A099254 -id:A099254 - cp's OEIS frontend

A049347 Period 3: repeat [1, -1, 0].

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

Offset: 0

Wolfdieter Lang

G.f. 1/cyclotomic(3, x) (the third cyclotomic polynomial).
Self-convolution yields (-1)^n*A099254(n). - R. J. Mathar, Apr 06 2008
Hankel transform of A099324. - Paul Barry, Aug 10 2009
A057083(n) = p(-1)  where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0..n. - Michael Somos, Apr 29 2012
a(n) appears, together with b(n) = A099837(n+3) in the formula 2*exp(2*Pi*n*I/3) = b(n) + a(n)*sqrt(3)*I, n >= 0, with I = sqrt(-1). See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014
The binomial transform is 1, 0, -1, -1, 0, 1, 1, 0, -1, -1.. (see A010891). The inverse binom. transform is 1, -2, 3, -3, 0, 9, -27, 54, -81.. (see A057682). - R. J. Mathar, Feb 25 2023

			G.f. = 1 - x + x^3 - x^4 + x^6 - x^7 + x^9 - x^10 + x^12 - x^13 + x^15 + ...

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 175.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012.
Elena Barcucci, Antonio Bernini, Stefano Bilotta and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (-1,-1).
Index entries for sequences related to Chebyshev polynomials.
Index to sequences related to inverse of cyclotomic polynomials

Cf. A001006, A010892, A057078, A057083, A102283, A106510, A130716, A143818, A194770.

Alternating row sums of A049310 (Chebyshev-S). [Wolfdieter Lang, Nov 04 2011]

&cat[[1,-1,0]: n in [0..90]]; // Vincenzo Librandi, Apr 03 2014

A049347 := proc(n)
    op(modp(n,3)+1,[1,-1,0]) ;
end proc:
seq(A049347(n),n=0..100) ; # R. J. Mathar, Aug 06 2016

Flatten[Table[{1, -1, 0}, {27}]] (* Alonso del Arte, Feb 07 2013 *)
CoefficientList[Series[1/Cyclotomic[3, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)
LinearRecurrence[{-1, -1}, {1, -1}, 90] (* Ray Chandler, Sep 15 2015 *)
Table[DirichletCharacter[3, 2, n + 1], {n, 0, 29}] (* Steven Foster Clark, May 29 2019 *)
Table[Mod[n + 2, 3] - 1, {n, 0, 20}] (* Michael Somos, Sep 24 2019 *)
Table[ChebyshevU[n, -1/2], {n, 0, 20}] (* Eric W. Weisstein, Jan 09 2024 *)
ChebyshevU[Range[0, 20], -1/2] (* Eric W. Weisstein, Jan 09 2024 *)

A049347(n) := block(
        [1,-1,0][1+mod(n,3)]
)$ /* R. J. Mathar, Mar 19 2012 */

{a(n) = n++; kronecker( -3, n)} /* Michael Somos, Oct 03 2006 */

{a(n) = [1, -1, 0][n%3 + 1]} /* Michael Somos, Oct 15 2008 */

a(n)=(n+2)%3-1 /* Jaume Oliver Lafont, Mar 24 2009 */

def A049347():
    x, y = 1, -1
    while True:
        yield x
        x, y = y, -x - y
a = A049347(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013

