A059260 -id:A059260 - cp's OEIS frontend

A113861 a(n) = (1/9)((6n - 7)*2^(n-1) - (-1)^n).

0, 1, 5, 15, 41, 103, 249, 583, 1337, 3015, 6713, 14791, 32313, 70087, 151097, 324039, 691769, 1470919, 3116601, 6582727, 13864505, 29127111, 61050425, 127693255, 266571321, 555512263, 1155763769, 2401006023, 4980969017, 10319851975, 21355531833, 44142719431

Offset: 1

N. J. A. Sloane, Jan 25 2006

This sequence is connected with the Collatz problem (see the sequences A045883 and A001045). - Michel Lagneau, Jan 13 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000
T. Etzion, On the stopping redundancy of Reed-Muller codes, IEEE Trans. Information Theory 52 (11) (2006) 4867-4879; arXiv:cs.IT/0511056.
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A102301.

Cf. A059260, A016813.

Join[{0},Numerator[CoefficientList[Series[(x+1)/((x-1)*(x^2+x-2)),{x,0,40}],x]]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)
LinearRecurrence[{3,0,-4},{0,1,5},40] (* Harvey P. Dale, Jun 16 2022 *)

a(n)=((6*n-7)<<(n-1)-(-1)^n)/9 \\ Charles R Greathouse IV, Jan 13 2012

a(n+1) - 2*a(n) = A001045(n+2), Jacobsthal numbers. - Paul Curtz, Jul 05 2008
3*a(n) - a(n+1) = -1, -2, 4*a(n). - Paul Curtz, Jul 05 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: x^2*(1+2*x)/((1+x)*(1-2*x)^2).
a(n) + a(n+1) = A014480(n-1). (End)
a(n) = 4*a(n-1) - 4*a(n-2) + (-1)^(n+1), n>2. - Gary Detlefs, Dec 19 2010
a(n) = 3*a(n-1) - 4*a(n-3), n>3. - Gary Detlefs, Dec 19 2010
a(n) = n*2^n - A045883(n). - Michel Lagneau, Jan 13 2012
Starting with "1" = triangle A059260 * A016813 as a vector, where A016813 = (4n + 1): [ 1, 5, 9, 13, ...]. - Gary W. Adamson, Mar 06 2012

A052543 Expansion of (1-x)/(1 - 3x - 2x^2 + 2*x^3).

Original entry on oeis.org

1, 2, 8, 26, 90, 306, 1046, 3570, 12190, 41618, 142094, 485138, 1656366, 5655186, 19308014, 65921682, 225070702, 768439442, 2623616366, 8957586578, 30583113582, 104417281170, 356502897518, 1217177027730, 4155702315886

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

From Andrew Woods, Jun 03 2013: (Start)
a(n) is the number of ways to tile a 2 X n square grid with 1 X 1, 1 X 2, 2 X 1, and 2 X 2 tiles. Solutions for a(2)=8:
.          _   _   _            
  | | | | || |   | |__| |_| || | ||_| |||
  ||| ||| |_| |_| ||| ||| |_| |||
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 478
Index entries for linear recurrences with constant coefficients, signature (3,2,-2).

Cf. A000129, A059260.

a:=[1,2,8];; for n in [4..30] do a[n]:=3*a[n-1]+2*a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, May 09 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1 -3*x-2*x^2+2*x^3) )); // G. C. Greubel, May 09 2019

spec := [S,{S=Sequence(Prod(Union(Z,Z),Union(Z,Sequence(Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[(1-x)/(1-3x-2x^2+2x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,2,-2},{1,2,8},30] (* Harvey P. Dale, Jan 23 2013 *)

my(x='x+O('x^30)); Vec((1-x)/(1-3*x-2*x^2+2*x^3)) \\ G. C. Greubel, May 09 2019

((1-x)/(1-3*x-2*x^2+2*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

G.f.: (1-x)/(1 - 3*x - 2*x^2 + 2*x^3).
a(n) = 3*a(n-1) + 2*a(n-2) - 2*a(n-3), with a(0)=1, a(1)=2, a(2)=8.
a(n) = Sum_{alpha = RootOf(1 -3*x -2*x^2 +2*x^3)} (1/98)*(13 + 25*alpha - 16*alpha^2)*alpha^(-n-1).
Equals triangle A059260 * the Pell sequence [1, 2, 5, 12, ...] as a vector. - Gary W. Adamson, Mar 06 2012
a(n) = A214997(n) - A214996(n). - Clark Kimberling, Nov 28 2012

More terms from James Sellers, Jun 06 2000

A128494 Coefficient table for sums of Chebyshev's S-Polynomials.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, -1, 1, 1, 1, -1, -2, 1, 1, 1, 2, -2, -3, 1, 1, 0, 2, 4, -3, -4, 1, 1, 0, -2, 4, 7, -4, -5, 1, 1, 1, -2, -6, 7, 11, -5, -6, 1, 1, 1, 3, -6, -13, 11, 16, -6, -7, 1, 1, 0, 3, 9, -13, -24, 16, 22, -7, -8, 1, 1, 0, -3, 9, 22, -24, -40, 22, 29, -8, -9, 1, 1, 1, -3, -12, 22, 46, -40, -62, 29, 37, -9, -10, 1, 1, 1, 4, -12

Offset: 0

Wolfdieter Lang, Apr 04 2007

See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.

