A052101 -id:A052101 - cp's OEIS frontend

A140495 Union of A052103, A052102 and A052101, uniqued and sorted.

0, 1, 2, 3, 6, 9, 12, 15, 21, 27, 36, 45, 63, 81, 99, 144, 180, 225, 324, 405, 513, 729, 918, 1161, 1647, 2079, 2619, 3726, 4698, 5913, 8424, 10611, 13365, 19035, 23976, 30213, 43011, 54189, 68283, 97200, 122472, 154305, 219672, 276777, 348705, 496449, 625482

Offset: 0

Paul Curtz, Jun 28 2008

The three sequences that are merged share the same recurrence, case p=3 in A140414.

The first differences are 1, 1, 1, 3, 3, 3, 3, 6, 6, 9, 9, 18, 18, 18, 45, 36, 45, 99, 81, 108...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,3).

Cf. A052101, A052102, A052103, A140414.

I:=[6,9,12,15,21,27,36,45,63]; [0,1,2,3] cat [n le 9 select I[n] else 3*(Self(n-3) -Self(n-6) +Self(n-9)): n in [1..51]]; // G. C. Greubel, Apr 15 2021

LinearRecurrence[{0,0,3,0,0,-3,0,0,3}, {0,1,2,3,6,9,12,15,21,27,36,45,63}, 50] (* G. C. Greubel, Apr 15 2021 *)

[( x*(1+2*x+3*x^2+6*x^9+3*x^5+3*x^10+9*x^11+3*x^3+3*x^4)/(1-3*x^3+3*x^6-3*x^9) ).series(x,n+1).list()[n] for n in (0..50)] # G. C. Greubel, Apr 15 2021

G.f.: x*(1+2*x+3*x^2+6*x^9+3*x^5+3*x^10+9*x^11+3*x^3+3*x^4)/(1-3*x^3+3*x^6-3*x^9).

Edited and extended by R. J. Mathar, Mar 02 2010

A052103 The third of the three sequences associated with the polynomial x^3 - 2.

Original entry on oeis.org

0, 0, 1, 3, 6, 12, 27, 63, 144, 324, 729, 1647, 3726, 8424, 19035, 43011, 97200, 219672, 496449, 1121931, 2535462, 5729940, 12949227, 29264247, 66134880, 149459580, 337766841, 763326423, 1725057486, 3898493712, 8810287947, 19910555163

Offset: 0

Ashok K. Gupta and Ashok K. Mittal (akgjkiapt(AT)hotmail.com), Jan 20 2000

nonn

If x^3 = 2 and n >= 0, then there are unique integers a, b, c such that (1 + x)^n = a + b*x + c*x^2. The coefficient c is a(n).

			G.f. = x^2 + 3*x^3 + 6*x^4 + 12*x^5 + 27*x^6 + 63*x^7 + 144*x^8 + ...

Maribel Díaz Noguera [Maribel Del Carmen Díaz Noguera], Rigoberto Flores, Jose L. Ramirez, and Martha Romero Rojas, Catalan identities for generalized Fibonacci polynomials, Fib. Q., 62:2 (2024), 100-111. See Table 3.
R. Schoof, Catalan's Conjecture, Springer-Verlag, 2008, pp. 17-18.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. Kumar Gupta and A. Kumar Mittal, Integer Sequences associated with Integer Monic Polynomial, arXiv:math/0001112 [math.GM], Jan 2000.
Index entries for linear recurrences with constant coefficients, signature (3,-3,3).

Cf. A052101, A052102.

I:=[0,0,1]; [n le 3 select I[n] else 3*(Self(n-1) -Self(n-2) +Self(n-3)): n in [1..41]]; // G. C. Greubel, Apr 15 2021

A052103:= n-> add(2^j*binomial(n, 3*j+2), j = 0..floor(1/3*n));
seq(A052103(n), n = 0..40); # G. C. Greubel, Apr 15 2021

LinearRecurrence[{3,-3,3}, {0,0,1}, 32] (* Ray Chandler, Sep 23 2015 *)

{a(n) = polcoeff( lift( Mod(1 + x, x^3 - 2)^n ), 2)} /* Michael Somos, Aug 05 2009 */

{a(n) = sum(k=0, n\3, 2^k * binomial(n, 3*k + 2))} /* Michael Somos, Aug 05 2009 */

{a(n) = if( n<0, 0, polcoeff( x^2 / (1 - 3*x + 3*x^2 - 3*x^3) + x * O(x^n), n))} /* Michael Somos, Aug 05 2009 */

[sum(2^j*binomial(n, 3*j+2) for j in (0..n//3)) for n in (0..40)] # G. C. Greubel, Apr 15 2021

a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3), n > 2.
a(n) = Sum_{0..floor(n/3)}, 2^k * binomial(n, 3*k+2). - Ralf Stephan, Aug 30 2004
From Paul Curtz, Mar 10 2008: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 6*a(n-3) - 3*a(n-4).
a(n) is binomial transform of 0, 0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32 (see A077958).
a(n) is a sequence identical to half its third differences. (End)
From R. J. Mathar, Apr 01 2008: (Start)
O.g.f.: x^2/(1 - 3*x + 3*x^2 - 3*x^3).
a(n+1) - a(n) = A052102(n). (End)

More terms from R. J. Mathar, Apr 01 2008

A052102 The second of the three sequences associated with the polynomial x^3 - 2.

