A099443 -id:A099443 - cp's OEIS frontend

A100047 A Chebyshev transform of the Fibonacci numbers.

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

Offset: 0

Paul Barry, Oct 31 2004

Multiplicative with a(p^e) = (-1)^(e+1) if p = 2, 0 if p = 5, 1 if p == 1 or 9 (mod 10), (-1)^e if p == 3 or 7 (mod 10). - David W. Wilson, Jun 10 2005
This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 1, P2 = -1, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 24 2014
From Peter Bala, Mar 24 2014: (Start)
This is the particular case P1 = 1, P2 = -1, Q = 1 of the following results:
Let P1, P2 and Q be integers. Let alpha and beta denote the roots of the quadratic equation x^2 - 1/2*P1*x + 1/4*P2 = 0. Let T(n,x;Q) denote the bivariate Chebyshev polynomial of the first kind defined by T(n,x;Q) = 1/2*( (x + sqrt(x^2 - Q))^n + (x - sqrt(x^2 - Q))^n ) (when Q = 1, T(n,x;Q) reduces to the ordinary Chebyshev polynomial of the first kind T(n,x)). Then we have
1) The sequence A(n) := ( T(n,alpha;Q) - T(n,beta;Q) )/(alpha - beta) is a linear divisibility sequence of the fourth order.
2) A(n) belongs to the 3-parameter family of fourth-order divisibility sequences found by Williams and Guy.
3) The o.g.f. of the sequence A(n) is the rational function x*(1 - Q*x^2)/(1 - P1*x + (P2 + 2*Q)*x^2 - P1*Q*x^3 + Q^2*x^4).
4) The o.g.f. is the Chebyshev transform of the rational function x/(1 - P1*x + P2*x^2), where the Chebyshev transform takes the function A(x) to the function (1 - Q*x^2)/(1 + Q*x^2)*A(x/(1 + Q*x^2)).
5) Let q = sqrt(Q) and set a = sqrt( q + (P2)/(4*q) + (P1)/2 ) and b = sqrt( q + (P2)/(4*q) - (P1)/2 ). Then the o.g.f. of the sequence A(n) is the Hadamard product of the rational functions x/(1 - (a + b)*x + q*x^2) and x/(1 - (a - b)*x + q*x^2). Thus A(n) is the product of two (usually, non-integer) Lucas-type sequences.
6) A(n) = the bottom left entry of the 2 X 2 matrix 2*T(n,1/2*M;Q), where M is the 2 X 2 matrix [0, -P2; 1, P1].
For examples of the above see A006238, A054493, A078070, A092184, A098306, A100048, A108196, A138573, A152090 and A218134. (End)

			A Chebyshev transform of the Fibonacci numbers A000045: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).
The denominator is the 10th cyclotomic polynomial.
G.f. = x + x^2 - x^3 - x^4 - x^6 - x^7 + x^8 + x^9 + x^11 + x^12 - x^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
Tian-Xiao He, Peter J.-S. Shiue, An Approach to the Construction of Linear Divisibility Sequences of Higher Orders, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
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 (1,-1,1,-1).

Other Chebyshev transforms: A011655, A099443, A100048.
Cf. A006238, A054493, A078070, A092184, A098306, A108196, A138573, A152090, A218134.
Cf. also A011558, A080891, A244895, A226162.

a := n -> (-1)^iquo(n, 5)*signum(mods(n, 5)):
seq(a(n), n=0..89); # after Michael Somos, Peter Luschny, Dec 30 2018

a[ n_] := {1, 1, -1, -1, 0, -1, -1, 1, 1, 0}[[Mod[ n, 10, 1]]]; (* Michael Somos, May 24 2015 *)
a[ n_] := (-1)^Quotient[ n, 5] Sign[ Mod[ n, 5, -2]]; (* Michael Somos, May 24 2015 *)
a[ n_] := (-1)^Quotient[n, 5] {1, 1, -1, -1, 0}[[Mod[ n, 5, 1]]]; (* Michael Somos, May 24 2015 *)
LinearRecurrence[{1, -1, 1, -1}, {0, 1, 1, -1}, 90] (* Jean-François Alcover, Jun 11 2019 *)

