A057079 -id:A057079 - cp's OEIS frontend

A130752 Binomial transform of periodic sequence (2, 3, 1).

2, 5, 9, 16, 31, 63, 128, 257, 513, 1024, 2047, 4095, 8192, 16385, 32769, 65536, 131071, 262143, 524288, 1048577, 2097153, 4194304, 8388607, 16777215, 33554432, 67108865, 134217729, 268435456, 536870911, 1073741823, 2147483648, 4294967297, 8589934593

Offset: 0

Paul Curtz, Jul 13 2007

The second sequence of "less twisted numbers"; this sequence, A130750 and A130755 form a "suite en trio" (cf. reference, p. 130).
First differences of A130750, second differences of A130755.
Sequence equals its third differences:
2.....5.....9....16....31....63...128...257...513..1024...
...3.....4.....7....15....32....65...129...256...511...
......1.....3.....8....17....33....64...127...255...
..........2.....5.....9....16....31....63...128...

P. Curtz, Exercise Book, manuscript, 1995.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A010882, A130755 (first differences), A130750 (second differences).

m:=31; S:=[ [2, 3, 1][(n-1) mod 3 +1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; /* Klaus Brockhaus, Aug 03 2007 */

a[n_] := 2^(n+1) + 2*Sin[n*Pi/3]/Sqrt[3]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 13 2012 *)
LinearRecurrence[{3,-3,2},{2,5,9},40] (* Harvey P. Dale, Jun 21 2017 *)

{m=31; v=vector(m); v[1]=2; v[2]=5; v[3]=9; for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} \\ Klaus Brockhaus, Aug 03 2007

{for(n=0, 30, print1(2^(n+1)+[0, 1, 1, 0, -1, -1][n%6+1], ","))} \\ Klaus Brockhaus, Aug 03 2007

Vec((2-x) / ((1-2*x)*(1-x+x^2)) + O(x^40)) \\ Colin Barker, Jan 20 2017

G.f.: (2 - x) / ((1 - 2*x)*(1 - x + x^2)).
a(0) = 2; a(1) = 5; a(2) = 9; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
a(n) = 2^(n+1) + A128834(n).
a(0) = 2; for n > 0, a(n) = 2*a(n-1) + A057079(n+1).
E.g.f.: 2*(sqrt(3)*exp(2*x) + sin(sqrt(3)*x/2)*exp(x/2))/sqrt(3). - Ilya Gutkovskiy, Jun 20 2016
a(n) = 2^(n+1) + (2*sin((Pi*n)/3))/sqrt(3). - Colin Barker, Jan 20 2017

Edited and extended by Klaus Brockhaus, Aug 03 2007

A192080 Expansion of 1/((1-x)^6 - x^6).

Original entry on oeis.org

1, 6, 21, 56, 126, 252, 463, 804, 1365, 2366, 4368, 8736, 18565, 40410, 87381, 184604, 379050, 758100, 1486675, 2884776, 5592405, 10919090, 21572460, 43144920, 87087001, 176565486, 357913941, 723002336, 1453179126, 2906358252

Offset: 0

Bruno Berselli, Jun 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Cf. A006090, A049016, A049017, A057079, A057083, A057681.

Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), A049016 (m=5), this sequence (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

m:=30; R:=PowerSeriesRing(Integers(),m); Coefficients(R!(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))));

CoefficientList[Series[1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), {x,0,50}], x] (* Vincenzo Librandi, Oct 15 2012 *)
LinearRecurrence[{6,-15,20,-15,6},{1,6,21,56,126},30] (* Harvey P. Dale, Feb 22 2017 *)

makelist(coeff(taylor(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), x, 0, n), x, n), n, 0, 29);

Vec(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))+O(x^99)) \\ Charles R Greathouse IV, Jun 23 2011

def A192080_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^6-x^6) ).list()
A192080_list(51) # G. C. Greubel, Apr 11 2023

a(n) = abs(A006090(n)) = (-1)^n * A006090(n).
G.f.: 1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)).
From G. C. Greubel, Apr 11 2023: (Start)
a(n) = (2^(n+5) + A010892(n) - 2*A010892(n-1) - 27*(A057083(n) - 2*A057083(n-1)))/6.
a(n) = (2^(n+5) + A057079(n+2) - 27*A057681(n+1))/6. (End)

