A076118 -id:A076118 - cp's OEIS frontend

A301674 Coordination sequence for node of type V1 in "krs" 2-D tiling (or net).

1, 4, 8, 14, 16, 26, 22, 34, 36, 38, 44, 54, 46, 62, 64, 62, 72, 82, 70, 90, 92, 86, 100, 110, 94, 118, 120, 110, 128, 138, 118, 146, 148, 134, 156, 166, 142, 174, 176, 158, 184, 194, 166, 202, 204, 182, 212, 222, 190, 230, 232, 206, 240, 250, 214, 258, 260

Offset: 0

N. J. A. Sloane, Mar 25 2018

Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 31 2023

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, bottom row, 2nd tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 6 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Reticular Chemistry Structure Resource (RCSR), The krs tiling (or net)
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
Rémy Sigrist, Illustration of first terms
Rémy Sigrist, PARI program for A301674
Index entries for linear recurrences with constant coefficients, signature (-1,0,2,2,0,-1,-1).

Cf. A301676.

Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

LinearRecurrence[{-1,0,2,2,0,-1,-1},{1,4,8,14,16,26,22,34,36},100] (* Paolo Xausa, Nov 15 2023 *)

PARI
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See Links section.
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(a) G.f. = -(2*x^8-x^7-5*x^6-18*x^5-20*x^4-20*x^3-12*x^2-5*x-1)/((x+1)*(x-1)^2*(x^2+x+1)^2). (b) Satisfies the recurrence {( - 2*n^5 - 13*n^4 - 22*n^3 + 7*n^2 + 30*n)*a(n) + ( - 2*n^5 - 13*n^4 - 25*n^3 + n^2 + 39*n)*a(n + 1) + ( - 6*n^2 + 6*n)*a(n + 2) + (2*n^5 + 7*n^4 + 7*n^3 - 7*n^2 - 9*n)*a(n + 3) + (2*n^5 + 7*n^4 + 4*n^3 - 7*n^2 - 6*n)*a(n + 4) = 0, a(0) = 1, a(1) = 4, a(2) = 8, a(3) = 14, a(4) = 16, a(5) = 26}. - N. J. A. Sloane, Mar 28 2018

Equivalent conjecture: 9*a(n) = 40*n -18*(-1)^n -6*(-1)^n*A076118(n+1) +6*A049347(n) -4*A049347(n-1). - R. J. Mathar, Apr 01 2018

More terms from Rémy Sigrist, Mar 28 2018

A099254 Self-convolution of A010892. The g.f. is 1/(Alexander polynomial of granny knot).

Original entry on oeis.org

1, 2, 1, -2, -4, -2, 3, 6, 3, -4, -8, -4, 5, 10, 5, -6, -12, -6, 7, 14, 7, -8, -16, -8, 9, 18, 9, -10, -20, -10, 11, 22, 11, -12, -24, -12, 13, 26, 13, -14, -28, -14, 15, 30, 15, -16, -32, -16, 17, 34, 17, -18, -36, -18, 19, 38, 19, -20, -40, -20, 21, 42, 21

Offset: 0

Paul Barry, Oct 08 2004

A granny knot sequence.

INVERTi transform of A077855: (1, 3, 6, 11, 20, 36, 64, 133, ...). - Gary W. Adamson, Jan 13 2017

A.H.M. Smeets, Table of n, a(n) for n = 0..20000
Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Row sums of array A128502.

Cf. A077855, A076118 (first differences).