{a(n) = (-1)^(n\5) * [0, 1, 1, -1, -1][n%5+1]}; /* Michael Somos, May 24 2015 */

{a(n) = (-1)^(n\5) * sign( centerlift( Mod(n, 5)))}; /* Michael Somos, May 24 2015 */

G.f.: x*(1 - x^2)/(1 - x + x^2 - x^3 + x^4).
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4).
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k *binomial(n-k, k)*A000045(n-2*k)/(n  -k).
From Peter Bala, Mar 24 2014: (Start)
a(n) = (T(n,alpha) - T(n,beta))/(alpha - beta), where alpha = (1 + sqrt(5))/4 and beta = (1 - sqrt(5))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = bottom left entry of the matrix T(n, M), where M is the 2 X 2 matrix [0, 1/4; 1, 1/2].
The o.g.f. is the Hadamard product of the rational functions x/(1 - 1/2*(sqrt(5) + 1)*x + x^2) and x/(1 - 1/2*(sqrt(5) - 1)*x + x^2). (End)
Euler transform of length 10 sequence [ 1, -2, 0, 0, -1, 0, 0, 0, 0, 1]. - Michael Somos, May 24 2015
a(n) = a(-n) = -a(n + 5) for all n in Z. - Michael Somos, May 24 2015
|A011558(n)| = |A080891(n)| = |a(n)| = A244895(n). - Michael Somos, May 24 2015

A105633 Row sums of triangle A105632.

Original entry on oeis.org

1, 2, 4, 9, 22, 57, 154, 429, 1223, 3550, 10455, 31160, 93802, 284789, 871008, 2681019, 8298933, 25817396, 80674902, 253106837, 796968056, 2517706037, 7977573203, 25347126630, 80738862085, 257778971504, 824798533933

Offset: 0

Paul D. Hanna, Apr 17 2005

nonn

Binomial transform of A007477. INVERT transform of A082582. First differences give A086581 and A025242 (offset 1). Is this sequence equal to A057580?
a(n) = the number of Dyck paths of semilength n+1 avoiding UUDU. a(n) = the number of Dyck paths of semilength n+1 avoiding UDUU = the number of binary trees without zigzag (i.e., with no node with a father, with a right son and with no left son). This sequence is the first column of the triangle A116424. E.g., a(2) = 4 because there exist four Dyck paths of semilength 3 that avoid UUDU: UDUDUD, UDUUDD, UUDDUD, UUUDDD, as well as four Dyck paths of semilength 3 that avoid UDUU: UDUDUD, UUDUDD, UUDDUD, UUUDDD. - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
The sequence beginning 1,1,2,4,9,... gives the diagonal sums of A130749, and has g.f. 1/(1-x-x^2/(1-x/(1-x-x^2/(1-x/(1-x-x^2/(1-... (continued fraction); and general term Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} binomial(n-k,j)*A090181(j,k). Its Hankel transform is A099443(n+1). - Paul Barry, Jun 30 2009
The number of plain lambda terms presented by de Bruijn indices, see Bendkowski et al. - Kellen Myers, Jun 15 2015
a(n) = the number of Dyck paths of semilength n+1 with no pairs of
consecutive valleys at the same height. Sergi Elizalde, Feb 25 2021

