (* Harvey P. Dale, Apr 30 2018 *)

G.f.: -x*(-1+4*x^3+x^2) / ( (3*x-1)*(2*x^3+x^2-1) ).
Recurrence Relations (a(n)=P909(n) as in referenced paper):
a(n) = a(n-2) + a(n-3) + 2*3^(n-2) + 2*3^(n-4) ; n >= 4
Closed-Form Expression:
Define:
λ1 = [1+sqrt(26/27)]^(1/3) +  [1-sqrt(26/27)]^(1/3)
λ2 = -0.5* λ1 + 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)}
λ3 = -0.5* λ1 - 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)}
AP = [(1/11)* λ2* λ3 - (3/11)*(λ2 + λ3) + (9/11)]/[( λ2 - λ1)*( λ3 - λ1)]
BP = [(1/11)* λ1* λ3 - (3/11)*(λ1 + λ3) + (9/11)]/[( λ1 - λ2)*( λ3 - λ2)]
CP = [(1/11)* λ1* λ2 - (3/11)*(λ1 + λ2) + (9/11)]/[( λ2 - λ3)*( λ1 - λ3)]
For any n > 0:
a(n) = (10/11)*3^(n-1) + AP* λ1^(n-1) + BP* λ2^(n-1) + CP* λ3^(n-1)
33*a(n) = 10*3^n -3*( A052947(n-2) -A052947(n-1) -4*A052947(n) ). - R. J. Mathar, Feb 05 2020

More terms from Harvey P. Dale, Apr 30 2018

A159284 Expansion of x(1+x)/(1-x^2-2x^3).

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 5, 9, 11, 19, 29, 41, 67, 99, 149, 233, 347, 531, 813, 1225, 1875, 2851, 4325, 6601, 10027, 15251, 23229, 35305, 53731, 81763, 124341, 189225, 287867, 437907, 666317, 1013641, 1542131, 2346275, 3569413, 5430537

Offset: 0

Creighton Dement, Apr 08 2009

a(n) is the number of composition of n+1 into parts congruent to 0 or 2 modulo 3. - Joerg Arndt, Apr 21 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
Matthias Beck and Neville Robbins, Variations on a Generatingfunctional Theme: Enumerating Compositions with Parts Avoiding an Arithmetic Sequence, arXiv:1403.0665 [math.NT], 2014.
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A159285, A159286, A159287, A159288.

I:=[0,1,1]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 27 2018

CoefficientList[Series[x (1+x)/(1-x^2-2x^3),{x,0,50}],x] (* or *) LinearRecurrence[ {0,1,2},{0,1,1},50] (* Harvey P. Dale, Jul 16 2013 *)

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[0;1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = abs(A078028(n-1)). - R. J. Mathar, Jul 05 2012
a(n) = a(n-2) + 2*a(n-3), a(0)=0, a(1) = a(2) =1. - G. C. Greubel, Apr 30 2017
a(n) = A052947(n-1)+A052947(n-2). - R. J. Mathar, Mar 23 2023

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A219946 Number A(n,k) of tilings of a k X n rectangle using right trominoes and 2 X 2 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 4, 4, 4, 4, 0, 1, 1, 0, 5, 0, 6, 0, 5, 0, 1, 1, 0, 6, 8, 16, 16, 8, 6, 0, 1, 1, 0, 13, 0, 37, 0, 37, 0, 13, 0, 1, 1, 0, 16, 16, 92, 136, 136, 92, 16, 16, 0, 1, 1, 0, 25, 0, 245, 0, 545, 0, 245, 0, 25, 0, 1

Offset: 0

Alois P. Heinz, Dec 01 2012

			A(4,4) = 6, because there are 6 tilings of a 4 X 4 rectangle using right trominoes and 2 X 2 tiles:
  .___.___. .___.___. .___.___. .___.___. .___.___. .___.___.
  | . | . | | ._|_. | | ._| . | | ._|_. | | ._|_. | | . |_. |
  |___|___| |_| . |_| |_| |___| |_| ._|_| |_|_. |_| |___| |_|
  | . | . | | |___| | | |___| | | |_| . | | . |_| | | |___| |
  |___|___| |___|___| |___|___| |___|___| |___|___| |___|___|
Square array A(n,k) begins:
  1,  1,  1,  1,   1,    1,     1,      1,       1,        1, ...
  1,  0,  0,  0,   0,    0,     0,      0,       0,        0, ...
  1,  0,  1,  2,   1,    4,     5,      6,      13,       16, ...
  1,  0,  2,  0,   4,    0,     8,      0,      16,        0, ...
  1,  0,  1,  4,   6,   16,    37,     92,     245,      560, ...
  1,  0,  4,  0,  16,    0,   136,      0,    1128,      384, ...
  1,  0,  5,  8,  37,  136,   545,   2376,   10534,    46824, ...
  1,  0,  6,  0,  92,    0,  2376,   5504,   71248,   253952, ...
  1,  0, 13, 16, 245, 1128, 10534,  71248,  652036,  5141408, ...
  1,  0, 16,  0, 560,  384, 46824, 253952, 5141408, 44013568, ...