A099254 := proc(n)
    option remember ;
    if n <= 3 then
        op(n+1,[1,2,1,-2]) ;
    else
        2*procname(n-1)-3*procname(n-2)+2*procname(n-3)-procname(n-4) ;
    end if;
end proc:
seq(A099254(n),n=0..80) ; # R. J. Mathar, Jul 08 2022

LinearRecurrence[{2, -3, 2, -1}, {1, 2, 1, -2}, 100] (* Jean-François Alcover, Sep 21 2022 *)

a0,a1,a2,a3,n = -2,1,2,1,3
print(0,a3)
print(1,a2)
print(2,a1)
print(3,a0)
while n < 20000:
    a0,a1,a2,a3,n = 2*a0-3*a1+2*a2-a3,a0,a1,a2,n+1
    print(n,a0) # A.H.M. Smeets, Sep 13 2018

def A099254(n):
    a, b = divmod(n,3)
    return (1+(b&1))*(-a-1 if a&1 else a+1) # Chai Wah Wu, Jan 31 2023

G.f.: 1/(1 - 2*x + 3*x^2 - 2*x^3 + x^4) = 1/(1 - x + x^2)^2.
a(n) = 4*sqrt(3)*sin(Pi*n/3 + Pi/3)/9 + 2*(n + 1)*sin(Pi*n/3 + Pi/6)/3.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*(n-k+1)*(-1)^k. - Paul Barry, Nov 12 2004
a(n) = 2*cos(2*Pi*(n + 2)/3)*(floor(n/3) + 1)*(-1)^(n+1). - Tani Akinari, Jul 01 2013
a(n) = (1/54)*(18*(n + 2)*(-1)^floor(n/3) + (3*n + 11)*(-1)^floor((n + 1)/3) - 9*(n + 1)*(-1)^floor((n + 2)/3) - 2*(3*n + 8)*(-1)^floor((n + 4)/3)). - John M. Campbell, Dec 23 2016
From A.H.M. Smeets, Sep 13 2018: (Start)
a(n) = 2*a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4) for n >= 4.
a(3*k) = a(3*k+2) = (-1)^k*(k + 1) for k >= 0.
a(3*k+1) = -(-1)^k*2*(k + 1) for k >= 0. (End)
Sum_{n>=0} 1/a(n) = 5*log(2)/2. - Amiram Eldar, May 10 2025

A050531 Number of multigraphs with loops on 3 nodes with n edges.

Original entry on oeis.org

1, 2, 6, 14, 28, 52, 93, 152, 242, 370, 546, 784, 1103, 1512, 2040, 2706, 3534, 4554, 5803, 7304, 9108, 11252, 13780, 16744, 20205, 24206, 28826, 34126, 40176, 47056, 54857, 63648, 73542, 84630, 97014, 110808, 126139, 143108, 161868, 182546, 205282

Offset: 0

Vladeta Jovovic, Dec 29 1999

a(n) is also the number of multigraphs (no loops allowed) on 3 nodes with n edges of two colors. - Geoffrey Critzer, Aug 10 2015

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).

Column k=3 of A290428.

Cf. A076118, A002620, A290428.

a076118:= 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):
f:= n -> ceil((-1)^n*a076118(n+1)/9+(-1)^n*n/32+(4009/4320)*n+(1/2)*n^2+(5/36)*n^3+(1/48)*n^4+(1/720)*n^5):
map(f, [$0..100]); # Robert Israel, Aug 07 2015

< 1/(1 - x^i), {i, 1, n^2 - n}], {x, 0, nn}], x] (* Geoffrey Critzer, Aug 07 2015 *)
CoefficientList[Series[(x^6 + x^4 + 2 x^3 + x^2 + 1)/((x^3 - 1)^2 (x^2 - 1)^2 (x - 1)^2), {x, 0, 45}], x] (* Vincenzo Librandi, Aug 08 2015 *)

Vec((x^6+x^4+2*x^3+x^2+1)/((x^3-1)^2*(x^2-1)^2*(x-1)^2) + O(x^40)) \\ Colin Barker, Jul 07 2019

G.f.: (x^6+x^4+2*x^3+x^2+1)/((x^3-1)^2*(x^2-1)^2*(x-1)^2).
a(n) = ceiling((-1)^n*A076118(n+1)/9+(-1)^n*n/32+(4009/4320)*n+(1/2)*n^2+(5/36)*n^3+(1/48)*n^4+(1/720)*n^5). - Robert Israel, Aug 07 2015
a(n) = (A+B+C)/6 where A = binomial(n+5,5); B = (n+2)*(n+3)*(n+4)/8 if n even, B = (n+1)*(n+3)*(n+5)/8 if n odd; C = 2*((n/3) + 1) if n divisible by 3, C = 0 if n not divisible by 3. - David Pasino, Jul 06 2019
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n>11. - Colin Barker, Jul 07 2019

A098493 Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.