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 22*x^4 + 57*x^5 + 154*x^6 + 429*x^7 + ...
with A(x)^2 = 1 + 4*x + 12*x^2 + 34*x^3 + 96*x^4 + 274*x^5 + 793*x^6 + ...
where A(x) = 1 + x*(2-x)*A(x) + x^2*(1-x)*A(x)^2.
The logarithm of the g.f. begins:
log(A(x)) = (1 + (1-x))*x + (1 + 2^2*(1-x) + (1-x)^2)*x^2/2 +
(1 + 3^2*(1-x) + 3^2*(1-x)^2 + (1-x)^3)*x^3/3 +
(1 + 4^2*(1-x) + 6^2*(1-x)^2 + 4^2*(1-x)^3 + (1-x)^4)*x^4/4 +
(1 + 5^2*(1-x) + 10^2*(1-x)^2 + 10^2*(1-x)^3 + 5^2*(1-x)^4 + (1-x)^5)*x^5/5 + ...
Explicitly,
log(A(x)) = 2*x + 4*x^2/2 + 11*x^3/3 + 32*x^4/4 + 97*x^5/5 + 301*x^6/6 + 947*x^7/7 + 3008*x^8/8 + 9623*x^9/9 + 30959*x^10/10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Permutations avoiding a simsun pattern, The Electronic Journal of Combinatorics (2020) Vol. 27, Issue 3, P3.45. (See a_n in Theorem 4.)
Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, Consecutive pattern-avoidance in Catalan words according to the last symbol, Univ. Bourgogne (France 2023).
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023. See pp. 3, 8.
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 10.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018. See p. 8.
Maciej Bendkowski, Quantitative aspects and generation of random lambda and combinatory logic terms, Ph.D. Thesis, Jagiellonian University, Kraków, 2017.
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, Combinatorics of λ-terms: a natural approach, arXiv:1609.08106 [cs.LO], 2016.
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, A Natural Counting of Lambda Terms, arXiv preprint arXiv:1506.02367 [cs.LO], 2015.
Maciej Bendkowski, K. Grygiel, and P. Tarau, Random generation of closed simply-typed lambda-terms: a synergy between logic programming and Boltzmann samplers, arXiv preprint arXiv:1612.07682 [cs.LO], 2016-2017.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019. See p. 12.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020. See p. 5.
Juan B. Gil and Michael D. Weiner, On pattern-avoiding Fishburn permutations, arXiv:1812.01682 [math.CO], 2018. See pp. 4, 6.
Katarzyna Grygiel and Pierre Lescanne, A natural counting of lambda terms, Preprint 2015.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.
A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741.
Robin D. P. Zhou, Pattern avoidance in revised ascent sequences, arXiv:2505.05171 [math.CO], 2025. See pp. 4, 20.

Cf. A105632, A057580, A116424, A216604. See also A258973.

a := n -> add((-1)^i*hypergeom([(i+1)/2, i/2+1, i-n-1], [1, 2], -4), i=0..n+1):
seq(simplify(a(n)), n=0..26); # Peter Luschny, May 03 2018

CoefficientList[Series[(1 - x - Sqrt[(1 - x)^2 - 4 x^2/(1 - x)])/(2 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)

{a(n)=local(X=x+x*O(x^n)); polcoeff(2/(1-X)/(1-X+sqrt((1-X)^2-4*X^2/(1-X))),n,x)}

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*(1-x)^(m-k) + x*O(x^n)))),n)} \\ Paul D. Hanna, Sep 12 2012