G.f.: 1/(1+x+x^2).
a(n) = +1 if n mod 3 = 0, a(n) = -1 if n mod 3 = 1, else 0.
a(n) = S(n, -1) = U(n, -1/2) (Chebyshev's U(n, x) polynomials.)
a(n) = 2*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/3. - Paul Barry, Mar 15 2004
a(n) = Sum_{k >= 0} (-1)^(n-k)*C(n-k, k).
Given g.f. A(x), then B(x) = x * A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v + 2*u*v. - Michael Somos, Oct 03 2006
Euler transform of length 3 sequence [-1, 0, 1]. - Michael Somos, Oct 03 2006
a(n) = b(n+1) where b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = 1 if p == 1 (mod 3), b(p^e) = (-1)^e if p == 2 (mod 3). - Michael Somos, Oct 03 2006
From Michael Somos, Oct 03 2006: (Start)
G.f.: (1 - x) /(1 - x^3).
a(n) = -a(1-n) = -a(n-1) - a(n-2) = a(n-3). (End)
From Michael Somos, Apr 29 2012: (Start)
G.f.: 1 / (1 + x / ( 1 - x / (1 + x))).
a(n) = (-1)^n * A010892(n).
a(n) * n! = A194770(n+1).
Revert transform of A001006. Convolution inverse of A130716. MOBIUS transform of A002324. EULER transform is A111317. BIN1 transform of itself. STIRLING transform is A143818(n+2). (End)
a(-n) = A057078(n). a(n) = A106510(n+1) unless n=0. - Michael Somos, Oct 15 2008
G.f. A(x) = 1/(1+x+x^2) = Q(0); Q(k) =  1- x/(1 - x^2/(x^2 - 1 + x/(x - 1 + x^2/(x^2 - 1/Q(k+1))))); (continued fraction 3 kind, 5-step ). - Sergei N. Gladkovskii, Jun 19 2012
a(n) = -1 + floor(67/333*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = -1 + floor(19/26*3^(n+1)) mod 3. - Hieronymus Fischer, Jan 03 2013
a(n) = ceiling((n-1)/3) - ceiling(n/3) + floor(n/3) - floor((n-1)/3). - Wesley Ivan Hurt, Dec 06 2013
a(n) = n + 1 - 3*floor((n+2)/3). - Mircea Merca, Feb 04 2014
a(n) = A102283(n+1) for all n in Z. - Michael Somos, Sep 24 2019
E.g.f.: exp(-x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Oct 26 2022

Edited by Charles R Greathouse IV, Mar 23 2010

A101950 Product of A049310 and A007318 as lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, -1, 1, 3, 1, -1, -2, 3, 4, 1, 0, -4, -2, 6, 5, 1, 1, -2, -9, 0, 10, 6, 1, 1, 3, -9, -15, 5, 15, 7, 1, 0, 6, 3, -24, -20, 14, 21, 8, 1, -1, 3, 18, -6, -49, -21, 28, 28, 9, 1, -1, -4, 18, 36, -35, -84, -14, 48, 36, 10, 1, 0, -8, -4, 60, 50, -98, -126, 6, 75, 45, 11, 1, 1, -4, -30, 20, 145, 36, -210

Offset: 0

Paul Barry, Dec 22 2004

A Chebyshev and Pascal product.
Row sums are n+1, diagonal sums the constant sequence 1 resp. A023434(n+1). Riordan array (1/(1-x+x^2),x/(1-x+x^2)).
Apart from signs, identical with A104562.
Subtriangle of the triangle given by [0,1,-1,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 27 2010
The Fi1 and Fi2 sums lead to A004525 and the Gi1 sums lead to A077889, see A180662 for the definitions of these triangle sums. - Johannes W. Meijer, Aug 06 2011
Also the convolution triangle of the inverse of 6th cyclotomic polynomial A010892. - Peter Luschny, Oct 08 2022

			Triangle begins:
   1,
   1, 1,
   0, 2, 1,
  -1, 1, 3, 1,
  -1,-2, 3, 4, 1,
  ...
Triangle [0,1,-1,1,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] begins : 1 ; 0,1 ; 0,1,1 ; 0,0,2,1 ; 0,-1,1,3,1 ; 0,-1,-2,3,4,1 ; ... - _Philippe Deléham_, Jan 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1325
Jerry Ray Dias, Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum - dendralene and radialene hydrocarbons, Croatica Chem. Acta, 77 (2004), 325-330. [p. 328].
Jonathan L. Gross, Toufik Mansour, Thomas W. Tucker, and David G. L. Wang, Root geometry of polynomial sequences. II: Type (1,0), J. Math. Anal. Appl. 441, No. 2, 499-528 (2016).

