A159284 -id:A159284 - cp's OEIS frontend

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

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

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)

A107852 Expansion of -x(x^2+1)(x+1)^2/((2x^3+x^2-1)(x^4+1)).

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 17, 22, 37, 58, 83, 134, 199, 298, 465, 694, 1061, 1626, 2451, 3750, 5703, 8650, 13201, 20054, 30501, 46458, 70611, 107462, 163527, 248682, 378449, 575734, 875813, 1332634, 2027283, 3084262, 4692551, 7138826, 10861073

Offset: 0

Creighton Dement, May 25 2005

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

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. A107849, A107850, A107851, A107853.

CoefficientList[Series[-x(x^2+1)(x+1)^2/((2x^3+x^2-1)(x^4+1)),{x,0, 50}],x] (* or *) LinearRecurrence[ {0,1,2,-1,0,1,2},{0,1,2,3,6,7,10},50] (* Harvey P. Dale, May 03 2024 *)

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 2,1,0,-1,2,1,0]^n*[0;1;2;3;6;7;10])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

concat(0, Vec(x*(1 + x)^2*(1 + x^2) / ((1 - x^2 - 2*x^3)*(1 + x^4)) + O(x^45))) \\ Colin Barker, Apr 30 2019

a(n) = 2*A159284(n) - A091337(n).

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

A107851 Expansion of g.f. x(-1-x-3x^2-x^3+2x^5)/((2x^3+x^2-1)*(x^4+1)).

Original entry on oeis.org

0, 1, 1, 4, 4, 5, 9, 10, 18, 29, 41, 68, 100, 149, 233, 346, 530, 813, 1225, 1876, 2852, 4325, 6601, 10026, 15250, 23229, 35305, 53732, 81764, 124341, 189225, 287866, 437906, 666317, 1013641, 1542132, 2346276, 3569413, 5430537, 8261962, 12569362

Offset: 0

Creighton Dement, May 25 2005

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

Index entries for linear recurrences with constant coefficients, signature (0,1,2,-1,0,1,2).

Cf. A107849, A107850, A107852.

CoefficientList[Series[x(-1-x-3x^2-x^3+2x^5)/((2x^3+x^2-1)(x^4+1)), {x,0,50}],x] (* or *) LinearRecurrence[{0,1,2,-1,0,1,2},{0,1,1,4,4,5,9},51] (* Harvey P. Dale, Jul 19 2011 *)

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 2,1,0,-1,2,1,0]^n*[0;1;1;4;4;5;9])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = A159284(n+1) + A132380(n+7).

a(0)=0, a(1)=1, a(2)=1, a(3)=4, a(4)=4, a(5)=5, a(6)=9, a(n)= a(n-2)+ 2*a(n-3)-a(n-4)+a(n-6)+2*a(n-7). - Harvey P. Dale, Jul 19 2011

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

A159286 Expansion of (x-1)^2/(1-x^2-2*x^3).

Original entry on oeis.org

1, -2, 2, 0, -2, 4, -2, 0, 6, -4, 6, 8, -2, 20, 14, 16, 54, 44, 86, 152, 174, 324, 478, 672, 1126, 1628, 2470, 3880, 5726, 8820, 13486, 20272, 31126, 47244, 71670, 109496, 166158, 252836, 385150, 585152

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, A159285, A159287, A159288.

I:=[1, -2, 2]; [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)^2/(1-x^2-2*x^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,1,2},{1,-2,2},40]  (* Harvey P. Dale, Apr 24 2011 *)

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

a(n) = A159288(n) - 3*A159287(n+1). - R. J. Mathar, Apr 10 2009
a(1)=1, a(2)=-2, a(3)=2, a(n) = 1*a(n-2) + 2*a(n-3) for n >= 3. - Harvey P. Dale, Apr 24 2011
a(n) = A078026(n)-A078026(n-1). - R. J. Mathar, Mar 23 2023

A198295 Riordan array (1, x*(1+x)/(1-x^3)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 3, 1, 0, 1, 2, 3, 4, 1, 0, 0, 4, 4, 6, 5, 1, 0, 1, 2, 9, 8, 10, 6, 1, 0, 1, 3, 9, 17, 15, 15, 7, 1, 0, 0, 6, 9, 24, 30, 26, 21, 8, 1, 0, 1, 3, 18, 26, 51, 51, 42, 28, 9, 1

Offset: 0

Philippe Deléham, Jan 26 2012

Triangle T(n,k), read by rows, given by (0, 1, -1, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Antidiagonals sums: see A159284.

