A052973 -id:A052973 - cp's OEIS frontend

A382683 Expansion of (1-x^2) / (1-x-3*x^2+x^3).

1, 1, 3, 5, 13, 25, 59, 121, 273, 577, 1275, 2733, 5981, 12905, 28115, 60849, 132289, 286721, 622739, 1350613, 2932109, 6361209, 13806923, 29958441, 65018001, 141086401, 306181963, 664423165, 1441882653, 3128970185, 6790194979, 14735222881, 31976837633

Offset: 0

Sean A. Irvine, Jun 02 2025

The number of walks of length n in the 4-vertex graph {{0,1}, {1,2}, {1,3}, {2,3}} starting at vertex 0 (see Example).

Also, a(n+1) is the number of such walks in the same graph starting at vertex 1.

			Consider walks starting at 0 in the following graph:
      2
     /|
  0-1 |
     \|
      3
The 5 walks of length 3 are 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-3-1, and 0-1-3-2.

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

Cf. A087640 (walks starting at 2).

Cf. A000079 (missing edge {0,1}), A108411 (missing edge {2,3}), A026581 (adding edge {0,2}), A000244 (K4).

a:= n-> (<<0|1|0>, <0|0|1>, <-1|3|1>>^n. <<1,1,3>>)[1,1]:
seq(a(n), n=0..32);  # Alois P. Heinz, Jun 04 2025

LinearRecurrence[{1,3,-1},{1,1,3},33] (* or *) CoefficientList[Series[ (1-x^2) / (1-x-3*x^2+x^3),{x,0,32}],x] (* James C. McMahon, Jun 02 2025 *)

a(n) = A052973(n) + A052973(n-1). a(n) = A087640(n+1) - A087640(n). - R. J. Mathar, Jun 03 2025

A087640 To obtain a(n+1), take the square of the n-th partial sum, minus the sum of the first n squared terms, then divide this difference by a(n); for all n>1, starting with a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 2, 5, 10, 23, 48, 107, 228, 501, 1078, 2353, 5086, 11067, 23972, 52087, 112936, 245225, 531946, 1154685, 2505298, 5437407, 11798616, 25605539, 55563980, 120581981, 261668382, 567850345, 1232273510, 2674156163, 5803126348

Offset: 0

Paul D. Hanna, Sep 15 2003

nonn

For n > 0, a(n) counts the number of walks of length n-1 starting at vertex A in the undirected graph with edge set {(A, B), (A, C), (B, C), (C, D)}. - Noah Niederklein, Jun 07 2025

Sean A. Irvine, Walks on Graphs
Index entries for linear recurrences with constant coefficients, signature (1,3,-1).

Cf. A052973.

CoefficientList[Series[(1-2x^2+x^3)/(1-x-3x^2+x^3),{x,0,40}],x] (* or *) LinearRecurrence[{1,3,-1},{1,1,2,5},40] (* Harvey P. Dale, Dec 06 2015 *)

{a(n) = if(n<=1,1,( sum(k=0, n-1, a(k))^2 - sum(k=0, n-1, a(k)^2) )/a(n-1))}
for(n=0,40,print1(a(n),", "))

a(n) = a(n-1) + 3a(n-2) - a(n-3) for n>3.
G.f.: (1-2x^2+x^3)/(1-x-3x^2+x^3).
G.f.: A052973(x)/(1+x-x^2).

A087956 a(n) is the square of the n-th partial sum minus the n-th partial sum of the squares, divided by a(n-1), for all n>=1, starting with a(0)=1, a(1)=3.

Original entry on oeis.org

1, 3, 2, 11, 14, 45, 76, 197, 380, 895, 1838, 4143, 8762, 19353, 41496, 90793, 195928, 426811, 923802, 2008307, 4352902, 9454021, 20504420, 44513581, 96572820, 209609143, 454814022, 987068631, 2141901554, 4648293425, 10086929456

Offset: 0

Paul D. Hanna, Sep 16 2003

			a(4) = 14 since ((1+3+2+11)^2 - (1^2+3^2+2^2+11^2))/11 = (17^2-135)/11 = 14.

Cf. A087640, A087955, A087957, A087958.

Essentially the same as A052973.

a(0)=1; a(1)=3; for(n=2,50,a(n)=((sum(k=0,n,a(k))^2-sum(k=0,n,a(k)^2))/a(n-1))

a(n) = a(n-1) + 3*a(n-2) - a(n-3) for n>3.

G.f.: (1+2*x-4*x^2+x^3)/(1-x-3*x^2+x^3).

A117316 Riordan array ((1-x)/(1-x-2x^2),x(1-x)/(1-x-2x^2)).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 2, 4, 0, 1, 6, 4, 6, 0, 1, 10, 16, 6, 8, 0, 1, 22, 28, 30, 8, 10, 0, 1, 42, 72, 54, 48, 10, 12, 0, 1, 86, 148, 158, 88, 70, 12, 14, 0, 1, 170, 336, 342, 288, 130, 96, 14, 16, 0, 1, 342, 716, 846, 648, 470, 180, 126, 16, 18, 0, 1

Offset: 0

Paul Barry, Mar 07 2006

Product of A007318 and alternating sign version of A073370. Row sums are A001333. Diagonal sums are A052973. First column is A078008. Convolution array for A078008.

			Triangle begins
1,
0, 1,
2, 0, 1,
2, 4, 0, 1,
6, 4, 6, 0, 1,
10, 16, 6, 8, 0, 1,
22, 28, 30, 8, 10, 0, 1,
42, 72, 54, 48, 10, 12, 0, 1

T(n,k)=sum{j=0..n-k,sum{i=0..k+1, C(k+1,i)C(k+j,j)C(n-i-j,n-k-i-j)(-1)^(i+j)2^(n-k-i-j)}}

T(n,k)=T(n-1,k)+T(n-1,k-1)+2*T(n-2,k)-T(n-2,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. - Philippe Deléham, Jan 04 2013

cp's OEIS Frontend

A382683 Expansion of (1-x^2) / (1-x-3*x^2+x^3).

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A087640 To obtain a(n+1), take the square of the n-th partial sum, minus the sum of the first n squared terms, then divide this difference by a(n); for all n>1, starting with a(0)=1, a(1)=1.

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A087956 a(n) is the square of the n-th partial sum minus the n-th partial sum of the squares, divided by a(n-1), for all n>=1, starting with a(0)=1, a(1)=3.

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PARI

Formula

A117316 Riordan array ((1-x)/(1-x-2x^2),x(1-x)/(1-x-2x^2)).

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