A100063 A Chebyshev transform of Jacobsthal numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1

Offset: 0

Paul Barry, Nov 02 2004

A Chebyshev transform of A001045(n+1): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))*A(x/(1+x^2)).
Also decimal expansion of 1111/9990. - Elmo R. Oliveira, Feb 18 2024
Also partial quotients of the continued fraction expansion of sqrt(5/2). - Hugo Pfoertner, Jan 10 2025

			G.f. = 1 + x + x^2 + 2*x^3 + x^4 + x^5 + 2*x^6 + x^7 + x^8 + 2*x^9 + ... - _Michael Somos_, Feb 20 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A000012, A001045, A100051, A061347, A057079, A154272.

Cf. A000032, A000045.

PadRight[{1},120,{2,1,1}] (* or *) LinearRecurrence[{0,0,1},{1,1,1,2},120] (* Harvey P. Dale, Jul 08 2015 *)
a[ n_] := If[n<1, Boole[n==0], {2, 1, 1}[[1+Mod[n, 3]]]]; (* Michael Somos, Feb 20 2024 *)

my(x='x+O('x^50)); Vec((1+x)(1+x^2)/(1-x^3)) \\ G. C. Greubel, May 03 2017

{a(n) = if(n<1, n==0, [2, 1, 1][n%3+1])}; /* Michael Somos, Feb 20 2024 */

contfrac(sqrt(5/2),,80) \\ Hugo Pfoertner, Jan 10 2025

G.f.: (1+x)(1+x^2)/(1-x^3).
a(n) = n*Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*A001045(n-2k+1)/(n-k).
Multiplicative with a(3^e) = 2, a(p^e) = 1 otherwise. - David W. Wilson, Jun 11 2005
Dirichlet g.f.: zeta(s)*(1+1/3^s). Dirichlet convolution of A154272 and A000012. - R. J. Mathar, Feb 07 2011
a(n) = 2 if n == 0 (mod 3) and n > 0, and a(n) = 1 otherwise. - Amiram Eldar, Nov 01 2022
a(n) = gcd(Fibonacci(n), Lucas(n)) = gcd(A000045(n), A000032(n)), for n >= 1. - Amiram Eldar, Jul 10 2023

A130750 Binomial transform of A010882.

Original entry on oeis.org

1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, 4096, 8191, 16383, 32768, 65537, 131073, 262144, 524287, 1048575, 2097152, 4194305, 8388609, 16777216, 33554431, 67108863, 134217728, 268435457, 536870913, 1073741824, 2147483647

Offset: 0

Paul Curtz, Jul 13 2007

The first sequence of "less twisted numbers"; this sequence, A130752 and A130755 form a "suite en trio" (cf. reference, p. 130).
First differences of A130755, second differences of A130752.
Sequence equals its third differences:
  1     3     8    17    33    64   127   255   512  1025
     2     5     9    16    31    63   128   257   513
        3     4     7    15    32    65   129   256
           1     3     8    17    33    64   127

P. Curtz, Exercise Book, manuscript, 1995.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A010882 (periodic (1, 2, 3)), A128834 (periodic (0, 1, 1, 0, -1, -1)), A057079 (periodic (1, 2, 1, -1, -2, -1)), A130752 (first differences), A130755 (second differences).

m:=31; S:=[ [1, 2, 3][(n-1) mod 3 +1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; // Klaus Brockhaus, Aug 03 2007

I:=[1,3,8]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018

CoefficientList[Series[(1+2*x^2)/((1-2*x)*(1-x+x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{3,-3,2}, {1,3,8}, 30] (* G. C. Greubel, Jan 15 2018 *)

{m=31; v=vector(m); v[1]=1; v[2]=3; v[3]=8; for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} \\ Klaus Brockhaus, Aug 03 2007

{for(n=0, 30, print1(2^(n+1)+[ -1, -1,0, 1, 1, 0][n%6+1], ","))} \\ Klaus Brockhaus, Aug 03 2007

G.f.: (1+2*x^2)/((1-2*x)*(1-x+x^2)).
a(0) = 1; a(1) = 3; a(2) = 8; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
a(n) = 2^(n+1) + A128834(n+4).
a(0) = 1; for n > 0, a(n) = 2*a(n-1) + A057079(n-1).

