A159287 -id:A159287 - 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

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

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

A107853 Expansion of g.f. x(x-1)(x+1)^3/((2x^3+x^2-1)(x^4+1)).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 7, 10, 13, 26, 33, 50, 83, 114, 183, 282, 413, 650, 977, 1474, 2275, 3426, 5223, 7978, 12077, 18426, 28033, 42578, 64883, 98642, 150039, 228410, 347325, 528490, 804145, 1223138, 1861123, 2831426, 4307399, 6553674, 9970253

Offset: 0

Creighton Dement, May 25 2005

Floretion Algebra Multiplication Program, FAMP Code: - 2basejforzapseq[(.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, A107851, A107852, A107854.

LinearRecurrence[{0,1,2,-1,0,1,2},{0,1,2,1,2,3,2},50] (* Harvey P. Dale, Jan 24 2018 *)

a(n) = 2*A159287(n) + A091337(n+6).

A167434 Diagonal sums of the Riordan array (1-4x+4x^2, x(1-2x)) (A167431).

Original entry on oeis.org

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

Offset: 0

Paul Barry, Nov 03 2009

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

Cf. A052947, A159287.

I:=[1,-4,5]; [n le 3 select I[n] else Self(n-2) - 2*Self(n-3): n in [1..40]]; // G. C. Greubel, Jun 27 2018

LinearRecurrence[{0, 1, -2}, {1, -4, 5}, 100] (* G. C. Greubel, Jun 13 2016 *)
CoefficientList[Series[(1-4x+4x^2)/(1-x^2+2x^3),{x,0,40}],x] (* Harvey P. Dale, Nov 08 2022 *)

x='x+O('x^40); Vec((1-4*x+4*x^2)/(1-x^2+2*x^3)) \\ G. C. Greubel, Jun 27 2018

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

a(n) = (-1)^n*A052947(n+4). - R. J. Mathar, Jun 24 2024

cp's OEIS Frontend

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

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

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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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A107853 Expansion of g.f. x*(x-1)*(x+1)^3/((2*x^3+x^2-1)*(x^4+1)).

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A167434 Diagonal sums of the Riordan array (1-4*x+4*x^2, x*(1-2*x)) (A167431).

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

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

A107853 Expansion of g.f. x(x-1)(x+1)^3/((2x^3+x^2-1)(x^4+1)).

A167434 Diagonal sums of the Riordan array (1-4x+4x^2, x(1-2x)) (A167431).