G.f.: A(x) = (1-x - sqrt((1-x)^2 - 4*x^2/(1-x)))/(2*x^2).
a(n) = 2*a(n-1) + Sum_{i=1..n-2} a(i)*(a(n-1-i) - a(n-2-i)). a(n) = Sum_{i=0..floor(n/2)} (-1)^i * binomial(n+1-i,i) * binomial(2*(n+1)-3*i, n-2*i) /(n+1-i). - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
G.f.: (1/(1-x)^2)c(x^2/(1-x)^3), where c(x) is the g.f. of A000108. - Paul Barry, May 22 2009
1/(1-x-x/(1-x^2/(1-x-x/(1-x^2/(1-x-x/(1-x^2/(1-... (continued fraction). - Paul Barry, Jun 30 2009
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} binomial(n-k,j)(0^(j+k)+(1/(j+0^j))*binomial(j,k)*binomial(j,k+1)). - Paul Barry, Jun 30 2009
G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x*(1-x)*A(x)). - Paul D. Hanna, Sep 12 2012
G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * (1-x)^k ). - Paul D. Hanna, Sep 12 2012
D-finite with recurrence: (n+2)*a(n) + (-4*n-3)*a(n-1) + (2*n+1)*a(n-2) + a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 26 2012
The recurrence is true, since by holonomic transformation, it can be computed formally using GFUN, associated with the equation: x^3 + x^2 - 2x + (x^3 + 3 x^2 -3x +1) A(x) + (x^5 + 2x^3 -4 x^2 + x) A'(x) = 0. - Pierre Lescanne, Jun 30 2015
G.f.: (1 - 1/(G(0)-x))/x^2 where G(k) =  1 + x/(1 + x/(x^2 - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2012
a(n) ~ 2^(n/3-1/6) * 3^(n+2) * (13+3*sqrt(33))^((n+1)/3) * sqrt(4*(2879 + 561*sqrt(33))^(1/3) + 8*(7822 + 1362*sqrt(33))^(1/3) - 91 - 21*sqrt(33)) / (((26+6*sqrt(33))^(2/3) - (26+6*sqrt(33))^(1/3) - 8)^(n+3/2) * (4*(26+6*sqrt(33))^(1/3) - (26+6*sqrt(33))^(2/3) + 8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Mar 13 2014
a(n) = Sum_{i=0..n+1} (-1)^i*hypergeom([(i+1)/2, i/2+1, i-n-1], [1, 2], -4). - Peter Luschny, May 03 2018

More terms from I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006

A100051 A Chebyshev transform of 1,1,1,...

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

Offset: 0

Paul Barry, Oct 31 2004

1, followed by period 6: repeat [1, -1, -2, -1, 1, 2]. - Joerg Arndt, Aug 28 2024
A Chebyshev transform of 1/(1-x): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).
Transform of 1/(1+x) under the mapping g(x)->((1+x)/(1-x))g(x/(1-x)^2). - Paul Barry, Dec 01 2004
Multiplicative with a(p^e) = -1 if p = 2; -2 if p = 3; 1 otherwise. - David W. Wilson, Jun 10 2005

			G.f. = 1 + x - x^2 - 2*x^3 - x^4 + x^5 + 2*x^6 + x^7 - x^8 - 2*x^9 - x^10 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (1,-1).

Cf. A011655, A057079, A087204, A099837, A099443, A100047, A100048, A100050.

Row sums of array A127677.

CoefficientList[Series[(1 - x^2)/(1 - x + x^2), {x,0,50}], x] (* G. C. Greubel, May 03 2017 *)
LinearRecurrence[{1,-1},{1,1,-1},80] (* Harvey P. Dale, Mar 25 2019 *)

{a(n) = - (n == 0) + [2, 1, -1, -2, -1, 1][n%6 + 1]}; /* Michael Somos, Mar 21 2011 */

From Paul Barry, Dec 01 2004: (Start)
G.f.: (1-x^2)/(1-x+x^2).
a(n) = a(n-1) - a(n-2), n>2.
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)/(n-k).
a(n) = Sum_{k=0..n} binomial(n+k, 2k)*(2n/(n+k))*(-1)^k, n>1. (End)
Moebius transform is length 6 sequence [1, -2, -3, 0, 0, 6].
Euler transform of length 6 sequence [1, -2, -1, 0, 0, 1].
a(n) = a(-n). a(n) = c_6(n) if n>1 where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
a(n) = A087204(n), n>0. - R. J. Mathar, Sep 02 2008
a(n) = A057079(n+1), n>0. Dirichlet g.f. zeta(s) *(1-2^(1-s)-3^(1-s)+6^(1-s)). - R. J. Mathar, Apr 11 2011