Edited and extended by Klaus Brockhaus, Aug 03 2007

A078070 Expansion of (1-x)/(1 + 2x + 2x^2 + x^3).

Original entry on oeis.org

1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3, 4, -3, 1, 0, 1, -3

Offset: 0

N. J. A. Sloane, Nov 17 2002

Period 6: repeat [1, -3, 4, -3, 1, 0].
The unsigned sequence is the r=3 member in the r-family of sequences S_r(n) defined in A092184 where more information can be found. |a(n)| = 2-2*T(n,1/2), with twice the Chebyshev polynomials of the first kind 2*T(n,x=1/2) = A057079(n+1) = S(n+1,1) + S(n,1) with S(n,1)= A010892(n).
Working with an offset of 1, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = -3, P2 = 2, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A100047, A254745.

a[ n_] := {-3, 4, -3, 1, 0, 1}[[Mod[ n, 6, 1]]]; (* Michael Somos, Aug 05 2015 *)
CoefficientList[Series[(1-x)/(1+2x+2x^2+x^3),{x,0,120}],x] (* or *) PadRight[ {},120,{1,-3,4,-3,1,0}] (* or *) LinearRecurrence[{-2,-2,-1},{1,-3,4},120] (* Harvey P. Dale, Jan 06 2016 *)

Vec((1-x)/(1+2*x+2*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

{a(n) = [1, -3, 4, -3, 1, 0][n%6 + 1]}; /* Michael Somos, Aug 05 2015 */

abs(a(n)) = 2 + 2*cos(Pi*n/3 - 2*Pi/3). - Paul Barry, Mar 14 2004
Euler transform of finite sequence [-3, 1, 1]. - Michael Somos, Sep 17 2004
a(n) = (n+1)*(Sum_{k=0..floor((n+1)/2)} (-1)^k*binomial(n-k+1, k)*(-1)^(n-2k+1)/(n-k+1)) + 2*(-1)^n; a(n) = 2*T(n+1, -1/2) + 2(-1)^n. - Paul Barry, Dec 12 2004
From Peter Bala, Mar 25 2014: (Start)
The following formulas assume an offset of 1.
Let {u_j(n)}, j = 0 or j = 1, be two Lucas sequences in the quadratic integer ring Z[w], where w = exp(2*Pi*i/3), defined by the recurrences u_j(0) = 0, u_j(1) = 1 and u_j(n) = (-1)^j*sqrt(3)*u(n-1) - u(n-2) for n >= 2. Then a(n) = u_0(n)*u_1(n).
Equivalently, a(n) = U(n-1,sqrt(3)/2)*U(n-1,-sqrt(3)/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = - ( ((sqrt(3) + i)/2)^n - ((sqrt(3) - i)/2)^n )*( ((-sqrt(3) + i)/2)^n - ((-sqrt(3) - i)/2)^n ) = w^n + w^(2*n) - 2*(-1)^n = 2*cos(2*n*Pi/3) - 2*(-1)^n.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -1/2; 1, -3/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
a(n) = a(n+6) = a(-2-n) for all n in Z. - Michael Somos, Aug 05 2015
a(n) = (-1)^n * A254745(n). - Michael Somos, Jul 16 2017

Chebyshev comment and related formulas from Wolfdieter Lang, Sep 10 2004

A155587 Expansion of (1 + x*c(x))/(1 - x), where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 3, 5, 10, 24, 66, 198, 627, 2057, 6919, 23715, 82501, 290513, 1033413, 3707853, 13402698, 48760368, 178405158, 656043858, 2423307048, 8987427468, 33453694488, 124936258128, 467995871778, 1757900019102, 6619846420554