Alois P. Heinz, Antidiagonals n = 0..35, flattened
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=0-10 give: A000012, A000007, A052947, A077957, A165799, A190759, A219947, A219948, A219949, A219950, A219951.

Main diagonal gives: A219952.

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;
         `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}, If[Max[l] > n , 0 , If [n == 0 || l == {},1 , If[Min[l] > 0, t = Min[l]; b[n-t, l-t], For[k = 1, k <= Length[l], k++, If[l[[k]] == 0 , Break[]]]; 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 -> 2, k+1 -> 2}]] + b[n, ReplacePart[l, {k -> 1, k+1 -> 2}]] + b[n, ReplacePart[l, {k -> 2, k+1 -> 1}]], 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_, ] := 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, Nov 26 2013, translated from Alois P. Heinz's Maple program *)

A110509 Riordan array (1, x(1-2x)).

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 0, -4, 1, 0, 0, 4, -6, 1, 0, 0, 0, 12, -8, 1, 0, 0, 0, -8, 24, -10, 1, 0, 0, 0, 0, -32, 40, -12, 1, 0, 0, 0, 0, 16, -80, 60, -14, 1, 0, 0, 0, 0, 0, 80, -160, 84, -16, 1, 0, 0, 0, 0, 0, -32, 240, -280, 112, -18, 1, 0, 0, 0, 0, 0, 0, -192, 560, -448, 144, -20, 1, 0, 0, 0, 0, 0, 0, 64, -672, 1120, -672, 180, -22, 1

Offset: 0

Paul Barry, Jul 24 2005

Inverse is Riordan array (1,xc(2x)) [A110510]. Row sums are A107920(n+1). Diagonal sums are (-1)^n*A052947(n).

			Rows begin
1;
0,  1;
0, -2,  1;
0,  0, -4,  1;
0,  0,  4, -6,  1;
0,  0,  0, 12, -8,   1;
0,  0,  0, -8, 24, -10, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[n_, k_] := (-2)^(n - k)*Binomial[k, n - k]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 29 2017 *)

for(n=0,25, for(k=0,n, print1((-2)^(n-k)*binomial(k, n-k), ", "))) \\ G. C. Greubel, Aug 29 2017

Number triangle: T(n, k) = (-2)^(n-k)*binomial(k, n-k).

T(n,k) = A109466(n,k)*2^(n-k). - Philippe Deléham, Oct 26 2008

A159287 Expansion of x^2/(1-x^2-2*x^3).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 4, 5, 6, 13, 16, 25, 42, 57, 92, 141, 206, 325, 488, 737, 1138, 1713, 2612, 3989, 6038, 9213, 14016, 21289, 32442, 49321, 75020, 114205, 173662, 264245, 402072, 611569, 930562, 1415713, 2153700, 3276837, 4985126, 7584237, 11538800

Offset: 0

Creighton Dement, Apr 08 2009

A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj').
From Greg Dresden, Nov 15 2024: (Start)
a(n) is the number of ways to tile a 2 X (n+1) board with L-shaped trominos and S-shaped quadrominos, where the first tile must be an upright L. For example, here are the a(7)=4 ways to tile a 2 X 8 board:
  ._____________.   ._____________.
  | |  |  |  | |   | |  |  |  | |
  |_|_||__|___|   |_|___|||___|
  ._____________.   ._____________.
  | |  | |  |  |   | |  |  | |  |
  |_|_|_|___||   |__|___||__|_| (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier.
Yüksel Soykan, A Study on Generalized Jacobsthal-Padovan Numbers, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 227-251.
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A159284, A159285, A159286, A159288.

Essentially the same as A052947.

I:=[0,0,1]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 27 2018

LinearRecurrence[{0, 1, 2}, {0, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, May 24 2011 *)
CoefficientList[Series[x^2/(1-x^2-2x^3),{x,0,50}],x] (* Harvey P. Dale, May 29 2021 *)

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[0;0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x^2/(1-x^2-2*x^3).
a(n) = A052947(n-2). - R. J. Mathar, Nov 10 2009
a(n) = a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, May 23 2023
From Greg Dresden, Nov 17 2024: (Start)
a(2*n+1) = 2*a(n)^2 + 2*a(n+1)*a(n+2).
a(3*n+1) = Sum_{i=1..n} a(3*i-2)*2^(n-i). (End)

A107849 Expansion of (1 + x)^2 / ((1 - x^2 - 2x^3)(1 + x^4)).