Original entry on oeis.org

1, 0, -1, -1, -1, 1, -1, 1, 2, -1, 0, 3, 0, -3, 1, 1, 2, -5, -2, 4, -1, 1, -2, -7, 6, 5, -5, 1, 0, -5, 0, 15, -5, -9, 6, -1, -1, -3, 12, 9, -25, 1, 14, -7, 1, -1, 3, 15, -18, -29, 35, 7, -20, 8, -1, 0, 7, 0, -42, 14, 63, -42, -20, 27, -9, 1, 1, 4, -22, -24, 85, 14, -112, 42

Offset: 0

Ralf Stephan, Sep 12 2004

Also, coefficients of polynomials that have values in A098495 and A094954.

			Triangle begins:
   1;
   0, -1;
  -1, -1, 1;
  -1,  1, 2, -1;
   0,  3, 0, -3, 1;
  ...

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.

Columns include A010892, -A076118. Diagonals include A033999, A038608, (-1)^n*A000096. Row sums are in A057077.

Cf. A098494 (diagonal polynomials), A085478, A244419.

A098493 := proc (n, k)
add((-1)^(k+binomial(n-j+1,2))*binomial(floor((1/2)*n+(1/2)*j),j)* binomial(j,k), j = k..n);
end proc:
seq(seq(A098493(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021

T(n,k)=if(k>n||k<0||n<0,0,if(k>=n-1,(-1)^n*if(k==n,1,-k),if(n==1,0,if(k==0,T(n-1,0)-T(n-2,0),T(n-1,k)-T(n-2,k)-T(n-1,k-1)))))

T(n, k) = A098489[n(n+1)/2, k] - A098490[n(n+1)/2, k].
Recurrence: T(n, k) = T(n-1, k)-T(n-1, k-1)-T(n-2, k); T(n, k)=0 for n<0, k>n, k<0; T(n, n)=(-1)^n; T(n, n-1)=(-1)^n*(1-n).
G.f.: (1-x)/(1+(y-1)*x+x^2). [Vladeta Jovovic, Dec 14 2009]
From Peter Bala, Jul 13 2021: (Start)
Riordan array ( (1 - x)/(1 - x + x^2), -x/(1 - x + x^2) ).
T(n,k) = (-1)^k * the (n,k)-th entry of Q^(-1)*P = Sum_{j = k..n} (-1)^(k+binomial(n-j+1,2))*binomial(floor((1/2)*n+(1/2)*j),j)* binomial(j,k), where P denotes Pascal's triangle A007318 and Q denotes triangle A061554 (formed from P by sorting the rows into descending order). (End)
From Peter Bala, Jun 26 2025: (Start)
n-th row polynomial R(n, x) = Sum_{k = 0..n} (-1)^k * binomial(n+k, 2*k) * (1 + x)^k.
R(n, 2*x + 1) = (-1)^n * Dir(n, x), where Dir(n,x) denotes the n-th row polynomial of the triangle A244419.
R(n, -1 - x) = b(n, x), where b(n, x) denotes the n-th row polynomial of the triangle A085478. (End)

A166711 Permutation of the integers: two positives, one negative.

