A104769 -id:A104769 - cp's OEIS frontend

A176971 Expansion of (1+x)/(1+x-x^3) in powers of x.

1, 0, 0, 1, -1, 1, 0, -1, 2, -2, 1, 1, -3, 4, -3, 0, 4, -7, 7, -3, -4, 11, -14, 10, 1, -15, 25, -24, 9, 16, -40, 49, -33, -7, 56, -89, 82, -26, -63, 145, -171, 108, 37, -208, 316, -279, 71, 245, -524, 595

Offset: 0

Roger L. Bagula, Apr 29 2010

Except for signs the sequence is the essentially same as A078013, A050935 and A104769.

Padovan sequence extended to negative indices. - Hugo Pfoertner, Jul 16 2017

			G.f. = 1 + x^3 - x^4 + x^5 - x^7 + 2*x^8 - 2*x^9 + x^10 + x^11 - 3*x^12 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See p. 12.
Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
Wikipedia, Padovan sequence.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1). [From _R. J. Mathar_, Apr 30 2010]

Cf. A000931, A005251, A050935, A078013, A104769, A182097.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)/(1+x-x^3))); // G. C. Greubel, Sep 25 2018

a[0] := 1; a[1] = 0; a[2] = 0;
a[n_] := a[n] = a[n - 2] + a[n - 3];
b = Table[a[n], {n, 0, 50}];
Table[b[[n]]^2 - b[[n - 1]]*b[[n + 1]], {n, 1, Length[b] - 1}]
a[ n_] := If[ n >= 0, SeriesCoefficient[ (1 + x) / (1 + x - x^3), {x, 0, n}], SeriesCoefficient[ 1 / (1 - x^2 - x^3), {x, 0, Abs@n}]]; (* Michael Somos, Dec 13 2013 *)

{a(n) = if( n>=0, polcoeff( (1 + x) / (1 + x - x^3) + x * O(x^n), n), polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^-n), -n))}; /* Michael Somos, Dec 13 2013 */

a(n) = A000931(n)^2 -A000931(n-1)*A000931(n+1).
a(n) = -a(n-1) +a(n-3). - R. J. Mathar, Apr 30 2010
a(n) = -A104769(n) - A104769(n+1). - Ralf Stephan, Aug 18 2013
G.f.: 1 / (1 - x^3 / (1 + x)). - Michael Somos, Dec 13 2013
a(n) = A182097(-n) for all n in Z. - Michael Somos, Dec 13 2013
A000931(n) = a(n)^2 - a(n-1) * a(n+1). - Michael Somos, Dec 13 2013
Binomial transform is A005251(n+1). - Michael Somos, Dec 13 2013

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

A247917 Expansion of 1 / (1 + x - x^3) in powers of x.

Original entry on oeis.org

1, -1, 1, 0, -1, 2, -2, 1, 1, -3, 4, -3, 0, 4, -7, 7, -3, -4, 11, -14, 10, 1, -15, 25, -24, 9, 16, -40, 49, -33, -7, 56, -89, 82, -26, -63, 145, -171, 108, 37, -208, 316, -279, 71, 245, -524, 595, -350, -174, 769, -1119, 945, -176, -943, 1888, -2064, 1121

Offset: 0

Michael Somos, Sep 26 2014

			G.f. = 1 - x + x^2 - x^4 + 2*x^5 - 2*x^6 + x^7 + x^8 - 3*x^9 + 4*x^10 + ...

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

Cf. A000931, A050935, A078013, A104769, A176971.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 + x - x^3)));  // G. C. Greubel, Aug 04 2018

CoefficientList[Series[1/(1 + x - x^3), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)
LinearRecurrence[{-1,0,1},{1,-1,1},60] (* Harvey P. Dale, Apr 10 2025 *)

{a(n) = if( n<0, n = -3-n; polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^n), n), polcoeff( 1 / (1 + x - x^3) + x * O(x^n), n))};

G.f.: 1 / (1 + x - x^3).
0 = a(n) - a(n+2) - a(n+3) for all n in Z.
a(-n) = A000931(n) for all n in Z.
a(n) = A176971(n+3) for all n in Z.
-a(n) = A104769(n+1) for all n in Z.
(-1)^n * a(n) = A050935(n+3) for all n in Z.
-(-1)^n * a(n) = A078013(n+3) for all n in Z.