This is a 'repetition triangle' based on a signed version of triangle A059260: a(2*p,2*k) = a(2*p+1,2*k) = A059260(p+k,2*k)*(-1)^(p+k) and a(2*p+1,2*k+1) = a(2*p+2,2*k+1) = A059260(p+k+1,2*k+1)*(-1)^(p+k), k >= 0.

			The triangle a(n,m) begins:
  n\m  0   1   2   3   4   5   6   7   8   9  10
   0:  1
   1:  1   1
   2:  0   1   1
   3:  0  -1   1   1
   4:  1  -1  -2   1   1
   5:  1   2  -2  -3   1   1
   6:  0   2   4  -3  -4   1   1
   7:  0  -2   4   7  -4  -5   1   1
   8:  1  -2  -6   7  11  -5  -6   1   1
   9:  1   3  -6 -13  11  16  -6  -7   1   1
  10:  0   3   9 -13 -24  16  22  -7  -8   1   1
... reformatted by _Wolfdieter Lang_, Oct 16 2012
Row polynomial S(1;4,x) = 1 - x - 2*x^2 + x^3 + x^4 = Sum_{k=0..4} S(k,x).
S(4,y)*S(5,y)/y = 3 - 13*y^2 + 16*y^4 - 7*y^6 + y^8, with y=sqrt(2+x) this becomes S(1;4,x).
From _Wolfdieter Lang_, Oct 16 2012: (Start)
S(1;4,x) = (1 - (S(5,x) - S(4,x)))/(2-x) = (1-x)*(2-x)*(1+x)*(1-x-x^2)/(2-x) = (1-x)*(1+x)*(1-x-x^2).
S(5,x) - S(4,x) = R(11,sqrt(2+x))/sqrt(2+x) = -1 + 3*x + 3*x^2 - 4*x^3 - x^4 + x^5. (End)

Wolfdieter Lang, First 15 rows
Per Alexandersson, Luis Angel González-Serrano, and Egor A. Maximenko, Mario Alberto Moctezuma-Salazar, Symmetric polynomials in the symplectic alphabet and their expression via Dickson-Zhukovsky variables, arXiv:1912.12725 [math.CO], 2019.
Per Alexandersson, Luis Angel González-Serrano, Egor A. Maximenko, and Mario Alberto Moctezuma-Salazar, Symmetric Polynomials in the Symplectic Alphabet and the Change of Variables z_j = x_j + x_j^(-1), The Elec. J. of Combinatorics (2021) Vol. 28, No. 1, #P1.56.

Row sums (signed): A021823(n+2). Row sums (unsigned): A070550(n).
Cf. A128495 for S(2; n, x) coefficient table.
The column sequences (unsigned) are, for m=0..4: A021923, A002265, A008642, A128498, A128499.
For m >= 1 the column sequences (without leading zeros) are of the form a(m, 2*k) = a(m, 2*k+1) = ((-1)^k)*b(m, k) with the sequences b(m, k), given for m=1..11 by A008619, A002620, A002623, A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808.

S(1;n,x) = Sum_{k=0..n} S(k,x) = Sum_{m=0..n} a(n,m)*x^m, n >= 0.
a(n,m) = [x^m](S(n,y)*S(n+1,y)/y) with y:=sqrt(2+x).
G.f. for column m: (x^m)/((1-x)*(1+x^2)^(m+1)), which shows that this is a lower diagonal matrix of the Riordan type, named (1/((1+x^2)*(1-x)), x/(1+x^2)).
From Wolfdieter Lang, Oct 16 2012: (Start)
a(n,m) = [x^m](1- (S(n+1,x) - S(n,x)))/(2-x). From the Binet - de Moivre formula for S and use of the geometric sum.
a(n,m) = [x^m](1- R(2*n+3,sqrt(2+x))/sqrt(2+x))/(2-x) with the monic integer T-polynomials R with coefficient triangle given in A127672. From the odd part of the bisection of the T-polynomials. (End)

A235501 Riordan array (1/(1-2*x^2), x/(1-x)).

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 3, 2, 1, 4, 3, 5, 3, 1, 0, 7, 8, 8, 4, 1, 8, 7, 15, 16, 12, 5, 1, 0, 15, 22, 31, 28, 17, 6, 1, 16, 15, 37, 53, 59, 45, 23, 7, 1, 0, 31, 52, 90, 112, 104, 68, 30, 8, 1, 32, 31, 83, 142, 202, 216, 172, 98, 38, 9, 1, 0, 63, 114, 225

Offset: 0

Philippe Deléham, Jan 11 2014

Row sums are A007179(n+1).

