A052102 -id:A052102 - 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

A052101 One of the three sequences associated with the polynomial x^3 - 2.

Original entry on oeis.org

1, 1, 1, 3, 9, 21, 45, 99, 225, 513, 1161, 2619, 5913, 13365, 30213, 68283, 154305, 348705, 788049, 1780947, 4024809, 9095733, 20555613, 46454067, 104982561, 237252321, 536171481, 1211705163, 2738358009, 6188472981, 13985460405

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 a is a(n).

			From the Schoof reference, pp. 17, 18: Set pi = 1 + sqrt[3]{2}. For every integer k >= 0, there are unique a_k,b_k,c_k in Q such that pi^k = a_k + b_k sqrt[3]{2} + c_k sqrt[3]{4}. The coefficients a_k,b_k,c_k are actually in Z:
     Coefficients a_k, b_k, c_k:
     k      0     1     2     3     4     5      6
     ----------------------------------------------
     a_k    1     1     1     3     9     21     45
     b_k    0     1     2     3     6     15     36
     c_k    0     0     1     3     6     12     27
     ----------------------------------------------
G.f. = 1 + x + x^2 + 3*x^3 + 9*x^4 + 21*x^5 + 45*x^6 + 99*x^7 + 225*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. A052102, A052103.

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

A052101 := n -> add(2^j*binomial(n, 3*j), j = 0..floor(n/3));
seq(A052101(n), n = 0..40); # G. C. Greubel, Apr 15 2021

LinearRecurrence[{3, -3, 3},{1, 1, 1},31] (* Ray Chandler, Sep 23 2015 *)

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

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

{a(n) = if( n<0, 0, polcoeff( (1 - 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) 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).
a(n)/a(n-1) tends to 2.259921049... = 1 + 2^(1/3) (a real root to (x - 1)^3 = 2 or x^3 - 3x^2 + 3x - 3 = 0). A 3 X 3 matrix corresponding to the latter polynomial is [0 1 0 / 0 0 1 / 3 -3 3]. Let the matrix = M. Then a(n) = the center term in M^n * [1, 1, 1]. M^[1, 1, 1] = [9, 21, 45], center term = a(4) - Gary W. Adamson, Mar 28 2004
a(n) = Sum_{0..floor(n/3)}, 2^k * binomial(n, 3*k). - Ralf Stephan, Aug 30 2004
From Paul Curtz, Mar 10 2008: (Start)
Equals the first differences of A052102.
Equals the second differences of A052103.
Equals the binomial transform of A077959.
a(n) = 4*a(n-1) - 6*a(n-2) + 6*a(n-3) - 3*a(n-4).
A052103 is binomial transform of c(n)=0, 1, 1, 0, 2, 2, 0, 4, 4, 0, 8, 8, ... b(n+1) - 2*b(n) is essentially 3*b(n). (End)
G.f.: (1 - x)^2 / (1 - 3*x + 3*x^2 - 3*x^3).

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

A102785 G.f.: (x-1)/(-2x^2 + 3x^3 + 2*x - 1).

Original entry on oeis.org

1, 1, 0, 1, 5, 8, 9, 17, 40, 73, 117, 208, 401, 737, 1296, 2321, 4261, 7768, 13977, 25201, 45752, 83033, 150165, 271520, 491809, 891073, 1613088, 2919457, 5285957, 9572264, 17330985, 31375313, 56805448

Offset: 0

Creighton Dement, Feb 11 2005

nonn

Inverse binomial transform of A078017. Inversion of A052102.

Floretion Algebra Multiplication Program, FAMP Code: 4jbasekseq[ (+ 'ii' + 'jj' + 'ij' + 'ji' + e)*x) ] where x is defined as 1/4 times the sum of all 16 floretion basis vectors.

Index entries for linear recurrences with constant coefficients, signature (2,-2,3).

Cf. A078017, A052102, A077952.

a(n):=sum(sum((sum(binomial(j,-3*k+2*j+i)*(-2)^(-3*k+2*j+i)*3^(k-j)*binomial(k,j),j,0,k))*binomial(n+k-i-1,k-1),i,k,n),k,1,n); /* Vladimir Kruchinin, May 05 2011 */

makelist(coeff(taylor((x-1)/(-2*x^2+3*x^3+2*x-1), x, 0, n), x, n), n, 0, 32); /* Bruno Berselli, May 30 2011 */

a(n+3) = 2a(n+2) - 2a(n+1) + 3a(n), a(0) = 1, a(1) = 1, a(2) = 0

a(n) = Sum(k=1..n, Sum(i=k..n, (Sum(j=0..k, binomial(j,-3*k+2*j+i)*(-2)^(-3*k+2*j+i)*3^(k-j)*binomial(k,j)))*binomial(n+k-i-1,k-1))), n > 0, a(0)=1. - Vladimir Kruchinin, May 05 2011

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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A052101 One of the three sequences associated with the polynomial x^3 - 2.

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

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A102785 G.f.: (x-1)/(-2*x^2 + 3*x^3 + 2*x - 1).

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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.

A102785 G.f.: (x-1)/(-2x^2 + 3x^3 + 2*x - 1).