Cf. A049310, A007318, A104562, A010892, A084938.

A101950 := proc(n,k) local j,k1: add((-1)^((n-j)/2)*binomial((n+j)/2,j)*(1+(-1)^(n+j))* binomial(j,k)/2, j=0..n) end: seq(seq(A101950(n,k),k=0..n), n=0..11); # Johannes W. Meijer, Aug 06 2011
# Uses function PMatrix from A357368. Adds a row on top and a column to the left.
PMatrix(10, n -> [0, 1, 1, 0, -1,-1][irem(n, 6) + 1]); # Peter Luschny, Oct 08 2022

T[0, 0] = 1; T[n_, k_] /; k>n || k<0 = 0; T[n_, k_] := T[n, k] = T[n-1, k-1]+T[n-1, k]-T[n-2, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2014, after Philippe Deléham *)

T(n, k) = Sum_{j=0..n} (-1)^((n-j)/2)*C((n+j)/2,j)*(1+(-1)^(n+j))*C(j,k)/2.
T(0,0) = 1, T(n,k) = 0,if k>n or if k<0, T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k). - Philippe Deléham, Jan 26 2010
p(n,x) = (x+1)*p(n-1,x)-p(n-2,x) with p(0,x) = 1 and p(1,x) = x+1 [Dias].
G.f.: 1/(1-x-x^2-y*x). - Philippe Deléham, Feb 10 2012
T(n,0) = A010892(n), T(n+1,1) = A099254(n), T(n+2,2) = A128504(n). - Philippe Deléham, Mar 07 2014
T(n,k) = C(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], 4) for n>=1. - Peter Luschny, Apr 25 2016

Typo in formula corrected and information added by Johannes W. Meijer, Aug 06 2011

A076118 a(n) = Sum_{k=n/2..n} k * (-1)^(n-k) * C(k,n-k).

Original entry on oeis.org

0, 1, 1, -1, -3, -2, 2, 5, 3, -3, -7, -4, 4, 9, 5, -5, -11, -6, 6, 13, 7, -7, -15, -8, 8, 17, 9, -9, -19, -10, 10, 21, 11, -11, -23, -12, 12, 25, 13, -13, -27, -14, 14, 29, 15, -15, -31, -16, 16, 33, 17, -17, -35, -18, 18, 37, 19, -19, -39, -20, 20, 41, 21, -21, -43, -22, 22, 45, 23, -23, -47, -24, 24, 49, 25, -25, -51, -26, 26

Offset: 0

Henry Bottomley, Oct 31 2002

Piecewise linear depending on residue modulo 6. Might be described as an inverse Catalan transform of the nonnegative integers.

Number of compositions of n consisting of at most two parts, all congruent to {0,2} mod 3 (offset 1). - Vladeta Jovovic, Mar 10 2005

			a(10) = -5*1 + 6*15 - 7*35 + 8*28 - 9*9 + 10*1 = -5 + 90 -245 + 224 - 81 + 10 = -7.

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Cf. A003881, A038608, A078028, A099254 (partial sums).

See A151842 for a version without signs.

A076118:=n->add(k*(-1)^(n-k)*binomial(k,n-k), k=floor(n/2)..n); seq(A076118(n), n=0..50); # Wesley Ivan Hurt, May 08 2014
f:= gfun:-rectoproc({a(n+4) = 2*a(n+3)-3*a(n+2)+2*a(n+1)-a(n), a(0)=0,a(1)=1,a(2)=1,a(3)=-1}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Aug 07 2015

Table[Sum[k*(-1)^(n - k)*Binomial[k, n - k], {k, Floor[n/2], n}], {n,
0, 50}] (* Wesley Ivan Hurt, May 08 2014 *)