Original entry on oeis.org

0, 1, 2, -1, 3, 4, -2, 5, 6, -3, 7, 8, -4, 9, 10, -5, 11, 12, -6, 13, 14, -7, 15, 16, -8, 17, 18, -9, 19, 20, -10, 21, 22, -11, 23, 24, -12, 25, 26, -13, 27, 28, -14, 29, 30, -15, 31, 32, -16, 33, 34, -17, 35, 36, -18, 37, 38, -19, 39, 40, -20, 41, 42, -21, 43, 44, -22, 45, 46

Offset: 0

Jaume Oliver Lafont, Oct 18 2009

Setting m=2 in
log(m) = Sum_{n>0} (n mod m - (n-1) mod m)/n [1]
yields the sum
log(2) = (1 -1/2) +(1/3 -1/4) +(1/5 -1/6)+...
Substituting every -1/d by 1/d - 2/d we obtain
log(2) = (1+1/2-1)+(1/3+1/4-1/2)+(1/5+1/6-1/3)+...
a(n) is the sequence of denominators of this modified sum with unit numerators, so
Sum_{k>0} 1/a(k) = log(2)
Substituting -1/d by -2/d + 1/d would yield another permutation (one positive, one negative, one positive) with the same sum of inverses.
Similar sequences (m positives, one negative) may be obtained for the logarithm of any integer m>0. A001057 is the case m=1, with sum of inverses log(1).
Equation [1] is a result of expanding log( Sum_{0<=k<=m-1} x^k ) at x=1 (see comment to A061347.)

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

Cf. A001057, A002162, A038608. Signed and shifted version of A009947.

LinearRecurrence[{0, 0, 2, 0, 0, -1}, {0, 1, 2, -1, 3, 4}, 100] (* G. C. Greubel, May 24 2016 *)
Join[{0},With[{nn=50},Riffle[Range[nn],Range[-1,-nn/2,-1],3]]] (* Harvey P. Dale, May 15 2023 *)