A122047 Degree of the polynomial P(n,x), defined by a Somos-6 type sequence: P(n,x)=(x^(n-1)P(n-1,x)P(n-5,x) + P(n-2,x)*P(n-4,x))/P(n-6,x), initialized with P(n,x)=1 at n<0.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 22, 31, 42, 55, 70, 88, 109, 133, 160, 190, 224, 262, 304, 350, 400, 455, 515, 580, 650, 725, 806, 893, 986, 1085, 1190, 1302, 1421, 1547, 1680, 1820, 1968, 2124, 2288, 2460, 2640, 2829, 3027

Offset: 0

Roger L. Bagula, Sep 13 2006

nonn

Maximum Wiener index of all maximal 5-degenerate graphs with n vertices. (A maximal 5-degenerate graph can be constructed from a 5-clique by iteratively adding a new 5-leaf (vertex of degree 5) adjacent to 5 existing vertices.) The extremal graphs are 5th powers of paths, so the bound also applies to 5-trees. - Allan Bickle, Sep 15 2022

Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004; arXiv:0807.2538 [nlin.SI], 2008. [Set a(n)=d(n+3) on p. 8]
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231 [quant-ph], 2008. [Eq 10a, lambda=5]

Cf. A014125, A122046.

The maximum Wiener index of all maximal k-degenerate graphs for k=1..6 are given in A000292, A002623, A014125, A122046, A122047 (this sequence), A175724, respectively.

p[n_] := p[n] = Cancel[Simplify[(x^(n - 1)p[n - 1]p[n - 5] + p[n - 2]*p[n - 4])/p[n - 6]]];p[ -6] = 1; p[ -5] = 1; p[ -4] = 1; p[ -3] = 1; p[ -2] = 1; p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]

Conjectures from R. J. Mathar, Jul 15 2008: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8);
o.g.f.: x^2/((x^4+x^3+x^2+x+1)(x-1)^4). (End)
Conjecture: a(n) = (A000292(n+1) - n - 2 - (-1)^floor((n-1)/5)*A099443(n+1))/5. - R. J. Mathar, Jul 15 2008
a(n+2) = A144679(n) + A144679(n-1) + A144679(n-2) + A144679(n-3) + A144679(n-4). - Johannes W. Meijer, May 20 2011
a(n) = floor((n^3 + 6*n^2 + 5*n)/30). - Allan Bickle, Sep 15 2022

Edited by N. J. A. Sloane, Jul 15 2008

a(22)-a(43) from R. J. Mathar, Jul 15 2008

A014019 Inverse of 10th cyclotomic polynomial.

Original entry on oeis.org

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

Offset: 0

Simon Plouffe

The g.f. is the image of the g.f. of Fib(n+1) by the transform A(x) -> (1/(1 + x^2)^2)*A(x/(1+x^2)). The denominator is associated with the knots 4_1 and 5_1 by their Alexander and Jones polynomials respectively. - Paul Barry, Oct 16 2004
For n >= 1, a(n) is the determinant of an n X n Toeplitz matrix M satisfying: M(i,j) = 1 if -1 <= j - i <= 3 and 0 otherwise. - Dmitry Efimov, Jun 23 2015
Period 10: repeat [1,1,0,0,0,-1,-1,0,0,0]. - Wesley Ivan Hurt, Jun 24 2015
This is the Riordan transform of {A000045(n+1)}, n >= 0, with the Riordan matrix A049310 (Chebyshev S) of the Bell type. See the first comment by Paul Barry. - Wolfdieter Lang, Feb 18 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dror Bar-Natan, The Rolfsen Knot Table
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1).
Index to sequences related to inverse of cyclotomic polynomials

Cf. A000045, A049310, A099443, A281861.