{a(n)=local(k=n%3); n=n\3; (-1)^n*((k>0)+n+(k==1)*n)} /* Michael Somos, Jul 14 2006 */

{a(n)=if(n<0, n=-1-n); polcoeff(x*(1-x)/(1-x+x^2)^2+x*O(x^n),n)} /* Michael Somos, Jul 14 2006 */

a(3n) = -a(3n-1) = A038608(n).
a(n) = ( 2n*sin((n+1/2)*Pi/3) + sin(n*Pi/3)/sin(Pi/3) )/3.
a(3n) = n*(-1)^n; a(3n+1) = (2n+1)*(-1)^n; a(3n+2) = (n+1)*(-1)^n.
a(n) = Sum{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*(n-k). - Paul Barry, Nov 12 2004
From Michael Somos, Jul 14 2006: (Start)
Euler transform of length 6 sequence [ 1, -2, -2, 0, 0, 2].
G.f.: x(1-x)/(1-x+x^2)^2 = x*(1-x^2)^2*(1-x^3)^2/((1-x)*(1-x^6)^2).
a(-1-n)=a(n). (End)
a(n+4) = 2*a(n+3)-3*a(n+2)+2*a(n+1)-a(n). - Robert Israel, Aug 07 2015
a(n) = A099254(n-1)-A099254(n-2). - R. J. Mathar, Apr 01 2018
Sum_{n>=1} 1/a(n) = Pi/4 (A003881). - Amiram Eldar, May 10 2025

A003082 Number of multigraphs with 4 nodes and n edges.

Original entry on oeis.org

1, 1, 3, 6, 11, 18, 32, 48, 75, 111, 160, 224, 313, 420, 562, 738, 956, 1221, 1550, 1936, 2405, 2958, 3609, 4368, 5260, 6279, 7462, 8814, 10356, 12104, 14093, 16320, 18834, 21645, 24783, 28272, 32158, 36442, 41187, 46410, 52151, 58443, 65345, 72864

Offset: 0

N. J. A. Sloane

Also, expansion of Molien series for representation Sym^2(R^n) of the automorphism group of the lattice D_3.

CRC Handbook of Combinatorial Designs, 1996, p. 650.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.19).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Axel Kleinschmidt and Valentin Verschinin, Tetrahedral modular graph functions, arXiv:1706.01889 [hep-th], 2017, p. 20.
P. Sarnak and A. Strömbergsson, Minima of Epstein's zeta function and heights of flat tori, Inventiones mathematicae, July 2006, Volume 165, Issue 1, pp 115-151.
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-2,-2,3,0,3,-2,-2,0,0,2,-1).

Cf. A001399, A014395 (5 nodes), A014396, A014397, A014398, row 4 of A192517.

Cf. A290778 (connected).

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x+x^2+x^4+x^6-x^7+x^8)/((1-x)^6*(1+x)^2*(1+x^2)*(1+x+x^2)^2) )); // G. C. Greubel, Nov 04 2022

CoefficientList[Series[PairGroupIndex[SymmetricGroup[4], s] /.Table[s[i] -> 1/(1 - x^i), {i, 1, 4}], {x, 0, 40}], x] (* Geoffrey Critzer, Nov 10 2011 *)
LinearRecurrence[{2,0,0,-2,-2,3,0,3,-2,-2,0,0,2,-1},{1,1,3,6,11,18,32,48,75,111, 160,224,313,420},50] (* Harvey P. Dale, Oct 09 2016 *)

Vec((x^8-x^7+x^6+x^4+x^2-x+1)/((x-1)^6*(x+1)^2*(x^2+1)*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Apr 02 2015

def A003082_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x+x^2+x^4+x^6-x^7+x^8)/((1-x)^6*(1+x)^2*(1+x^2)*(1+x+x^2)^2) ).list()
A003082_list(50) # G. C. Greubel, Nov 04 2022