PARI
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a(n)=(2*(n+1)\3)*(1-3/2*!(n%3))
```

a(n)=if(n>=0,[ -n\3, 2*(n\3)+1, 2*(n\3)+2][n%3+1]) \\ Jaume Oliver Lafont, Nov 14 2009

G.f.: (x*(1+2*x-x^2+x^3)/((1-x)^2*(1+x+x^2)^2)).
a(0)=0, a(1)=1, a(2)=2, a(3)=-1, a(4)=3, a(5)=4, a(n)=2*a(n-3)-a(n-6), n>=6.
a(n) = (n+1)/3 +2*A049347(n)/3 -(-1)^n*A076118(n+1). - R. J. Mathar, Oct 30 2009

Corrected by Jaume Oliver Lafont, Oct 22 2009

frac keyword removed by Jaume Oliver Lafont, Nov 02 2009

A028289 Expansion of (1+x^2+x^3+x^5)/((1-x)(1-x^3)(1-x^4)(1-x^6)).

Original entry on oeis.org

1, 1, 2, 4, 5, 7, 11, 13, 17, 23, 27, 33, 42, 48, 57, 69, 78, 90, 106, 118, 134, 154, 170, 190, 215, 235, 260, 290, 315, 345, 381, 411, 447, 489, 525, 567, 616, 658, 707, 763, 812, 868, 932, 988, 1052, 1124, 1188

Offset: 0

N. J. A. Sloane

C. Ahmed, P. Martin, and V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015-2019.
B. N. Cyvin et al., Enumeration of conjugated hydrocarbons: Hollow hexagons revisited, Structural Chem., 6 (1995), 85-88, equations (6) and (22).
W. C. Huffman, The biweight enumerator of self-orthogonal binary codes, Discr. Math. Vol. 26 1979, pp. 129-143.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-2,-2,1,1,1,-1).

A117373 := proc(n) op(1+(n mod 6),[1,-2,-3,-1,2,3]) ; end proc:
A076118 := proc(n) coeftayl( x*(1-x)/(1-x+x^2)^2,x=0,n) ; end proc:
A028289 := proc(n) 1/108*n^3 +1/8*n^2 +55/108*n +29/48 +1/16*(-1)^n -2*(-1)^n*A117373(n+2)/27 +(-1)^n*A076118(n+1)/9; end proc:
seq(A028289(n),n=0..20) ; # R. J. Mathar, Mar 22 2011

CoefficientList[Series[(1+x^2+x^3+x^5)/((1-x)(1-x^3)(1-x^4) (1-x^6)),{x,0,50}],x]  (* Harvey P. Dale, Apr 20 2011 *)

Vec((1+x^2+x^3+x^5)/((1-x)*(1-x^3)*(1-x^4)*(1-x^6))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

G.f.: 1 / ( (1+x)*(1+x+x^2)^2*(x-1)^4 ). - R. J. Mathar, Mar 22 2011

A122765 Triangle read by rows: Let p(k, x) = x*p(k-1, x) - p(k-2, x). Then T(k,x) = dp(k,x)/dx.

Original entry on oeis.org

1, -1, 2, -2, -2, 3, 2, -6, -3, 4, 3, 6, -12, -4, 5, -3, 12, 12, -20, -5, 6, -4, -12, 30, 20, -30, -6, 7, 4, -20, -30, 60, 30, -42, -7, 8, 5, 20, -60, -60, 105, 42, -56, -8, 9, -5, 30, 60, -140, -105, 168, 56, -72, -9, 10

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 22 2006

Based on the coefficients of derivatives of the polynomials in A130777.

			Triangle begins as:
   1;
  -1,   2;
  -2,  -2,   3;
   2,  -6,  -3,   4;
   3,   6, -12,  -4,   5;
  -3,  12,  12, -20,  -5,   6;
  -4, -12,  30,  20, -30,  -6,   7;
   4, -20, -30,  60,  30, -42,  -7,   8;
   5,  20, -60, -60, 105,  42, -56,  -8,  9;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Cf. A000217, A026741, A046854, A066170, A076118, A122766, A130777, A165202.

A122765:= func< n,k | k*(-1)^Binomial(n-k+1, 2)*Binomial(Floor((n+k)/2), k) >;
[A122765(n,k): k in [1..n], n in [1..14]]; // G. C. Greubel, Dec 30 2022

(* First program *)
p[0,x]=1; p[1,x]=x-1; p[k_,x_]:= p[k, x]= x*p[k-1,x] -p[k-2,x]; a = Table[Expand[p[n, x]], {n, 0, 10}]; Table[CoefficientList[D[a[[n]], x], x], {n, 2, 10}]//Flatten
(* Second program *)
T[n_, k_]:= k*(-1)^Binomial[n-k+1,2]*Binomial[Floor[(n+k)/2], k];
Table[T[n, k], {n,14}, {k,n}]//Flatten (* G. C. Greubel, Dec 30 2022 *)

tpol(n) = if (n<=0, 1, if (n==1, x-1, x*tpol(n-1) -tpol(n-2)));
lista(nn) = {for(n=0, nn, pol = deriv(tpol(n)); for (k=0, poldegree(pol), print1(polcoeff(pol, k), ", ");););} \\ Michel Marcus, Feb 07 2014