&cat[[1,1,0,0,0,-1,-1,0,0,0]: n in [0..15]]; // Vincenzo Librandi, Apr 03 2014

with(numtheory,cyclotomic); c := n->series(1/cyclotomic(n,x),x,80);

CoefficientList[Series[1/Cyclotomic[10, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)

Vec(1/polcyclo(10)+O(x^99)) \\ Charles R Greathouse IV, Mar 24 2014

G.f.: 1/(1 - x + x^2 - x^3 + x^4). - Paul Barry, Oct 16 2004

a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4), n > 4. - Wesley Ivan Hurt, Jun 24 2015

A100048 A Chebyshev transform of the Pell numbers.

Original entry on oeis.org

0, 1, 2, 2, 4, 9, 16, 29, 56, 106, 198, 373, 704, 1325, 2494, 4698, 8848, 16661, 31376, 59089, 111276, 209554, 394634, 743177, 1399552, 2635641, 4963450, 9347186, 17602652, 33149377, 62427024, 117562789, 221394656, 416931194, 785166286

Offset: 0

Paul Barry, Oct 31 2004

A Chebyshev transform of the Pell numbers A000129: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).

This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 2, P2 = -1, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. - Peter Bala, Mar 24 2014

G. C. Greubel, 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,-1,2,-1).

Cf. A099443, A011655, A100047.

LinearRecurrence[{2,-1,2,-1},{0,1,2,2},40] (* Harvey P. Dale, Jun 07 2015 *)

my(x='x+O('x^50)); Vec(x*(1-x^2)/(1-2*x+x^2-2*x^3+x^4)) \\ G. C. Greubel, Aug 08 2017

G.f.: x(1-x^2)/(1-2x+x^2-2x^3+x^4).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - a(n-4).
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*A000129(n-2*k)/(n-k).
From Peter Bala, Mar 24 2014: (Start)
a(n) = |u(n)|^2, where {u(n)} is the Lucas-type sequence defined by the recurrence u(0) = 0, u(1) = 1 and u(n) = P*u(n-1) - u(n-2) for n >= 2, where P = 1/2*(sqrt(7) + i).
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(2))/2 and beta = (1 - sqrt(2))/2 and T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1/4; 1, 1].
The o.g.f. is the Hadamard product of the rational functions x/(1 - 1/2*(sqrt(7) + i)*x + x^2) and x/(1 - 1/2*(sqrt(7) - i)*x + x^2). (End)

A156174 Period 5: repeat [1,-1,1,-1,0].

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 06 2009

C(n) := a(n+4) appears in the formula 2*exp(2*Pi*n*i/5) = (A(n) + B(n)*phi) + (C(n) + D(n)*phi)*sqrt(2 + phi)*i, with the golden section phi, i = sqrt(-1) and A(n) = A164116(n+5), B(n) = A080891(n) and D(n) = A010891(n+3) for n >= 0. See a comment on A164116(n+5). - Wolfdieter Lang, Feb 26 2014

With offset 1 this is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = -1, y = 1, z = 1. - Michael Somos, Oct 17 2018

			G.f. = 1 - x + x^2 - x^3 + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 + ...

Arthur Gill, Linear Sequential Circuits, McGraw-Hill, 1966, Eq. (17-10).

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1).
C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

Cf. A010874, A011558 (this read mod 2), A099443, A198517.

A156174:=n->[1,-1,1,-1,0][(n mod 5)+1]: seq(A156174(n), n=0..100); # Wesley Ivan Hurt, May 31 2015

CoefficientList[Series[(1 + x^2)/(1 + x + x^2 + x^3 + x^4), {x, 0, 100}], x] (* Wesley Ivan Hurt, May 31 2015 *)
a[ n_] := { -1, 1, -1, 0, 1}[[Mod[n, 5, 1]]]; (* Michael Somos, Jun 17 2015 *)
a[ n_] := (-1)^Mod[n, 5] Sign @ Mod[n + 1, 5]; (* Michael Somos, Jun 17 2015 *)