			Triangle begins (0<=k<=n):
1
0, 1
2, 1, 1
0, 3, 2, 1
4, 3, 5, 3, 1
0, 7, 8, 8, 4, 1
8, 7, 15, 16, 12, 5, 1
0, 15, 22, 31, 28, 17, 6, 1

Cf. Columns: A077957, A052551, A077866.
Diagonals: A000012, A001477, A022856.
Cf. Similar sequences: A059260, A191582.

T(n,n)=1, T(2n,0)=2^n, T(2n+1,0)=0, T(n,k)=T(n-1,k-1)+T(n-1,k) for 0

T(n,k)=T(n-1,k)+T(n-1,k-1)+2*T(n-2,k)-T(n-3,k)-2*T(n-3,k-1), T(0,0)=1, T(1,0)=0, T(1,1)=1, T(n,k)=0 if k<0 or if k>n.

T(n,n)=1, T(n+1,n)=n, T(n+2,n)=n*(n+1)/2 + 2.

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(3*x + 2*x^2/2! + x^3/3!) = 3*x + 8*x^2/2! + 16*x^3/3! + 28*x^4/4! + 45*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

A239230 Expansion of -xlog'(-sqrt(12x+2sqrt(1-4x)+2)/4+sqrt(1-4*x)/4+5/4).

Original entry on oeis.org

0, 1, 1, 4, 9, 36, 112, 428, 1505, 5692, 21026, 79806, 301488, 1151866, 4403778, 16929474, 65204353, 251947668, 975366094, 3784197606, 14705937794, 57242631464, 223121176224, 870805992278, 3402485053664, 13308485156086, 52104519751272, 204176144516818

Offset: 0

Vladimir Kruchinin, May 22 2014

nonn

Cf. A055113.

a:= n-> add((-1)^(k+n)*binomial(2*n-k-1, n-1)*hypergeom([k-n, (n+1)/2, n/2], [n, n+1], 4), k=1..n);
seq(round(evalf(a(n), 32)), n=0..24); # Peter Luschny, May 22 2014

a(n):=n*sum(sum(binomial(n+2*j-1,j+n-1)*(-1)^(k+j+n)*binomial(2*n-k,j+n), j,0,n-k)/(2*n-k), k,1,n);

G.f.: A(x) = x*F'(x)/(1-F(x)), where F(x) is g.f. of A055113.
a(n) = n * Sum_{k=1..n} (Sum_{j=0..n-k} C(n+2*j-1,j+n-1) * (-1)^(k+j+n) * C(2*n-k,j+n)) / (2*n-k).
a(n) = Sum_{k=1..n} (-1)^(k+n) * C(2*n-k-1,n-1) * hypergeom([k-n, n/2+1/2, n/2], [n, n+1], 4). - Peter Luschny, May 22 2014
Conjecture D-finite with recurrence +24*(365025561*n-1672569283)*(n-1)*(n-2)*(2*n-1)*a(n) -4*(n-2)*(27129169947*n^3-209577621466*n^2+463278020461*n-314084557758)*a(n-1) +2*(-28823853823*n^4+487259692534*n^3-3105214937957*n^2+8814274338098*n-9143920331436)*a(n-2) +4*(276083065830*n^4-4172118623320*n^3+24824880820695*n^2-69263721795041*n+75832154222148)*a(n-3) +(-587491214125*n^4+9941738070620*n^3-75070680472775*n^2+281912285021344*n-413197788157152)*a(n-4) +2*(-1186924847911*n^4+26108844767699*n^3-211936472383904*n^2+757584729548632*n-1009721693733312)*a(n-5) +4*(2*n-11)*(328530544924*n^3-5280431217363*n^2+28334632524947*n-50473913356356)*a(n-6) -72*(4143100547*n-18456753180)*(n-6)*(2*n-11)*(2*n-13)*a(n-7)=0. - R. J. Mathar, Jul 27 2022
Conjecture: a(n) = Sum_{i=0..n-1} A059260(2*(n-1),n+i-1)*A009766(n+i-1,i)*(-1)^(n+i-1) = n*Sum_{i=0..n-1} binomial(n+2*i, i)/(n+2*i)*(-1)^(n+i-1)*[x^(n+i-1)] (1+(x+1)^(2*n-1))/(x+2) for n >= 0. - Mikhail Kurkov, Feb 18 2025

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A113861 a(n) = (1/9)*((6*n - 7)*2^(n-1) - (-1)^n).

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A052543 Expansion of (1-x)/(1 - 3*x - 2*x^2 + 2*x^3).

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A128494 Coefficient table for sums of Chebyshev's S-Polynomials.

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A235501 Riordan array (1/(1-2*x^2), x/(1-x)).

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A239230 Expansion of -x*log'(-sqrt(12*x+2*sqrt(1-4*x)+2)/4+sqrt(1-4*x)/4+5/4).

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A113861 a(n) = (1/9)((6n - 7)*2^(n-1) - (-1)^n).

A052543 Expansion of (1-x)/(1 - 3x - 2x^2 + 2*x^3).

A239230 Expansion of -xlog'(-sqrt(12x+2sqrt(1-4x)+2)/4+sqrt(1-4*x)/4+5/4).