			Triangle begins:
1
0, 1
0, 1, 1
0, 0, 2, 1
0, 1, 1, 3, 1
0, 1, 2, 3, 4, 1
0, 0, 4, 4, 6, 5, 1
0, 1, 2, 9, 8, 10, 6, 1
0, 1, 3, 9, 17, 15, 15, 7, 1

A. Luzón, D. Merlini, M. A. Morón, R. Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87.

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

Cf. Columns: A000007, A011655, A186731, A185395, A185292.

Cf. Diagonals: A000012, A001477, A161680, A000125.

Sum_{k, 0<=k<=n} T(n,k) = A001590(n+2), n>0.
Sum_{k, 0<=k<=n}T(n,k)*(-1)^(n-k) = A078056(n-1), n>0.
T(n,n) = A000012(n), T(n+1,n) = A001477(n) = n, T(n+2,n) = A161680(n) = A000217(n-1); T(n+3,n) = A000125(n-1), n>=1.
G.f.: (-1+x)*(1+x+x^2)/(-1+x^3+x*y+x^2*y). - R. J. Mathar, Aug 11 2015

A237714 Expansion of (1 + x)/(1 - x^2 - 2*x^5).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 11, 13, 21, 23, 35, 45, 61, 87, 107, 157, 197, 279, 371, 493, 685, 887, 1243, 1629, 2229, 2999, 4003, 5485, 7261, 9943, 13259, 17949, 24229, 32471, 44115, 58989, 80013, 107447, 144955, 195677, 262933, 355703, 477827, 645613, 869181, 1171479, 1580587

Offset: 0

Sergio Falcon, Feb 12 2014

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

Cf. A000129, A122522, A159284.

For[j = 0, j < 5, j++, a[j] = 1]
For[j = 5, j < 51, j++, a[j] = 2 a[j - 5] + a[j - 2]]
Table[a[j], {j, 0, 50}]
CoefficientList[Series[(1 + x)/(1 - x^2 - 2 x^5), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)
LinearRecurrence[{0,1,0,0,2},{1,1,1,1,1},70] (* Harvey P. Dale, Nov 24 2024 *)

Vec( (1 + x)/(1 - x^2 - 2*x^5) + O(x^66) ) \\ Joerg Arndt, Feb 24 2014

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(n) = 2*a(n-5) + a(n-2) for n>=5.
a(2n) = sum_{j=0}^{n/5}C(n-3j,2j)*2^(2j)+sum_{j=0}^{(n-3)/5} C(n-2-3j,2j+1)*2^(2j+1).
a(2n+1) = sum_{j=0}^{n/5}C(n-3j,2j)*2^(2j)+sum_{j=0}^{(n-2)/5} C(n-1-3j,2j+1)*2^(2j+1).

A172358 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c are partial products of a sequence defined in comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 3, 9, 9, 3, 1, 1, 5, 15, 45, 15, 5, 1, 1, 9, 45, 135, 135, 45, 9, 1, 1, 11, 99, 495, 495, 495, 99, 11, 1, 1, 19, 209, 1881, 3135, 3135, 1881, 209, 19, 1, 1, 29, 551, 6061, 18183, 30305, 18183, 6061, 551, 29, 1

Offset: 0

Roger L. Bagula, Feb 01 2010

Start from the sequence A159284 and its partial products c(n) = 1, 1, 1, 1, 3, 9, 45, 405, 4455, 84645, 2454705, ... . Then T(n,k) = round( c(n)/(c(k)*c(n-k)) ).