a(n)=[1,-1,1,-1,0][n%5+1] \\ Charles R Greathouse IV, Oct 28 2011

{a(n) = (-1)^(n%5) * sign((n+1)%5)}; /* Michael Somos, Jun 17 2015 */

G.f.: (1+x^2)/(1 + x + x^2 + x^3 + x^4).
Sum_{i=0..n} a(i) = A198517(n). - Bruno Berselli, Nov 02 2011
From Wesley Ivan Hurt, May 31 2015: (Start)
a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) = 0 for n > 4.
a(n) = Sum_{i=0..3} A011558(n+2+i)*(-1)^i. (End)
Euler transform of length 5 sequence [-1, 1, 0, -1, 1]. - Michael Somos, Jun 17 2015
G.f.: (1-x)*(1-x^4)/((1-x^2)*(1-x^5)). - Michael Somos, Jun 17 2015
a(n) = -a(-2-n) = a(n+5) for all n in Z. - Michael Somos, Jun 17 2015
a(n) = (2/5) * (cos(4*(n-2)*Pi/5) + cos(2*n*Pi/5) + cos(4*n*Pi/5) - cos(2*(n-3)*Pi/5) - cos(4*(n-3)*Pi/5) - cos(2*(n-1)*Pi/5) - cos(4*(n-1)*Pi/5) - cos((2*n+1)*Pi/5)). - Wesley Ivan Hurt, Sep 26 2018
a(n) = (-1)^n * A099443(n). - Michael Somos, Oct 17 2018
a(5*n) = a(5*n + 2) = 1, a(5*n + 1) = a(5*n + 3) = -1, a(5*n + 4) = 0 for all n in Z. - Michael Somos, Nov 27 2019

A100050 A Chebyshev transform of n.

Original entry on oeis.org

0, 1, 2, 0, -4, -5, 0, 7, 8, 0, -10, -11, 0, 13, 14, 0, -16, -17, 0, 19, 20, 0, -22, -23, 0, 25, 26, 0, -28, -29, 0, 31, 32, 0, -34, -35, 0, 37, 38, 0, -40, -41, 0, 43, 44, 0, -46, -47, 0, 49, 50, 0, -52, -53, 0, 55, 56, 0, -58, -59, 0, 61, 62, 0, -64, -65, 0, 67, 68, 0, -70, -71, 0, 73, 74, 0, -76, -77, 0, 79, 80, 0, -82, -83, 0, 85, 86, 0

Offset: 0

Paul Barry, Oct 31 2004

A Chebyshev transform of x/(1-x)^2: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).

			x + 2*x^2 - 4*x^4 - 5*x^5 + 7*x^7 + 8*x^8 - 10*x^10 - 11*x^11 + 13*x^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Cf. A165202 (partial sums).

Cf. A099837, A099443, A011655, A100047, A100048, A100051, A091684.

LinearRecurrence[{2, -3, 2, -1}, {0, 1, 2, 0},50] (* G. C. Greubel, Aug 08 2017 *)

{a(n) = n * (-1)^(n\3) * sign( n%3)} /* Michael Somos, Mar 19 2011 */

{a(n) = local(A, p, e); if( abs(n)<1, 0, A = factor(abs(n)); prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p==2, -(-2)^e, (kronecker( -12, p) * p)^e))))} /* Michael Somos, Mar 19 2011 */

[lucas_number1(n,2,1)*lucas_number1(n,1,1) for n in range(0,88)] # Zerinvary Lajos, Jul 06 2008

Euler transform of length 6 sequence [ 2, -3, -2, 0, 0, 2]. - Michael Somos, Mar 19 2011
a(n) is multiplicative with a(2^e) = -(-2)^e if e>0, a(3^e) = 0^e, a(p^e) = p^e if p == 1 (mod 6), a(p^e) = (-p)^e if p == 5 (mod 6). - Michael Somos, Mar 19 2011
G.f.: x*(1 - x^2)^3 *(1 - x^3)^2 / ((1 - x)^2 *(1 - x^6)^2) = x *(1 + x)^2 *(1 - x^2) / (1 + x^3)^2. - Michael Somos, Mar 19 2011
a(3*n) = 0, a(3*n + 1) = (-1)^n * (3*n + 1), a(3*n + 2) = (-1)^n * (3*n + 2). a(-n) = a(n). - Michael Somos, Mar 19 2011
G.f.: x(1-x^2)/(1-x+x^2)^2.
a(n) = 2*a(n-1) -3*a(n-2) +2*a(n-3) -a(n-4).
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*(n-2k)/(n-k).