Offset: 0

Paul Barry, Jan 24 2009

Row sums of A155586.
Hankel transform is A057079(n+2).
From Petros Hadjicostas, Aug 03 2020: (Start)
To prove R. J. Mathar's conjecture, note that the o.g.f. of the sequence implies (Sum_{n >= 0} a(n)*x^n)*(1 - x) = 1 + x*c(x); i.e., a(0) + Sum_{n >= 1} (a(n) - a(n-1))*x^n = 1 + Sum_{n >= 1} C(n-1)*x^n, where C(n) = A000108(n) (Catalan numbers).
Thus, C(n-1) = a(n) - a(n-1) (for n >= 1), and hence C(n) = a(n+1) - a(n). Since 2*(2*n - 1)*C(n-1) = (n + 1)*C(n), we get (n + 1)*a(n+1) + (-5*n + 1)*a(n) + 2*(2*n - 1)*a(n-1) = 0. The last equation implies R. J. Mathar's conjecture. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Gerard Cohen and Jean-Pierre Flori, On a generalized combinatorial conjecture involving addition mod 2^k - 1, IACR, Report 2011/400.
Jean-Pierre Flori, Fonctions booléennes, courbes algébriques et multiplication complexe, Thesis, ParisTech, Feb 03 2012.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Cf. A000108, A014138, A057079, A155586.

Partial sums of A120588.

a155587 n = a155587_list !! n
a155587_list = scanl (+) 1 a000108_list  -- Reinhard Zumkeller, Mar 01 2013

CatalanNumber := n -> binomial(2*n, n)/(n+1):
a := n -> ((3 - I*sqrt(3)))/2 - CatalanNumber(n)*hypergeom([1, n+1/2], [n+2], 4):
seq(simplify(a(n)), n=0..26); # Peter Luschny, Aug 04 2020

a(n) = 1 + Sum_{k=0..n-1} A000108(k).
Conjecture: n*a(n) + (6-5*n)*a(n-1) + 2*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 15 2011
a(n) = A014138(n-1) + 2 for n > 0. - Reinhard Zumkeller, Mar 01 2013 [Corrected by Petros Hadjicostas, Aug 03 2020]
a(n+1) - a(n) = A000108(n). - Petros Hadjicostas, Aug 04 2020
a(n) = ((3 - i*sqrt(3)))/2 - CatalanNumber(n)*hypergeom([1, n + 1/2], [n + 2], 4). - Peter Luschny, Aug 04 2020

A192174 Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k.

Original entry on oeis.org

-1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, -1, 1, 0, -2, 0, -1, 0, 1, 0, -3, 0, 0, 0, 1, 1, 0, -4, 0, 2, 0, 2, 0, 1, 0, -5, 0, 5, 0, 2, 0, -1, 1, 0, -6, 0, 9, 0, 0, 0, -3, 0, 1, 0, -7, 0, 14, 0, -5, 0, -5, 0, 1, 1, 0, -8, 0, 20, 0, -14, 0, -5, 0, 4, 0

Offset: 0

Paul Curtz, Jun 24 2011

Consider the Catalan triangle A009766 antisymmetrically extended by a mirror along the diagonal (see also A176239):
0, -1, -1, -1, -1,  -1,  -1,   -1,
1,  0, -1, -2, -3,  -4,  -5,   -6,
1,  1,  0, -2, -5,  -9, -14,  -20,
1,  2,  2,  0, -5, -14, -28,  -48,
1,  3,  5,  5,  0, -14, -42,  -90,
1,  4,  9, 14, 14,   0, -42, -132,
1,  5, 14, 28, 42,  42,   0, -132,
1,  6, 20, 48, 90, 132, 132,    0.
The rows in this array are essentially the columns of T(n,k).

			Triangle begins
  -1;      # -1
   1,  0;      # x
   1,  0,  1;      # x^2+1
   1,  0,  0,  0;      # x^3
   1,  0, -1,  0, -1;      # x^4-x^2-1
   1,  0, -2,  0, -1,  0;
   1,  0, -3,  0,  0,  0,  1;
   1,  0, -4,  0,  2,  0,  2,  0;
   1,  0, -5,  0,  5,  0,  2,  0, -1;
   1,  0, -6,  0,  9,  0,  0,  0, -3,  0;
   1,  0, -7,  0, 14,  0, -5,  0, -5,  0,  1;
   1,  0, -8,  0, 20,  0,-14,  0, -5,  0,  4,  0;
   1,  0, -9,  0, 27,  0,-28,  0,  0,  0,  9,  0, -1;

Cf. A023443, A080956 and A000096, A129936 and A005586, A005587.