G.f.: (1-x+x^2+x^4+x^6-x^7+x^8)/((1-x)^6*(1+x)^2*(1+x^2)*(1+x+x^2)^2).
a(n) = 2*a(n-1) - 2*a(n-4) - 2*a(n-5) + 3*a(n-6) + 3*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-13) - a(n-14). - Wesley Ivan Hurt, Apr 20 2021
a(n) = (1/17280)*((3 + n)*(3175 + 2088*n + 564*n^2 + 72*n^3 + 6*n^4 + 945*(-1)^n) + 540*I^n*(1 + (-1)^n)) + (1/27)*(3*ChebyshevU(n, -1/2) + 2*ChebyshevU(n-1, -1/2) + 3*(-1)^n*(A099254(n) - A099254(n-1))). - G. C. Greubel, Nov 04 2022

Entry improved by comments from Vladeta Jovovic, Dec 23 1999

A077855 Expansion of 1/((1-2x+x^2-x^3)(1-x)).

Original entry on oeis.org

1, 3, 6, 11, 20, 36, 64, 113, 199, 350, 615, 1080, 1896, 3328, 5841, 10251, 17990, 31571, 55404, 97228, 170624, 299425, 525455, 922110, 1618191, 2839728, 4983376, 8745216, 15346785, 26931731, 47261894, 82938843, 145547524, 255418100, 448227520, 786584465

Offset: 0

N. J. A. Sloane, Nov 17 2002

a(n) is the number of binary words of length n+2 such that there is at least one run of 0's and every run of 0's is of length >=2. a(1)=3 because we have: {0,0,0}, {0,0,1}, {1,0,0}. - Geoffrey Critzer, Jan 12 2013
INVERT transform of A099254: (1, 2, 1, -2, -4, -2, 3, 6, 3, ...). - Gary W. Adamson, Jan 11 2017
a(n) is the number of nonempty subsets A of {1, 2, ..., n+1} with the property that every element in A has at least one consecutive neighbor also in A. That is, for every element x in A, either x-1 is in A or x+1 is in A. - MingKun Yue, Mar 07 2025

Seiichi Manyama, Table of n, a(n) for n = 0..4092
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-1).

Cf. A018918, A099254, A005314 (first differences).

nn=40; a=x^2/(1-x); Drop[CoefficientList[Series[(a+1)/(1-x a/(1-x))/(1-x)-1/(1-x), {x,0,nn}], x], 2] (* Geoffrey Critzer, Jan 12 2013 *)
LinearRecurrence[{3, -3, 2, -1}, {1, 3, 6, 11}, 36] (* or *)
CoefficientList[ Series[1/(x^4 - 2 x^3 + 3 x^2 - 3 x + 1), {x, 0, 35}], x] (* Robert G. Wilson v, Nov 25 2016 *)

Vec((1-x)^(-1)/(1-2*x+x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

G.f.: 1/((1-2*x+x^2-x^3)*(1-x)).
a(n) = A005251(n+4) - 1. a(n+1) - a(n) = A005314(n+2). - R. J. Mathar, Sep 19 2008
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4). - Seiichi Manyama, Nov 25 2016
a(n) = Sum_{i=1..(n+3)} binomial((n+3)-i, (n+3)-3*i). - Wesley Ivan Hurt, Jul 07 2020
a(n) ~ (48 - 11*r + 29*r^2) / (23 * r^n), where r = 0.569840290998... is the root of the equation r*(2 - r + r^2) = 1. - Vaclav Kotesovec, Apr 15 2024
From MingKun Yue, Mar 07 2025: (Start)
a(n) = 2*a(n-1) - a(n-2) + a(n-3) + 1.
a(n) = a(n-1) + Sum_{i=1..(n-3)} a(i) + n. (End)

A058031 a(n) = n^4 - 2n^3 + 3n^2 - 2*n + 1, the Alexander polynomial for reef and granny knots.