A163812 Expansion of (1 - x^5) * (1 - x^6) / ((1 - x) * (1 - x^10)) in powers of x.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 04 2009

			G.f. = 1 + x + x^2 + x^3 + x^4 - x^6 - x^7 - x^8 - x^9 + x^11 + x^12 + x^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1).

Cf. A099443, A163811, A163818.

a[ n_] := Boole[n == 0] + (-1)^Quotient[n, 5] Sign@Mod[n, 5]; (* Michael Somos, Jun 17 2015 *)

{a(n) = (n==0) + [0, 1, 1, 1, 1, 0, -1, -1, -1, -1][n%10 + 1]};

Euler transform of length 10 sequence [ 1, 0, 0, 0, -1, -1, 0, 0, 0, 1].
a(5*n) = 0 unless n=0.
a(n) = -a(-n) unless n=0. a(n+5) = -a(n) unless n=0 or n=-5.
G.f.: (1 + x^2 + x^4) / (1 - x + x^2 - x^3 + x^4).
a(n) = (-1)^n * A163818(n). Convolution inverse of A163811.
G.f.: A(x) = 1 / (1 - x / ( 1 + x^4 / (1 + x^2))) = 1 + x / (1 - x / (1 + x^3 / (1 + x^2 / (1 + x / (1 - x))))). - Michael Somos, Jan 03 2013
a(n) = A099443(n-1), n>0. - R. J. Mathar, Aug 05 2009

A330025 a(n) = (-1)^floor(n/5) * sign(mod(n, 5)).

Original entry on oeis.org

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

Offset: 0

Michael Somos, Nov 27 2019

This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 1, z = 1. - Michael Somos, Mar 17 2020

			G.f. = x + x^2 + x^3 + x^4 - x^6 - x^7 - x^8 - x^9 + x^11 + x^12 + ...

Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1).

Cf. A011558, A099443, A163812, A332828.

a[ n_] := (-1)^Quotient[n, 5] Sign@Mod[n, 5];

PARI
```
{a(n) = (-1)^(n\5) * sign(n%5)};
```

Euler transform of length 10 sequence [1, 0, 0, -1, -1, 0, 0, 0, 0, 1].
G.f.: x * (1 + x) * (1 + x^2) / (1 + x^5).
a(n) = A099443(n-1). a(n) = A163812(n) except n=0.
a(n) = (-1)^floor(n/5) * A011558(n) for all n in Z.
0 = a(n)*a(n+4) - a(n+1)*a(n+3) + a(n+2)^2 = a(n)*a(n+5) - a(n+1)*a(n+4) + a(n+2)*a(n+3) for all n in Z. - Michael Somos, Mar 17 2020

cp's OEIS Frontend

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A122047 Degree of the polynomial P(n,x), defined by a Somos-6 type sequence: P(n,x)=(x^(n-1)*P(n-1,x)*P(n-5,x) + P(n-2,x)*P(n-4,x))/P(n-6,x), initialized with P(n,x)=1 at n<0.

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A100048 A Chebyshev transform of the Pell numbers.

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A156174 Period 5: repeat [1,-1,1,-1,0].

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A122047 Degree of the polynomial P(n,x), defined by a Somos-6 type sequence: P(n,x)=(x^(n-1)P(n-1,x)P(n-5,x) + P(n-2,x)*P(n-4,x))/P(n-6,x), initialized with P(n,x)=1 at n<0.