Cf. A194084. - Paul Curtz, Aug 16 2011

p:= proc(n,x) option remember: if n=0 then -1 elif n=1 then x elif n>=2 then x*procname(n-1,x)-procname(n-2,x) fi: end: A192174 := proc(n,k): coeff(p(n,x),x,n-k): end: seq(seq(A192174(n,k),k=0..n), n=0..11); # Johannes W. Meijer, Aug 21 2011

Sum_{k=0..n} T(n,k) = A057079(n-1).

Apparently T(3s,2s-2) = (-1)^(s+1)*A000245(s), s >= 1.

A207606 Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.

Original entry on oeis.org

1, 2, 3, 2, 4, 7, 2, 5, 16, 11, 2, 6, 30, 36, 15, 2, 7, 50, 91, 64, 19, 2, 8, 77, 196, 204, 100, 23, 2, 9, 112, 378, 540, 385, 144, 27, 2, 10, 156, 672, 1254, 1210, 650, 196, 31, 2, 11, 210, 1122, 2640, 3289, 2366, 1015, 256, 35, 2, 12, 275, 1782, 5148, 8008

Offset: 1

Clark Kimberling, Feb 19 2012

As triangle T(n,k) with 0 <= k <= n, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

			First five rows:
  1;
  2;
  3,  2;
  4,  7,  2;
  5, 16, 11,  2;
Triangle (2, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, ...), 0 <= k <= n, begins:
  1;
  2,   0;
  3,   2,   0;
  4,   7,   2,   0;
  5,  16,  11,   2,   0;
  6,  30,  36,  15,   2,   0;
  7,  50,  91,  64,  19,   2,   0;
  8,  77, 196, 204, 100,  23,   2,   0;

G. C. Greubel, Rows n = 1..101 of the triangle, flattened

Cf. A207607.

T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=0 then n+2
    elif k=n then 2
    else 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k)
      fi; end:
1, seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Mar 15 2020

(* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207606 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207607 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, n+2, If[k==n, 2, 2*T[n-1, k] - T[n-2, k] + T[n-1, k-1] ]]]; Join[{1}, Table[T[n, k], {n, 0, 10}, {k, 0, n}]]//Flatten (* G. C. Greubel, Mar 15 2020 *)

from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else x*u(n - 1, x) + (x + 1)*v(n - 1, x)
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==1): return n+1
    elif (k==n): return 2
    else: return 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k)
[1]+[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 15 2020

u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = x*u(n-1,x) + (x+1)v(n-1,x), where u(1,x)=1, v(1,x)=1.
As triangle T(n,k) with 0 <= k <= n: g.f.: (1-y*x)/(1-(2+y)*x+x^2). - Philippe Deléham, Mar 03 2012
As triangle T(n,k) with 0 <= k <= n: Sum_{k=0..n} T(n,k)*x^k = A132677(n), A000034(n)*A057077(n), A057079(n), A000027(n+1), A001519(n+1), A001075(n), A002310(n), A038725(n), A172968(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Mar 03 2012
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Mar 03 2012
T(n,k) = C(n+k-1,2*k+1) + 2*C(n+k-1,2*k), where C is binomial. - Yuchun Ji, May 23 2019
T(n,k) = T(n-1,k) + A207607(n-1,k). - Yuchun Ji, May 28 2019

A234044 Period 7: repeat [2, -2, 1, 0, 0, 1, -2].

Original entry on oeis.org

2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2, 1, 0, 0, 1, -2, 2, -2

Offset: 0

Wolfdieter Lang, Feb 27 2014

This is a member of the six sequences which appear for the instance N=7 of the general formula 2*exp(2*Pi*n*I/N) = R(n, x^2-2) + x*S(n-1, x^2-2)*s(N)*I, for n >= 0, with I = sqrt(-1), s(N) = sqrt(2-x)*sqrt(2+x), x = rho(N) := 2*cos(Pi/N) and R and S are the monic Chebyshev polynomials whose coefficient tables are given in A127672 and A049310. If powers x^k with k >= delta(N) = A055034(N) enter in R or x*S then C(N, x), the minimal polynomial of x = rho(N) (see A187360) is used for a reduction. If delta(N) = 2 it may happen that sqrt(2+x) or sqrt(2-x) is an integer in the number field Q(rho(N)). See the N=5 case comment on A164116.