Original entry on oeis.org

0, 1, 2, 3, 6, 15, 36, 81, 180, 405, 918, 2079, 4698, 10611, 23976, 54189, 122472, 276777, 625482, 1413531, 3194478, 7219287, 16315020, 36870633, 83324700, 188307261, 425559582, 961731063, 2173436226, 4911794235, 11100267216, 25085727621

Offset: 0

Ashok K. Gupta and Ashok K. Mittal (akgjkiapt(AT)hotmail.com), Jan 20 2000

If x^3 = 2 and n >= 0, then there are unique integers a, b, c such that (1 + x)^n = a + b*x + c*x^2. The coefficient b is a(n).

			G.f.: = x + 2*x^2 + 3*x^3 + 6*x^4 + 15*x^5 + 36*x^6 + 81*x^7 + 180*x^8 + ...

Maribel Díaz Noguera [Maribel Del Carmen Díaz Noguera], Rigoberto Flores, Jose L. Ramirez, and Martha Romero Rojas, Catalan identities for generalized Fibonacci polynomials, Fib. Q., 62:2 (2024), 100-111. See Table 3.
R. Schoof, Catalan's Conjecture, Springer-Verlag, 2008, pp. 17-18.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. Kumar Gupta and A. Kumar Mittal, Integer Sequences associated with Integer Monic Polynomial, arXiv:math/0001112 [math.GM], Jan 2000.
Index entries for linear recurrences with constant coefficients, signature (3,-3,3).

Cf. A052101, A052103.

[n le 3 select n-1 else 3*(Self(n-1) -Self(n-2) +Self(n-3)): n in [1..40]]; // G. C. Greubel, Apr 15 2021

A052102:= n-> add(2^j*binomial(n, 3*j+1), j=0..floor(n/3)); seq(A052102(n), n=0..40); # G. C. Greubel, Apr 15 2021

LinearRecurrence[{3,-3,3}, {0,1,2}, 32] (* Ray Chandler, Sep 23 2015 *)

{a(n) = polcoeff( lift( Mod(1 + x, x^3 - 2)^n ), 1)} /* Michael Somos, Aug 05 2009 */

{a(n) = sum(k=0, n\3, 2^k * binomial(n, 3*k + 1))} /* Michael Somos, Aug 05 2009 */

{a(n) = if( n<0, 0, polcoeff( (x - x^2) / (1 - 3*x + 3*x^2 - 3*x^3) + x * O(x^n), n))} /* Michael Somos, Aug 05 2009 */

[sum(2^j*binomial(n, 3*j+1) for j in (0..n//3)) for n in (0..40)] # G. C. Greubel, Apr 15 2021

a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3), n > 2.
a(n) = Sum_{0..floor(n/3)}, 2^k * binomial(n, 3*k+1). - Ralf Stephan, Aug 30 2004
From R. J. Mathar, Apr 01 2008: (Start)
O.g.f.: x*(1 - x)/(1 - 3*x + 3*x^2 - 3*x^3).
a(n+1) - a(n) = A052101(n). (End)

A097122 Expansion of (1-x)^2/((1-x)^3 - 3*x^3).

Original entry on oeis.org

1, 1, 1, 4, 13, 31, 70, 169, 421, 1036, 2521, 6139, 14998, 36661, 89545, 218644, 533941, 1304071, 3184966, 7778449, 18996733, 46394716, 113307745, 276726019, 675833686, 1650553981, 4031064961, 9844867684, 24043624093, 58720529071

Offset: 0

Paul Barry, Jul 25 2004

Maribel Díaz Noguera [Maribel Del Carmen Díaz Noguera], Rigoberto Flores, Jose L. Ramirez, and Martha Romero Rojas, Catalan identities for generalized Fibonacci polynomials, Fib. Q., 62:2 (2024), 100-111. See Table 3.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,4).

Cf. A024493, A052101, A100136.

CoefficientList[Series[(1-x)^2/((1-x)^3-3x^3),{x,0,40}],x]

a(n) = sum(k=0, n\3, binomial(n, 3*k) * 3^k); \\ Michel Marcus, Oct 11 2021

G.f.: (1-2*x+x^2)/(1-3*x+3*x^2-4*x^3).
a(n) = 3*a(n-1) - 3*a(n-2) + 4*a(n-3).
a(n) = Sum_{k=0..floor(n/3)} binomial(n, 3k) * 3^k.

A100138 a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 2^(n-5k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 66, 144, 336, 832, 2144, 5632, 14852, 38968, 101312, 260736, 664704, 1681152, 4226056, 10578080, 26407648, 65838848, 164095360, 409129472, 1020795408, 2549137824, 6371133120, 15935185792, 39878810624, 99837958144

Offset: 0

Paul Barry, Nov 06 2004

Binomial transform of 1,1,1,1,1,1,3,3,9,9,21,... with g.f. (1-x)^2(1+x)^2/(1-3x^2+3x^4-3x^6)=(1+x)(1-x^2)^2/((1-x^2)^3-2x^6).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8,0,0,2).