def A122765(n, k): return k*(-1)^binomial(n-k+1, 2)*binomial(((n+k)//2), k)
flatten( [[A122765(n,k) for k in range(1,n+1)] for n in range(1,15)] ) # G. C. Greubel, Dec 30 2022

From G. C. Greubel, Dec 30 2022: (Start)
T(n, k) = coefficient [x^k]( p(n, x) ), where p(n,x) = (2/(x^2-4))*((n+1)*chebyshev_T(n+1,x/2) -n*chebyshev_T(n,x/2) - (x/2)*(chebyshev_U(n,x/2) - chebyshev_U(n-1,x/2))).
T(n, k) = k*(-1)^binomial(n-k+1, 2)*binomial(floor((n+k)/2), k).
T(n, n) = n.
T(n, n-1) = -(n-1).
T(n, n-2) = -2*A000217(n-2).
T(n, n-3) = 2*A000217(n-3).
T(n, 1) = (-1)^binomial(n, 2)*floor((n+1)/2).
T(n, 2) = 2*(-1)^binomial(n-1, 2)*binomial(floor((n+2)/2), 2).
Sum_{k=1..n} T(n, k) = A076118(n).
Sum_{k=1..n} (-1)^k*T(n, k) = (-1)^(n-1)*A165202(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = [n=1] - [n=2].
Sum_{k=1..floor((n+1)/2)} (-1)^k*T(n-k+1, k) = (-1)^binomial(n+1, 2)*b(n), where b(n) = 4^floor(n/4)*A026741(n/2) if n is even and b(n) = 4^floor((n-1)/4)*A026741((n-1)/4) if n is odd. (End)

Name corrected and more terms from Michel Marcus, Feb 07 2014

A151842 a(3n) = n, a(3n+1) = 2n+1, a(3n+2) = n+1.

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

Offset: 0

Shane Geiger (shane.geiger(AT)gmail.com), Jul 14 2009

Take a list of numbers (like 0,1,2,3,4,5,...) and then pair them up like this: (0,1)(1,2),(2,3),(3,4)... Then sum each pair, and insert the sum between the numbers, like this: (0,1,1), (1,3,2), (2,5,3), ... Finally, remove the parentheses: 0,1,1,1,3,2,2,5,3,...
This mirrors the pattern used to make a dragon curve fractal. You take two points, then find one to insert between them. In the next iteration, you take those three points and find two numbers to insert between them. (Rather than summing the two numbers, a different function is used to find a point relative to two other points.)
a(n) is the number of rises in all compositions of n + 2 with parts in {1,2} and adjacent differences in {-1,1}. - John Tyler Rascoe, Apr 29 2025

			G.f. = x + x^2 + x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 5*x^7 + 3*x^8 + 3*x^9 + ... - _Michael Somos_, Aug 12 2009

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

See A076118 for a version with signs.

Cf. A002264, A003881, A008133.

I:=[0,1,1,1,3,2]; [n le 6 select I[n] else 2*Self(n-3)-Self(n-6): n in [1..80]]; // Vincenzo Librandi, Feb 14 2015

CoefficientList[Series[x (1 + x) (1 + x^2) / ((x - 1)^2 (1 + x + x^2)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Feb 14 2015 *)

{a(n) = kronecker(9, n) + (n\3) * [1, 2, 1][n%3 + 1]} /* Michael Somos, Aug 12 2009 */

def pairup(x): return [x[i:i+2] for i in range(len(x)-1)]
def combine(vals): return sum(vals)
def expand(L,fn): return [(x[0],fn(x),x[1]) for x in pairup(L)]
L = list(range(20))
print(expand(L,combine))

From R. J. Mathar, Jul 14 2009: (Start)
G.f.: x*(1+x)*(1+x^2)/((x-1)^2*(1+x+x^2)^2).
a(n) = 2*a(n-3) - a(n-6). (End)
From Michael Somos, Aug 12 2009: (Start)
G.f.: x * (1 - x^4) / ((1 - x) * (1 - x^3)^2).
Euler transform of length 4 sequence [ 1, 0, 2, -1]. (End)
-a(n) = a(-1-n). - Michael Somos, Nov 11 2013
From Ridouane Oudra, Nov 23 2024: (Start)
a(n) = 5*n/6 + n^2/2 - n^3/3 + (2*n^2 - n - 3/2)*floor(n/3) - (3*n + 3/2)*floor(n/3)^2.
a(n) = t(n+2)*t(n+3) - t(n)*t(n+1), where t(n) = floor(n/3) = A002264(n).
a(n) = A008133(n+2) - A008133(n). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 (A003881). - Amiram Eldar, May 10 2025