For N=7 with delta(7) = 3, and C(7, x) = x^3 - x^2 - 2*x + 1 the final result becomes 2*exp(2*Pi*n*I/7) = (a(n) + b(n)*x + c(n)*x^2) + (A(n) + B(n)*x + C(n)*x^2)*s(7)*I, with x = rho(7) = 2*cos(Pi/7), a(n) the present sequence, b(n) = A234045(n), c(n) = A234046(n), A(n) = A238468(n), B(n) = A238469(n) and C(n) = A238470(n). The a, b, c and A, B, C brackets are integers in Q(rho(7)).

			n = 4: 2*exp(8*Pi*I/7) = (2-16*x^2+20*x^4-8*x^6+x^8) + (4*x+10*x^3-6*x^5+x^7)*s(7)*I, reduced with C(7, x) = x^3 - x^2 - 2*x + 1 = 0 this becomes = (-x) + (-1)*s(7)*I with x= 2*cos(Pi/7) and s(7) = 2*sin(Pi/7).The power basis coefficients are thus (a(4), b(4), c(4)) = (0, -1, 0) and (A(4), B(4), C(4)) = (-1, 0, 0).

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

Cf. A234045, A234046, A238468, A238469, A238470, A099837 (N=3), A056594 (N=4), A164116 (N=5), A057079 (N=6).

&cat [[2, -2, 1, 0, 0, 1, -2]^^20]; // Wesley Ivan Hurt, Jul 16 2016

seq(op([2, -2, 1, 0, 0, 1, -2]), n=0..20); # Wesley Ivan Hurt, Jul 16 2016

PadRight[{}, 100, {2, -2, 1, 0, 0, 1, -2}] (* Wesley Ivan Hurt, Jul 16 2016 *)

a(n)=[2, -2, 1, 0, 0, 1, -2][n%7+1] \\ Charles R Greathouse IV, Jul 17 2016

G.f.: (2 - 2*x + x^2 + x^5 - 2*x^6)/(1 - x^7).
a(n+7) = a(n) for n>=0, with a(0) = -a(1) = -a(6) = 2, a(3) = a(4) =0 and a(2) = a(5) = 1.
From Wesley Ivan Hurt, Jul 16 2016: (Start)
a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) = 0 for n>5.
a(n) = (1/7) * Sum_{k=1..6} 2*cos((2k)*n*Pi/7) - 2*cos((2k)*(1+n)*Pi/7) + cos((2k)*(2+n)*Pi/7) + cos((2k)*(5+n)*Pi/7) - 2*cos((2k)*(6+n)*Pi/7).
a(n) = 2 + 4*floor(n/7) - 3*floor((1+n)/7) + floor((2+n)/7) - floor((4+n)/7) + 3*floor((5+n)/7) - 4*floor((6+n)/7). (End)

A079757 Periodic sequence 1, 0, -2, 3, -2, 0, ...

Original entry on oeis.org

1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0, 1, 0, -2, 3, -2, 0

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jan 10 2003

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1).

Cf. A057079.

&cat[[1,0,-2,3,-2,0]: n in [0..20]]; // G. C. Greubel, Mar 27 2024

PadRight[{}, 120, {1,0,-2,3,-2,0}] (* G. C. Greubel, Mar 27 2024 *)

def A079757(n): return [1,0,-2,3,-2,0][n%6]
[A079757(n) for n in range(121)] # G. C. Greubel, Mar 27 2024

a(n) = A057079(2*n+1) - (-1)^floor((2*n+1)/2).
G.f.: (1+2*x)/(1+2*x+2*x^2+x^3).
a(n) = -2*a(n-1) - 2*a(n-2) - a(n-3).
a(n) = |Real(2*(sqrt(3/4)*i - 1/2)^n - (-1)^n )|, where i = sqrt(-1). - Ralf Stephan, Mar 07 2003

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