			Triangle begins as:
  1;
  1,  1;
  1,  1,   1;
  1,  1,   1,    1;
  1,  3,   3,    3,     1;
  1,  3,   9,    9,     3,     1;
  1,  5,  15,   45,    15,     5,     1;
  1,  9,  45,  135,   135,    45,     9,    1;
  1, 11,  99,  495,   495,   495,    99,   11,   1;
  1, 19, 209, 1881,  3135,  3135,  1881,  209,  19,  1;
  1, 29, 551, 6061, 18183, 30305, 18183, 6061, 551, 29, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A172353 (q=1), this sequence (q=2), A172359 (q=4), A172360 (q=5).

f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], f[n-2, q] + q*f[n-3, q]];
c[n_, q_]:= Product[f[j, q], {j,n}];
T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 09 2021 *)

@CachedFunction
def f(n,q): return fibonacci(n) if (n<3) else f(n-2, q) + q*f(n-3, q)
def c(n,q): return product( f(j,q) for j in (1..n) )
def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 09 2021

T(n, k, q) = round(c(n,q)/(c(k,q)*c(n-k,q))), where c(n,q) = Product_{j=1..n} f(j,q), f(n, q) = f(n-2, q) + q*f(n-3, q), f(0,q) = 0, f(1,q) = f(2,q) = 1, and q = 2. - G. C. Greubel, May 09 2021

Definition corrected to give integral terms by G. C. Greubel, May 09 2021

A237716 7-distance Pell sequence.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 13, 15, 23, 25, 37, 39, 55, 65, 85, 111, 135, 185, 213, 295, 343, 465, 565, 735, 935, 1161, 1525, 1847, 2455, 2977, 3925, 4847, 6247, 7897, 9941, 12807, 15895, 20657, 25589, 33151, 41383, 53033, 66997

Offset: 0

Sergio Falcon, Feb 12 2014

			a(7)=2a(0)+a(5)=3; a(8)=2a(1)+a(6)=3; a(9)=2a(2)+a(7)=5.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,0,2).

Cf. A000129, A122522, A159284, A237714.

For[j = 0, j < 7, j++, a[j] = 1]
For[j = 7, j < 51, j++, a[j] = 2 a[j - 7] + a[j - 2]]
Table[a[j], {j, 0, 50}]
CoefficientList[Series[(1 + x)/(1 - x^2 - 2 x^7), {x,0,50}], x] (* G. C. Greubel, May 01 2017 *)

Vec((1+x)/(1-x^2-2*x^7)+O(x^99)) \\ Charles R Greathouse IV, Mar 06 2014

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=1, a(6)=1; a(n) = 2*a(n-7) + a(n-2) for n>=7.
G.f.: (1 + x)/(1 - x^2 - 2*x^7).
a(2*n) = Sum_{j=0..n/7} binomial(n-5*j, 2*j)*2^(2*j) + Sum_{j=0..(n-4)/7} binomial(n-3-5*j, 2*j+1)*2^(2*j+1).
a(2*n+1) = Sum_{j=0..n/7} binomial(n-5*j, 2*j)*2^(2*j) + Sum_{j=0..(n-3)/7} binomial(n-2-5*j, 2*j+1)*2^(2*j+1).

cp's OEIS Frontend

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

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

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A107852 Expansion of -x*(x^2+1)*(x+1)^2/((2*x^3+x^2-1)*(x^4+1)).

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A107851 Expansion of g.f. x*(-1-x-3*x^2-x^3+2*x^5)/((2*x^3+x^2-1)*(x^4+1)).

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

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A159286 Expansion of (x-1)^2/(1-x^2-2*x^3).

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A198295 Riordan array (1, x*(1+x)/(1-x^3)).

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A107852 Expansion of -x(x^2+1)(x+1)^2/((2x^3+x^2-1)(x^4+1)).

A107851 Expansion of g.f. x(-1-x-3x^2-x^3+2x^5)/((2x^3+x^2-1)*(x^4+1)).

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