More terms from Vincenzo Librandi, Feb 14 2015

A104555 Expansion of x*(1 - x)/(1 - x + x^2)^3.

Original entry on oeis.org

0, 1, 2, 0, -5, -7, 0, 12, 15, 0, -22, -26, 0, 35, 40, 0, -51, -57, 0, 70, 77, 0, -92, -100, 0, 117, 126, 0, -145, -155, 0, 176, 187, 0, -210, -222, 0, 247, 260, 0, -287, -301, 0, 330, 345, 0, -376, -392, 0, 425, 442, 0, -477, -495

Offset: 0

Paul Barry, Mar 14 2005

Image of C(n+1,2) under the Riordan array (1, x*(1-x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,7,-6,3,-1).

Cf. A076118, A095130.

R:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( x*(1-x)/(1-x+x^2)^3 )); // G. C. Greubel, Jan 01 2023

S:=(j,n)->sum(k^j,k=1..n):seq((S(5,n+1)mod S(3,n+1))/((n+1)*(n+2))*(-1)^floor((n mod 6)/4), n=1..40). # Gary Detlefs, Oct 31 2011

CoefficientList[Series[x*(1-x)/(1-x+x^2)^3, {x,0,60}], x]  (* Harvey P. Dale, Apr 13 2011 *)

def A104555_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)/(1-x+x^2)^3 ).list()
A104555_list(60) # G. C. Greubel, Jan 01 2023

a(n) = 3*a(n-1) - 6*a(n-2) + 7*a(n-3) - 6*a(n-4) + 3*a(n-5) - a(n-6).
a(n) = Sum_{k=0..n} binomial(k, n-k)(-1)^(n-k)*k(k+1)/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*(n-k)(n-k+1)/2.
a(3*n) = 0, a(3*n-2) = n*(3*n - 1)/2, a(3*n-1) = n*(3*n + 1)/2. - Ralf Stephan, May 20 2007
a(n) = ((Sum_{k=1..n+1} k^5) mod (Sum_{k=1..n+1} k^3))/((n+1)*(n+2))*(-1)^floor((n mod 6)/4). - Gary Detlefs, Oct 31 2011

A383549 Number of rises in all compositions of n with parts in {1,2,3} and adjacent differences in {-1,1}.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 5, 3, 9, 11, 10, 24, 21, 30, 50, 43, 75, 93, 96, 161, 170, 215, 312, 323, 456, 574, 639, 906, 1046, 1276, 1710, 1935, 2501, 3135, 3642, 4760, 5699, 6893, 8823, 10401, 12952, 16079, 19104, 24002, 29097, 35165, 43865, 52628, 64503, 79363, 95329

Offset: 0

John Tyler Rascoe, Apr 29 2025

A rise is any pair of parts (p_{i-1},p_i) with p_{i-1} < p_i.

By reversal a(n) is also the number of descents in all compositions of n of this kind.

			For n = 6 the following compositions have 5 rises: (1,2,1,2), (1,2,3), (2,1,2,1), (3,2,1).

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,2,-1,0,-2,0,-1).

Cf. A011782, A076118, A124760, A151842, A173258, A214247, A220062, A238343.

A_x(N) = {my(x='x+O('x^N)); concat([0,0,0], Vec(x^3*(1 + x^2)^2*(1 + x + x^3)/(1 - x^3 - x^5)^2))}
A_x(40)

G.f.: x^3*(1 + x^2)^2*(1 + x + x^3)/(1 - x^3 - x^5)^2.

cp's OEIS Frontend

A301674 Coordination sequence for node of type V1 in "krs" 2-D tiling (or net).

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A099254 Self-convolution of A010892. The g.f. is 1/(Alexander polynomial of granny knot).

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A050531 Number of multigraphs with loops on 3 nodes with n edges.

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A098493 Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.

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A166711 Permutation of the integers: two positives, one negative.

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A028289 Expansion of (1+x^2+x^3+x^5)/((1-x)(1-x^3)(1-x^4)(1-x^6)).

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A122765 Triangle read by rows: Let p(k, x) = x*p(k-1, x) - p(k-2, x). Then T(k,x) = dp(k,x)/dx.

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