A099254 -id:A099254 - cp's OEIS frontend

A147611 The 3rd Witt transform of A000027.

0, 0, 0, 0, 2, 7, 18, 42, 84, 153, 264, 429, 666, 1001, 1456, 2061, 2856, 3876, 5166, 6783, 8778, 11214, 14168, 17710, 21924, 26910, 32760, 39582, 47502, 56637, 67122, 79112, 92752, 108207, 125664, 145299, 167310, 191919, 219336, 249795, 283556

Offset: 0

R. J. Mathar, Nov 08 2008

a(n) is the number of binary Lyndon words of length n+3 having 3 blocks of 0's, see Math.SE. - Andrey Zabolotskiy, Nov 16 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Pieter Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO].
Felix Pahl, Find the number of n-length Lyndon words on alphabet {0,1} with k blocks of 0's. (answer), Mathematics StackExchange, 2020.
Index entries for linear recurrences with constant coefficients, signature (4,-6,6,-9,12,-9,6,-6,4,-1).

Cf. A006584 (2nd Witt transform of A000027), A049347, A099254, A147618.

R:=PowerSeriesRing(Integers(), 50); [0,0,0,0] cat Coefficients(R!( x^4*(2-x+2*x^2)/((1-x)^6*(1+x+x^2)^2) )); // G. C. Greubel, Oct 24 2022

CoefficientList[Series[x^4(2 -x+ 2*x^2)/((1-x)^6*(1 +x +x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 13 2012 *)

def A147611_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^4*(2-x+2*x^2)/((1-x)^6*(1+x+x^2)^2) ).list()
A147611_list(50) # G. C. Greubel, Oct 24 2022

G.f.: x^4*(2-x+2*x^2)/((1-x)^6*(1+x+x^2)^2).

a(n) = (1/27)*((3*A049347(n) + A049347(n-1)) - 3*(-1)^n*(A099254(n) - A099254(n- 1)) + n*(3*n^4 - 15*n^2 - 28)/40). - G. C. Greubel, Oct 24 2022

A167177 Expansion of 1/((1 +x +x^2)^2 *(1 +x^2 +x^3)^3).

Original entry on oeis.org

1, -2, -2, 5, 5, -7, -13, 2, 29, 19, -47, -68, 43, 151, 31, -246, -237, 267, 611, -34, -1078, -707, 1327, 2149, -701, -4118, -1760, 5611, 6904, -4361, -14463, -3123, 21453, 20320, -20510, -47501, -426, 76389, 54711, -84119, -147200, 30748, 256922, 132152, -315913, -432648, 196632

Offset: 0

Roger L. Bagula, Oct 29 2009

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2, -6, -11, -19, -27, -34, -38, -36, -30, -21, -12, -5, -1).

Cf. A099254.

a = {t^2 + t + 1, tau^2 + tau + 1, x^3 + x + 1, y^3 + y + 1, z^3 + z + 1} /. y -> x /. z -> x /. t -> x /. tau -> x
p[x_] = Product[a[[n]], {n, 1, 5}]
q[x_] = Expand[x^13*p[1/x]]
Table[ SeriesCoefficient[ Series[1/q[x], {x, 0, 30}], n], {n, 0, 30}]
CoefficientList[Series[1/((1 + x + x^2)^2*(1 + x^2 + x^3)^3), {x, 0, 100}], x] (* G. C. Greubel, Jun 04 2016 *)

a(n) = -2*a(n-1) -6*a(n-2) -11*a(n-3) -19*a(n-4) -27*a(n-5) -34*a(n-6) -38*a(n-7) -36*a(n-8) -30*a(n-9) -21*a(n-10) -12*a(n-11) -5*a(n-12) -a(n-13).

A188146 Three interleaved 1st-order polynomials: a(3n) = 1+4n, a(1+3n) = 3+4n, a(2+3*n) = 1+n.