A104771 Expansion of g.f. (1-x+x^2)/(1+x-x^3).

Original entry on oeis.org

1, -2, 3, -2, 0, 3, -5, 5, -2, -3, 8, -10, 7, 1, -11, 18, -17, 6, 12, -29, 35, -23, -6, 41, -64, 58, -17, -47, 105, -122, 75, 30, -152, 227, -197, 45, 182, -379, 424, -242, -137, 561, -803, 666, -105, -698, 1364, -1469, 771, 593, -2062, 2833, -2240, 178, 2655, -4895, 5073, -2418, -2477, 7550, -9968

Offset: 0

Creighton Dement, Mar 24 2005

A floretion-generated sequence.

Floretion Algebra Multiplication Program, FAMP Code: Define A = + .5'i + .5'j + .5'k + .5e and B = + .5'i + .5i' + .5'ii' + .5e. Then (a(n)) = jesloop(infty)-jesfor[A*B], ForType: 1A.

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

Cf. A104769, A104770.

Recurrence: a(n+3) = a(n) - a(n+2); a(0) = 1, a(1) = -2, a(2) = 3.

a(n+1) - a(n) = ((-1)^(n+1))*a(n+5); a(n) = A104769(n) + A104770(n).

Edited by Ralf Stephan, Apr 05 2009

A104770 Expansion of g.f. (1+x^2)/(1+x-x^3).

Original entry on oeis.org

1, -1, 2, -1, 0, 2, -3, 3, -1, -2, 5, -6, 4, 1, -7, 11, -10, 3, 8, -18, 21, -13, -5, 26, -39, 34, -8, -31, 65, -73, 42, 23, -96, 138, -115, 19, 119, -234, 253, -134, -100, 353, -487, 387, -34, -453, 840, -874, 421, 419, -1293, 1714, -1295, 2, 1712, -3007, 3009, -1297, -1710, 4719, -6016

Offset: 0

Creighton Dement, Mar 24 2005

A floretion-generated sequence.

Floretion Algebra Multiplication Program, FAMP Code: Define A = + .5'i + .5'j + .5'k + .5e and B = + .5'i + .5i' + .5'ii' + .5e. Then (a(n)) = jesloop(infty)-jesleftfor[A*B], ForType: 1A.

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

Cf. A104769, A104771.

CoefficientList[Series[(1+x^2)/(1+x-x^3),{x,0,60}],x] (* or *) LinearRecurrence[ {-1,0,1},{1,-1,2},70] (* Harvey P. Dale, Jan 27 2013 *)

Recurrence: a(n+3) = a(n) - a(n+2); a(0) = 1, a(1) = -1, a(2) = 2.
a(n+1) - a(n) = ((-1)^(n+1))*a(n+5); a(n) = A104771(n) - A104769(n).
a(n+1) = -(A104769(n) + A104769(n+2)), n>=0. - Ralf Stephan, Apr 05 2009

Edited by Ralf Stephan, Apr 05 2009

A257543 Expansion of 1 / (1 - x^5 - x^8 + x^9) in powers of x.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 1, -1, 1, 0, 0, 2, -2, 1, 1, -2, 4, -3, 1, 3, -6, 7, -3, -2, 9, -13, 11, -1, -11, 22, -23, 12, 10, -33, 46, -35, 2, 43, -78, 81, -37, -41, 122, -159, 118, 4, -162, 281, -277, 114, 167, -443, 558, -391, -52, 610, -1001, 949, -338, -662

Offset: 0

Michael Somos, Apr 28 2015

			G.f. = 1 + x^5 + x^8 - x^9 + x^10 + 2*x^13 - 2*x^14 + x^15 + x^16 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,0,0,1,-1).