Original entry on oeis.org

1, 1, 9, 49, 169, 441, 961, 1849, 3249, 5329, 8281, 12321, 17689, 24649, 33489, 44521, 58081, 74529, 94249, 117649, 145161, 177241, 214369, 257049, 305809, 361201, 423801, 494209, 573049, 660969, 758641, 866761, 986049, 1117249, 1261129, 1418481, 1590121

Offset: 0

Jason Earls, Nov 21 2000

"The standard knot invariant, in the pre-Jones era of knot theory, was the Alexander polynomial, invented in 1926. This assigns to each knot a polynomial in a variable t, which can be calculated by following a standard procedure." See Courant and Robbins, p. 503.

First differences are in A105374. - Wesley Ivan Hurt, Apr 18 2016

Richard Courant and Herbert Robbins, What Is Mathematics?, 2nd Ed. 1996, pp. 501-505.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A002061, A002378, A002522, A062938, A099254, A105374.

[(n^2 - n + 1)^2 : n in [0..50]]; // Wesley Ivan Hurt, Jun 19 2014

A058031:=n->(n^2 - n + 1)^2; seq(A058031(n), n=0..50); # Wesley Ivan Hurt, Jun 19 2014

Table[(n^2 - n + 1)^2, {n, 0, 50}] (* Wesley Ivan Hurt, Jun 19 2014 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 1, 9, 49, 169}, 50] (* Vincenzo Librandi, Apr 11 2017 *)

a(n)=n^4-2*n^3+3*n^2-2*n+1 \\ Charles R Greathouse IV, Jun 19 2014

lista(nn) = for(n=0, nn, print1((n^2-n+1)^2, ", ")); \\ Altug Alkan, Apr 16 2016

def a(n): return n**4 - 2*n**3 + 3*n**2 - 2*n + 1 # Indranil Ghosh, Apr 06 2017

G.f.: (1-4*x+14*x^2+4*x^3+9*x^4)/(1-x)^5. - Colin Barker, Jan 17 2012
a(n) = (n^2-n+1)^2. - Carmine Suriano, Feb 16 2012
3*a(n+3) = A062938(n) + A062938(n+1) + A062938(n+2). - Bruno Berselli, Feb 16 2012
a(n) = (n-2)*(n-1)*n*(n+1) + (2*n-1)^2. - Charlie Marion, Apr 11 2013
a(n) = A002061(n)^2. - Richard R. Forberg, Sep 03 2013
a(n) = (n*(n-1))^2 + (n-1)^2 + n^2, sum of three squares. - Carmine Suriano, Jun 16 2014
a(n) = A002378(A002378(n-1))+A002378(n-1)+1, where A002378(-1)=0. [Bruno Berselli, May 28 2015]
E.g.f.: exp(x)*(1 + 4*x^2 + 4*x^3 + x^4). - Ilya Gutkovskiy, Apr 16 2016
a(n) = (n-1)^4 + 2*(n-1)^3 + 3*(n-1)^2 + 2*(n-1) + 1. - Bruce J. Nicholson, Apr 07 2017
For n>0  a(n) = A002522(n)*A002522(n-1) - 1. - Bruce J. Nicholson, Jul 02 2017

Name corrected by Andrey Zabolotskiy, Nov 21 2017

A128504 Row sums of array A128503 (second convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials).

Original entry on oeis.org

1, 3, 3, -2, -9, -9, 3, 18, 18, -4, -30, -30, 5, 45, 45, -6, -63, -63, 7, 84, 84, -8, -108, -108, 9, 135, 135, -10, -165, -165, 11, 198, 198, -12, -234, -234, 13, 273, 273, -14, -315, -315, 15, 360, 360, -16, -408, -408, 17, 459, 459

Offset: 0

Wolfdieter Lang Apr 04 2007

Second convolution of A010892.
Convolution of A099254 with A010892.
a(n) equals the coefficient of x^2 of the characteristic polynomial of the (n+2)X(n+2) tridiagonal matrix with 1's along the main diagonal, the superdiagonal, and the subdiagonal (see Mathematica code below). [John M. Campbell, Jul 10 2011]

Tewodros Amdeberhan, George E. Andrews, and Roberto Tauraso, Further study on MacMahon-type sums of divisors, arXiv:2409.20400 [math.NT], 2024. See p. 18.