Original entry on oeis.org

1, 3, 1, 5, 7, 2, 9, 11, 3, 13, 15, 4, 17, 19, 5, 21, 23, 6, 25, 27, 7, 29, 31, 8, 33, 35, 9, 37, 39, 10, 41, 43, 11, 45, 47, 12, 49, 51, 13, 53, 55, 14, 57, 59, 15, 61, 63, 16, 65, 67, 17, 69, 71, 18, 73, 75, 19, 77, 79, 20, 81, 83, 21, 85, 87, 22, 89, 91, 23, 93, 95, 24, 97, 99, 25, 101, 103, 26, 105, 107, 27, 109, 111

Offset: 0

Paul Curtz, Mar 22 2011

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

CoefficientList[Series[(1+3x+x^2+3x^3+x^4)/((x-1)^2(1+x+x^2)^2), {x,0,85}],x] (* Harvey P. Dale, Apr 09 2011 *)

Vec((1+3*x+x^2+3*x^3+x^4 ) / ((x-1)^2*(1+x+x^2)^2) + O(x^100)) \\ Colin Barker, Mar 06 2017

a(n)= 2*a(n-3) - a(n-6).
a(3*n) + a(1+3*n) + a(2+3*n) = 5+9*n.
a(n) = n + 1 - (-1)^n*A099254(n-1). - R. J. Mathar, Mar 31 2011
G.f.: ( 1+3*x+x^2+3*x^3+x^4 ) / ( (x-1)^2*(1+x+x^2)^2 ). - R. J. Mathar, Mar 31 2011
a(n) = (9*(n+1) + sqrt(3)*(3*n+4)*sin((2*Pi*n)/3) + 3*n*cos((2*Pi*n)/3)) / 9. - Colin Barker, Mar 06 2017

A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 2, 0, 3, 5, 1, -1, 5, 10, 4, -2, -1, 8, 20, 12, -4, -4, 0, 13, 38, 31, -4, -13, -2, 1, 21, 71, 73, 3, -33, -11, 3, 1, 34, 130, 162, 34, -74, -42, 6, 6, 0, 55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Offset: 0

Philippe Deléham, Dec 06 2011

Row-reversed variant of A123585. Row sums: 2^n.

			Triangle begins:
1
1, 1
2, 2, 0
3, 5, 1, -1
5, 10, 4, -2, -1
8, 20, 12, -4, -4, 0
13, 38, 31, -4, -13, -2, 1
21, 71, 73, 3, -33, -11, 3, 1
34, 130, 162, 34, -74, -42, 6, 6, 0
55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Cf. Columns: A000045, A001629, A129707.

Diagonals: A010892, A099254, Antidiagonal sums: A158943.

G.f.: 1/(1-(1+y)*x+(y+1)*(y-1)*x^2).
T(n,0) = A000045(n+1).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0)= 1 and T(n,k)= 0 if nSum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A090591(n), (-1)^n*A106852(n), A000007(n), A000045(n+1), A000079(n), A057083(n), A190966(n+1) for n = -3, -2, -1, 0, 1, 2, 3 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A010892(n), A000079(n), A030195(n+1), A180222(n+2) for x = 0, 1, 2, 3 respectively.

A238988 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, -1, 0, 2, 1, -1, -1, 1, 2, 1, 0, -1, -2, 1, 3, 1, 1, 0, -4, -2, 3, 3, 1, 1, 1, -2, -4, -2, 3, 4, 1, 0, 1, 3, -2, -9, -2, 6, 4, 1, -1, 0, 6, 3, -9, -9, 0, 6, 5, 1, -1, -1, 3, 6, 3, -9, -15, 0, 10, 5, 1, 0, -1, -4, 3, 18, 3, -24, -15, 5, 10, 6, 1

Offset: 0

Philippe Deléham, Mar 07 2014

T(n,0) = T(n+1,1) = A010892(n), T(n+2,2) = T(n+3,3) = A099254(n), T(n+4,4) = T(n+5,5) = A128504(n).