Cf. A104769, A233522, A247917, A247918.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 / ((1-x^4)*(1+x^4-x^5)))); // G. C. Greubel, Aug 02 2018

a[ n_] := SeriesCoefficient[ If[ n >= 0, 1 / (1 - x^5 - x^8 + x^9), -x^9 /(1 - x - x^4 + x^9)], {x, 0, Abs@n}];

{a(n) = if( n>=0, polcoeff( 1 / (1 - x^5 - x^8 + x^9) + x * O(x^n), n), polcoeff( -x^9 / (1 - x - x^4 + x^9) + x * O(x^-n), -n))};

G.f.: 1 / ((1 - x^4) * (1 + x^4 - x^5)) = (1 + x) / ((1 + x^3) * (1 - x^4) * (1 + x - x^3)).
a(n) = a(n-5) + a(n-8) - a(n-9) for all n in Z.
a(n) - a(n+2) - a(n+3) has period 12.
a(n) - a(n+12) = A104769(n+5) = -A247917(n+4) for all n in Z.
a(n) + a(n+1) = A247918(n) for all n in Z.
a(n) = -A233522(-9 - n) for all n in Z.

A109581 E.g.f.: x/(1+x-x^3).

Original entry on oeis.org

0, 1, -2, 6, 0, -120, 1440, -10080, 40320, 362880, -10886400, 159667200, -1437004800, 0, 348713164800, -9153720576000, 146459529216000, -1067062284288000, -25609494822912000, 1338096104497152000, -34060628114472960000, 510909421717094400000

Offset: 0

Roger L. Bagula, Jun 29 2005

sign

Sequence having -G as o.g.f. is A104769.

G:=x/(1+x-x^3): Gser:=series(G,x=0,26): 0,seq(n!*coeff(Gser,x^n),n=1..23); # yields signed sequence

g[x_] = x/(-1 - x + x^3) h[x_, n_] = Dt[g[x], {x, n}] a[x_] = Table[h[x, n], {n, 0, 25}]; b = a[0]
With[{nn=30},CoefficientList[Series[x/(1+x-x^3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 21 2024 *)

D-finite with recurrence a(n) + n*a(n-1) - n*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Aug 20 2021

A109582 Expansion of e.g.f.: -1/(1+x-x^3).

Original entry on oeis.org

-1, 1, -2, 0, 24, -240, 1440, -5040, -40320, 1088640, -14515200, 119750400, 0, -24908083200, 610248038400, -9153720576000, 62768369664000, 1422749712384000, -70426110763008000, 1703031405723648000, -24329020081766400000, -51090942171709440000

Offset: 0

Roger L. Bagula, Jun 29 2005

sign

Cf. A050935, A104769.

g:=-1/(1+x-x^3):gser:=series(g,x=0,26): -1,seq(n!*coeff(gser,x^n),n=1..22);

E.g.f.: -1/(1+x-x^3).
a(n) = n!*A104769(n+1). - R. J. Mathar, Aug 20 2021
D-finite with recurrence a(n) +n*a(n-1) -n*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Aug 20 2021

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A176971 Expansion of (1+x)/(1+x-x^3) in powers of x.

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A247917 Expansion of 1 / (1 + x - x^3) in powers of x.

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A104771 Expansion of g.f. (1-x+x^2)/(1+x-x^3).

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A104770 Expansion of g.f. (1+x^2)/(1+x-x^3).

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A257543 Expansion of 1 / (1 - x^5 - x^8 + x^9) in powers of x.

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A109581 E.g.f.: x/(1+x-x^3).

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A109582 Expansion of e.g.f.: -1/(1+x-x^3).

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