Cf. A010892, A099254, A128503.

Table[Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2] + KroneckerDelta[#1, #2 - 1] + KroneckerDelta[#1, #2 + 1] &, {n + 2, n + 2}], x], x^2], {n, 0, 70}] (* John M. Campbell, Jul 10 2011 *)

Vec(1/(1-x+x^2)^3+O(x^66)) \\ Joerg Arndt, Jul 02 2013

a(n) = Sum_{m=0..floor(n/2)} A128503(n,m).
G.f.: 1/(1-x+x^2)^3.
a(n) = (floor(n/3)+1)*(floor(n/3)-floor((n-1)/3)+(3/2)*(floor(n/3)+2)*(3*floor((n+1)/3)-n))*(-1)^n. - Tani Akinari, Jul 03 2013

A123585 Triangle T(n,k), 0<=k<=n, given by [1, -1, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 0, 2, 2, -1, 1, 5, 3, -1, -2, 4, 10, 5, 0, -4, -4, 12, 20, 8, 1, -2, -13, -4, 31, 38, 13, 1, 3, -11, -33, 3, 73, 71, 21, 0, 6, 6, -42, -74, 34, 162, 130, 34, -1, 3, 24, 0, -130, -146, 128, 344, 235, 55, -1, -4, 21, 72, -50, -352

Offset: 0

Philippe Deléham, Nov 13 2006

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

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A010892, A099254, A000045, A000079 (row sums), A001629, A129707.

CoefficientList[CoefficientList[Series[1/(1 - (1 + y)*x + (1 - y^2)*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 16 2017 *)

Sum_{k,0<=k<=n} T(n,k) = 2^n = A000079(n).
T(n,0) = A010892(n).
T(n,n) = Fibonacci(n+1) = A000045(n+1).
T(n+1,1) = A099254(n).
T(n+1,n) = A001629(n+2).
Sum_{k, 0<=k<=[n/2]} T(n-k,k) = A003269(n).
T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-2) - T(n-2,k), n>0.
Sum_{k, 0<=k<=n} x^k*T(n,k) = (-1)^n*A003683(n+1), (-1)^n*A006130(n), A000007(n), A010892(n), A000079(n), A030195(n+1) for x=-3, -2, -1, 0, 1, 2 respectively . - Philippe Deléham, Dec 01 2006
T(n+2,n) = A129707(n+1).- Philippe Deléham, Dec 18 2011
G.f.: 1/(1-(1+y)*x+(1-y^2)*x^2). - Philippe Deléham, Dec 18 2011

A165202 Expansion of (1+x)/(1 - x + x^2)^2.

Original entry on oeis.org

1, 3, 3, -1, -6, -6, 1, 9, 9, -1, -12, -12, 1, 15, 15, -1, -18, -18, 1, 21, 21, -1, -24, -24, 1, 27, 27, -1, -30, -30, 1, 33, 33, -1, -36, -36, 1, 39, 39, -1, -42, -42, 1, 45, 45, -1, -48, -48, 1, 51, 51, -1, -54, -54, 1, 57, 57, -1, -60, -60, 1

Offset: 0

Paul Barry, Sep 07 2009

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

Cf. A100050 (first differences).