Triangle T(n,k) = A101950(n - floor((k+1)/2),floor(k/2)).

			Triangle begins:
1;
1, 1;
0, 1, 1;
-1, 0, 2, 1;
-1, -1, 1, 2, 1;
0, -1, -2, 1, 3, 1;
1, 0, -4, -2, 3, 3, 1;
1, 1, -2, -4, -2, 3, 4, 1;
0, 1, 3, -2, -9, -2, 6, 4, 1;
-1, 0, 6, 3, -9, -9, 0, 6, 5, 1;
-1, -1, 3, 6, 3, -9, -15, 0, 10, 5, 1;
0, -1, -4, 3, 18, 3, -24, -15, 5, 10, 6, 1;
1, 0, -8, -4, 18, 18, -6, -24, -20, 5, 15, 6, 1;

Indranil Ghosh, Rows 0..100, flattened

Cf. A101950

nmax=11; Flatten[CoefficientList[Series[CoefficientList[Series[(1 + x*y)/(1 - x + x^2 - x^2*y^2), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 14 2017 *)

G.f.: (1 + x*y)/(1 - x + x^2 - x^2*y^2).
T(n,k) = T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k = 0..n} T(n,k)*x^k = A000007(n), A010892(n), A040000(n), A105476(n+1) for x = -1, 0, 1, 2 respectively.

A115054 G.f.: (x^3+6*x+2)^2/(x^2+x+1)^2.

Original entry on oeis.org

4, 16, -8, -36, 72, -36, -63, 126, -63, -90, 180, -90, -117, 234, -117, -144, 288, -144, -171, 342, -171, -198, 396, -198, -225, 450, -225, -252, 504, -252, -279, 558, -279, -306, 612, -306, -333, 666, -333, -360, 720, -360, -387, 774, -387, -414, 828, -414, -441, 882, -441, -468, 936, -468, -495, 990, -495

Offset: 0

Roger L. Bagula, Feb 28 2006

q=3 coefficient expansion of hierarchical lattice renormalization polynomial.

Auto-convolution of the sequence 2,4,-6,3,3,-6,3,3,.. (period length 3). [From R. J. Mathar, Mar 09 2009]

Peitgen and Richter, The Beauty of Fractals, Springer-Verlag, New York, 1986, page 146

Index entries for linear recurrences with constant coefficients, signature (-2,-3,-2,-1).

G:=(x^3+6*x+2)^2/(x^2+x+1)^2: Gser:=series(G,x=0,55): seq(coeff(Gser,x,n),n=0..50);

q=3 b = 9*Flatten[{{4/9}, Abs[Table[Coefficient[ Series[((x^3 + 3*(q - 1)*x + (q - 1)*(q - 2))/(3*x^2 + 3*( q - 2)*x + q^2 - 3*q + 3))^2, {x, 0, 30}], x^n], {n, 1, 30}]]}]

a(n) = 18*A131713(n)-27*(-1)^n*A099254(n), n>2. [From R. J. Mathar, Mar 09 2009]

Edited by N. J. A. Sloane, Apr 16 2006

Previous Showing 21-26 of 26 results.

cp's OEIS Frontend

A147611 The 3rd Witt transform of A000027.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A167177 Expansion of 1/((1 +x +x^2)^2 *(1 +x^2 +x^3)^3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A188146 Three interleaved 1st-order polynomials: a(3*n) = 1+4*n, a(1+3*n) = 3+4*n, a(2+3*n) = 1+n.

Views

Author

Keywords

Links

Programs

Mathematica

PARI

Formula

A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A238988 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A115054 G.f.: (x^3+6*x+2)^2/(x^2+x+1)^2.

Views

Author

Keywords

Comments

References

Links

Programs

Maple

Mathematica

Formula

Extensions

A188146 Three interleaved 1st-order polynomials: a(3n) = 1+4n, a(1+3n) = 3+4n, a(2+3*n) = 1+n.