Original entry on oeis.org

1, 2, 2, 4, 5, 6, 12, 16, 25, 42, 58, 92, 141, 206, 324, 488, 737, 1138, 1714, 2612, 3989, 6038, 9212, 14016, 21289, 32442, 49322, 75020, 114205, 173662, 264244, 402072, 611569, 930562, 1415714, 2153700, 3276837, 4985126, 7584236, 11538800

Offset: 0

Creighton Dement, May 25 2005

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

Cf. A107850, A107851, A107852.

CoefficientList[Series[(1 + x)^2 / ((1 - x^2 - 2*x^3)*(1 + x^4)),{x,0,39}],x] (* James C. McMahon, Feb 19 2024 *)

Vec((1 + x)^2 / ((1 - x^2 - 2*x^3)*(1 + x^4)) + O(x^45)) \\ Colin Barker, Apr 30 2019

a(n) = A052947(n+2) + A014017(n+6). - Ralf Stephan, Nov 30 2010

a(n) = a(n-2) + 2*a(n-3) - a(n-4) + a(n-6) + 2*a(n-7) for n>6. - Colin Barker, Apr 30 2019

A107850 Expansion of g.f. (x^2+x+1)(2x^2+2x+1)(x-1)^2/((1-x^2-2x^3)(x^4+1)).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 3, 6, 7, 13, 17, 24, 41, 57, 91, 142, 207, 325, 489, 736, 1137, 1713, 2611, 3990, 6039, 9213, 14017, 21288, 32441, 49321, 75019, 114206, 173663, 264245, 402073, 611568, 930561, 1415713, 2153699, 3276838, 4985127, 7584237, 11538801

Offset: 0

Creighton Dement, May 25 2005

Floretion Algebra Multiplication Program, FAMP Code: 1lesforzapseq[(.5i' + .5j' + .5'ki' + .5'kj')*(.5'i + .5'j + .5'ik' + .5'jk')], 1vesforzap = A000004

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,2,-1,0,1,2).

Cf. A107849, A107851, A107852.

CoefficientList[Series[(x^2+x+1)(2x^2+2x+1)(x-1)^2/((1-x^2-2x^3)(x^4+1)),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,2,-1,0,1,2},{1,1,1,0,1,1,3},50] (* Harvey P. Dale, Dec 26 2015 *)

a(n) = A052947(n-1)+A118831(n+6). - R. J. Mathar, Apr 18 2008

a(0)=1, a(1)=1, a(2)=1, a(3)=0, a(4)=1, a(5)=1, a(6)=3, a(n)=a(n-2)+ 2*a(n-3)- a(n-4)+a(n-6)+2*a(n-7). - Harvey P. Dale, Dec 26 2015

A159285 Expansion of (1+3x)/(1-x^2-2x^3).

Original entry on oeis.org

1, 3, 1, 5, 7, 7, 17, 21, 31, 55, 73, 117, 183, 263, 417, 629, 943, 1463, 2201, 3349, 5127, 7751, 11825, 18005, 27327, 41655, 63337, 96309, 146647, 222983, 339265, 516277, 785231, 1194807, 1817785, 2765269, 4207399, 6400839, 9737937, 14815637

Offset: 0

Creighton Dement, Apr 08 2009

A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj')

G. C. Greubel, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier [broken link]
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A159284, A159286, A159287, A159288.

I:=[1,3,1]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 27 2018

LinearRecurrence[{0, 1, 2}, {1, 3, 1}, 50] (* G. C. Greubel, Jun 27 2018 *)
CoefficientList[Series[(1+3x)/(1-x^2-2x^3),{x,0,40}],x] (* Harvey P. Dale, Jan 21 2019 *)

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[1;3;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = A052947(n) + 3*A052947(n-1). - R. J. Mathar, Mar 23 2023

cp's OEIS Frontend

A128099 Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).

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A159288 Expansion of (1 + x + x^2)/(1 - x^2 - 2*x^3).

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A183111 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [RED ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; BLUE] pre-colored puzzle.

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A159284 Expansion of x*(1+x)/(1-x^2-2*x^3).

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A219946 Number A(n,k) of tilings of a k X n rectangle using right trominoes and 2 X 2 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A110509 Riordan array (1, x(1-2x)).

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A159287 Expansion of x^2/(1-x^2-2*x^3).

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A159284 Expansion of x(1+x)/(1-x^2-2x^3).

A107849 Expansion of (1 + x)^2 / ((1 - x^2 - 2x^3)(1 + x^4)).

A107850 Expansion of g.f. (x^2+x+1)(2x^2+2x+1)(x-1)^2/((1-x^2-2x^3)(x^4+1)).

A159285 Expansion of (1+3x)/(1-x^2-2x^3).