Hankel transform of A165201.

a:=[1,3,3,-1];; for n in [5..70] do a[n]:=2*a[n-1]-3*a[n-2]+ 2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jul 18 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x)/(1-x+x^2)^2 )); // G. C. Greubel, Jul 18 2019

LinearRecurrence[{2,-3,2,-1}, {1,3,3,-1}, 70] (* G. C. Greubel, Jul 18 2019 *)
(-1)^Quotient[#-1,3]{1,1+#,#}[[Mod[#,3,1]]]&/@Range[0, 10] (* Federico Provvedi, Jul 18 2021 *)

my(x='x+O('x^70)); Vec((1+x)/(1-x+x^2)^2) \\ G. C. Greubel, Jul 18 2019

((1+x)/(1-x+x^2)^2).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 18 2019

a(n) = cos(Pi*n/3) + sin(Pi*n/3)*(2n/3 + 1)*sqrt(3).

a(n) = A099254(n) + A099254(n-1). - R. J. Mathar, May 02 2013

A380853 Number of ways to place six distinct positive integers on a triangle, three on the corners and three on the sides such that the sum of the three values on each side is n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 13, 14, 25, 37, 47, 58, 89, 98, 126, 159, 188, 219, 276, 303, 362, 423, 478, 536, 633, 688, 781, 881, 973, 1068, 1211, 1301, 1443, 1589, 1724, 1866, 2066, 2202, 2396, 2598, 2790, 2986, 3250, 3439, 3699, 3967, 4219, 4480, 4819, 5071

Offset: 1

Derek Holton and Alex Holton, Feb 06 2025

Solutions differing by only rotation or reflections are not counted separately.

If the numbers do not need to be distinct and rotations and reflections are counted separately we get A019298(n-2). If the numbers do not need to be distinct but rotations and reflections do not count separately we get A006918(n-2). If the six numbers must be distinct and reflections and rotations count separately we get 6*a(n). - R. J. Mathar, Feb 27 2025

			The a(9) = 1 solution is:
       1
     5   6
   3   4   2

R. J. Mathar, Illustrations - Examples (2025)
Index entries for linear recurrences with constant coefficients, signature (-1,0,2,3,2,0,-3,-4,-3,0,2,3,2,0,-1,-1).

Cf. A342467, A380105.

A380853 := proc(n)
    -3412+2353*n+30*n^3-480*n^2+(135*n-900)*(-1)^n ;
    %-144*(2*A010891(n)+A010891(n-1)+2*A010891(n-2)) ;
    %-160*(17*A049347(n)+8*A049347(n-1)) ;
    %-360*A057077(n) ;
    %+480*(-1)^n*(A099254(n)-A099254(n-1)) ;
    %/720 ;
end proc:
seq(A380853(n),n=1..40) ; # R. J. Mathar, Feb 27 2025

a(n) = my(c=0, t); for(x=3, n-5, t=n-x; for(y=2, min(x-1, t-1), for(z=1, y-1, if(#Set([x, y, z, t-y, t-z, n-y-z])==6, c++)))); c; \\ Jinyuan Wang, Feb 07 2025

G.f.: x^9*(1 + 4*x + 8*x^2 + 16*x^3 + 18*x^4 + 18*x^5 + 15*x^6 + 10*x^7)/((1 - x)^4*(1 + 2*x + 2*x^2 + x^3)^2*(1 + x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6)). - Stefano Spezia, Feb 08 2025

A380105(n) = a(n)-a(n-3). - R. J. Mathar, Mar 13 2025

cp's OEIS Frontend

A049347 Period 3: repeat [1, -1, 0].

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A101950 Product of A049310 and A007318 as lower triangular matrices.

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A076118 a(n) = Sum_{k=n/2..n} k * (-1)^(n-k) * C(k,n-k).

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A003082 Number of multigraphs with 4 nodes and n edges.

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A077855 Expansion of 1/((1-2*x+x^2-x^3)*(1-x)).

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A058031 a(n) = n^4 - 2*n^3 + 3*n^2 - 2*n + 1, the Alexander polynomial for reef and granny knots.

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A077855 Expansion of 1/((1-2x+x^2-x^3)(1-x)).

A058031 a(n) = n^4 - 2n^3 + 3n^2 - 2*n + 1, the Alexander polynomial for reef and granny knots.