Cf. A052101, A100132, A100134, A100137, A100139.

Table[Sum[Binomial[n-3k,3k]2^(n-5k),{k,0,Floor[n/6]}],{n,0,30}] (* or *) LinearRecurrence[{6,-12,8,0,0,2},{1,2,4,8,16,32},30] (* Harvey P. Dale, Dec 30 2019 *)

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

a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) + 2*a(n-6).

A093406 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3) + a(n-4).

Original entry on oeis.org

1, 3, 11, 31, 71, 145, 289, 601, 1321, 2979, 6683, 14743, 32111, 69697, 151777, 332113, 728689, 1598883, 3503627, 7668079, 16774775, 36704017, 80343361, 175916521, 385196761, 843365379, 1846290395, 4041672871, 8847607391, 19368919297, 42403014721, 92830645537

Offset: 1

Gary W. Adamson, Mar 28 2004

a(n)/a(n-1) tends to 2.189207115... = 1 + 2^(1/4) = 1 + A010767.

			a(4) = 31, since M^4 * [1,1,1,1] = [3, 11, 31, 71].

E. J. Barbeau, Polynomials, Springer-Verlag NY Inc, 1989, p. 136.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,1).

Cf. A010767, A052101.

LinearRecurrence[{4,-6,4,1},{1,3,11,31},40] (* Harvey P. Dale, Jul 22 2013 *)

We use a 4 X 4 matrix corresponding to the characteristic polynomial (x - 1)^4 - 2 = 0 = x^4 - 4x^3 + 6x^2 - 4x - 1 = 0, being [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / 1 4 -6 4]. Let the matrix = M. Perform M^n * [1, 1, 1, 1]. a(n) = the third term from the left, (the other 3 terms being offset members of the series).

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)+a(n-4). G.f.: -x*(x^3+5*x^2-x+1)/ (x^4+4*x^3-6*x^2+4*x-1). [Colin Barker, Oct 21 2012]

Corrected by T. D. Noe, Nov 08 2006

New name using recurrence from Colin Barker, Joerg Arndt, Apr 15 2021

A247584 a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + 3*a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 13, 43, 113, 253, 509, 969, 1849, 3719, 8009, 18027, 40897, 91257, 198697, 423777, 894081, 1886011, 4007301, 8594411, 18560081, 40181493, 86872293, 187197193, 402060793, 861827743, 1846685729, 3960390059, 8504658049, 18283290609, 39325827729

Offset: 0

Alexander Samokrutov, Sep 20 2014

a(n)/a(n-1) tends to 2.1486... = 1 + 2^(1/5), the real root of the polynomial x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 3.
If x^5 = 2 and n >= 0, then there are unique integers a, b, c, d, g  such that (1 + x)^n = a + b*x + c*x^2 + d*x^3 + g*x^4. The coefficient a is a(n) (from A052102). - Alexander Samokrutov, Jul 11 2015
If x=a(n), y=a(n+1), z=a(n+2), s=a(n+3), t=a(n+4) then x, y, z, s, t satisfies Diophantine equation (see link). - Alexander Samokrutov, Jul 11 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..48 from Alexander Samokrutov)
Alexander Samokrutov, Diophantine equation
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,3).

The following sequences belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A052101, A052102, A052103, A063727, A083098, A083099, A083100, A084057, A093406, A247344.

Cf. A005531.

[n le 5 select 1 else 5*Self(n-1) -10*Self(n-2) +10*Self(n-3) -5*Self(n-4) +3*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Jul 11 2015

m:=50; S:=series( (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 15 2021

LinearRecurrence[{5,-10,10,-5,3}, {1,1,1,1,1}, 50] (* Vincenzo Librandi, Jul 11 2015 *)

makelist(sum(2^k*binomial(n,5*k), k, 0, floor(n/5)), n, 0, 50); /* Alexander Samokrutov, Jul 11 2015 */

Vec((1-x)^4/(1-5*x+10*x^2-10*x^3+5*x^4-3*x^5) + O(x^100)) \\ Colin Barker, Sep 22 2014

[sum(2^j*binomial(n, 5*j) for j in (0..n//5)) for n in (0..50)] # G. C. Greubel, Apr 15 2021

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 3*a(n-5).
a(n) = Sum_{k=0...floor(n/5)} (2^k*binomial(n,5*k)). - Alexander Samokrutov, Jul 11 2015
G.f.: (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5). - Colin Barker, Sep 22 2014

cp's OEIS Frontend

A140495 Union of A052103, A052102 and A052101, uniqued and sorted.

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A052103 The third of the three sequences associated with the polynomial x^3 - 2.

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A052102 The second of the three sequences associated with the polynomial x^3 - 2.

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A097122 Expansion of (1-x)^2/((1-x)^3 - 3*x^3).

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A100138 a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 2^(n-5k).

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A093406 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) + a(n-4).

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A093406 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3) + a(n-4).

A247584 a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + 3*a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.