A133592 -id:A133592 - cp's OEIS frontend

A122950 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 3, 5, 0, 0, 0, 1, 7, 8, 0, 0, 0, 0, 4, 15, 13, 0, 0, 0, 0, 1, 12, 30, 21, 0, 0, 0, 0, 0, 5, 31, 58, 34, 0, 0, 0, 0, 0, 1, 18, 73, 109, 55, 0, 0, 0, 0, 0, 0, 6, 54, 162, 201, 89, 0, 0, 0, 0, 0, 0, 1, 25, 145, 344, 365, 144, 0, 0, 0, 0, 0, 0, 0, 7, 85, 361, 707

Offset: 0

Philippe Deléham, Oct 25 2006

Skew triangle associated with the Fibonacci numbers.

			Triangle begins:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 1, 3;
  0, 0, 0, 3, 5;
  0, 0, 0, 1, 7, 8;
  0, 0, 0, 0, 4, 15, 13;
  0, 0, 0, 0, 1, 12, 30, 21;
  0, 0, 0, 0, 0,  5, 31, 58,  34;
  0, 0, 0, 0, 0,  1, 18, 73, 109,  55;
  0, 0, 0, 0, 0,  0,  6, 54, 162, 201,  89;
  0, 0, 0, 0, 0,  0,  1, 25, 145, 344, 365, 144;
  0, 0, 0, 0, 0,  0,  0,  7,  85, 361, 707, 655, 233;

H. Fuks and J. M. G. Soto, Exponential convergence to equilibrium in cellular automata asymptotically emulating identity, arXiv preprint arXiv:1306.1189 [nlin.CG], 2013.

Cf. A055830 (another version).

T[0, 0] = T[1, 1] = 1; T[, 0] = T[, 1] = 0; T[n_, n_] := Fibonacci[n+1]; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + T[n-2, k-1] + T[n-2, k-2]; T[, ] = 0;
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2018 *)

Sum_{k=0..n} T(n,k) = A011782(n).
Sum_{n>=k} T(n,k) = A001333(k).
T(n,k) = 0 if k < 0 or if k > n, T(0,0) = 1, T(2,1) = 0, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2).
T(n,n) = Fibonacci(n+1) = A000045(n+1).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A133592(n), A133594(n), A133642(n), A133646(n), A133678(n), A133679(n), A133680(n), A133681(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Jan 03 2008
G.f.: (1-y*x^2)/(1-y*x-y*(y+1)*x^2). - Philippe Deléham, Nov 26 2011

A384711 Expansion of (1+x) / (1-2x-6x^2).

Original entry on oeis.org

1, 3, 12, 42, 156, 564, 2064, 7512, 27408, 99888, 364224, 1327776, 4840896, 17648448, 64342272, 234575232, 855204096, 3117859584, 11366943744, 41441044992, 151083752448, 550813774848, 2008130064384, 7321142777856, 26691065942016, 97308988551168

Offset: 0

Sean A. Irvine, Jun 07 2025

Number of walks of length n in the graph K_{1,1,1,2} starting at vertex 0 when the edges are given by {{0,1}, {0,3}, {0,4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}.

Also, by symmetry, the number of walks of length n starting at vertex 2 in the same graph.

			a(2)=12 because we have the walks 0-1-0, 0-1-2, 0-1-3, 0-1-4, 0-3-0, 0-3-1, 0-3-2, 0-3-4, 0-4-0, 0-4-1, 0-4-2, 0-4-3.

Sean A. Irvine, Walks on Graphs.

Cf. A133592, A384712 (vertices 1, 3, 4).

a:= n-> (<<0|1|0|1|1>, <1|0|1|1|1>, <0|1|0|1|1>, <1|1|1|0|1>, <1|1|1|1|0>>^n. <<1, 1, 1, 1, 1>>)[1, 1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+x) / (1-2*x-6*x^2), {x, 0, 32}], x]

a(n) = 3*A133592(n)/2 for n>0.

A384712 Expansion of (1+2x) / (1-2x-6*x^2).

Original entry on oeis.org

1, 4, 14, 52, 188, 688, 2504, 9136, 33296, 121408, 442592, 1613632, 5882816, 21447424, 78191744, 285068032, 1039286528, 3788981248, 13813681664, 50361250816, 183604591616, 669376688128, 2440380925952, 8897021980672, 32436329517056, 118254790918144

Offset: 0

Sean A. Irvine, Jun 07 2025

Number of walks of length n in the graph K_{1,1,1,2} starting at vertex 1 when the edges are given by {{0,1}, {0,3}, {0,4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}.

Also, by symmetry, the number of walks of length n starting at vertex 3 (or 4) in the same graph.

			a(2)=14 because we have the walks 1-0-1, 1-0-3, 1-0-4, 1-2-1, 1-2-3, 1-2-4, 1-3-0, 1-3-1, 1-3-2, 1-3-4, 1-4-0, 1-4-1, 1-4-2, 1-4-3.

Sean A. Irvine, Walks on Graphs.

Cf. A133592, A384711 (vertices 0, 2).

a:= n-> (<<0|1|0|1|1>, <1|0|1|1|1>, <0|1|0|1|1>, <1|1|1|0|1>, <1|1|1|1|0>>^n. <<1, 1, 1, 1, 1>>)[2, 1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+2*x) / (1-2*x-6*x^2), {x, 0, 32}], x]

a(n) = A133592(n+1)/2.

cp's OEIS Frontend

A122950 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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A384711 Expansion of (1+x) / (1-2x-6x^2).

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A384712 Expansion of (1+2x) / (1-2x-6*x^2).

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cp's OEIS Frontend

A122950 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 0, 1, -1, 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

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Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A384711 Expansion of (1+x) / (1-2*x-6*x^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A384712 Expansion of (1+2*x) / (1-2*x-6*x^2).

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

Formula

A384711 Expansion of (1+x) / (1-2x-6x^2).

A384712 Expansion of (1+2x) / (1-2x-6*x^2).