A118425 -id:A118425 - cp's OEIS frontend

A289207 a(n) = max(0, n-2).

0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 0

Jean-François Alcover and Paul Curtz, Jun 28 2017

This simple sequence is such that there is one and only one array of differences D(n,k) where the first and the second upper subdiagonal is a(n).
The rows of this array are existing sequences of the OEIS, prepended with zeros:
row 0 is A118425,
row 1 is A006478,
row 2 is A001629,
row 3 is A010049,
row 4 is A006367,
row 5 is not in the OEIS.
It can be observed that a(n) is an autosequence of the first kind whose second kind mate is A199969. In addition, the structure of the array D(n,k) shows that the first row is an autosequence.
For n = 1 to 8, rows with only one leading zero are also autosequences.

			Array of differences begin:
   0,   0,   0,   0,  0,   0,  0,  1,  4, 12, 30, 68, ...
   0,   0,   0,   0,  0,   0,  1,  3,  8, 18, 38, 76, ...
   0,   0,   0,   0,  0,   1,  2,  5, 10, 20, 38, 71, ...
   0,   0,   0,   0,  1,   1,  3,  5, 10, 18, 33, 59, ...
   0,   0,   0,   1,  0,   2,  2,  5,  8, 15, 26, 46, ...
   0,   0,   1,  -1,  2,   0,  3,  3,  7, 11, 20, 34, ...
   0,   1,  -2,   3, -2,   3,  0,  4,  4,  9, 14, 24, ...
   1,  -3,   5,  -5,  5,  -3,  4,  0,  5,  5, 10, 16, ...
  -4,   8, -10,  10, -8,   7, -4,  5,  0,  6,  6, 17, ...
  12, -18,  20, -18, 15, -11,  9, -5,  6,  0,  7,  7, ...
  ...

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

Essentially the same as A023444. Cf. A001477, A118425, A006478, A001629, A010049, A006367, A199969.

a[n_] := Max[0, n - 2];
D[n_, k_] /; k == n + 1 := a[n]; D[n_, k_] /; k == n + 2 := a[n]; D[n_, k_] /; k > n + 2 := D[n, k] = Sum[D[n + 1, j], {j, 0, k - 1}]; D[n_, k_] /; k <= n := D[n, k] = D[n - 1, k + 1] - D[n - 1, k];
Table[D[n, k], {n, 0, 11}, {k, 0, 11}]

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

A118424 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 001 (n,k>=0).

Original entry on oeis.org

1, 2, 4, 7, 1, 12, 4, 20, 12, 33, 30, 1, 54, 68, 6, 88, 144, 24, 143, 291, 77, 1, 232, 568, 216, 8, 376, 1080, 552, 40, 609, 2012, 1318, 156, 1, 986, 3688, 2988, 520, 10, 1596, 6672, 6504, 1552, 60, 2583, 11941, 13702, 4266, 275, 1, 4180, 21180, 28104, 11000

Offset: 0

Emeric Deutsch, Apr 27 2006

Row n has 1+floor(n/3) terms. Sum of entries in row n is 2^n (A000079). T(n,0)=A000071(n+3)=fibonacci(n+3)-1. T(n,1)=A118425(n). Sum(k*T(n,k),k=0..n-1)=(n-2)*2^(n-3) (A001787).

			T(7,2) = 6 because we have 0bb, 1bb, b0b, b1b, bb0 and bb1, where b=001.
Triangle starts:
1;
2;
4;
7,   1;
12,  4;
20, 12;
33, 30, 1;

Alois P. Heinz, Rows n = 0..250, flattened

Cf. A000071, A000079, A001787, A118425.

G:=1/(1-2*z+(1-t)*z^3): Gser:=simplify(series(G,z=0,20)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 17 do seq(coeff(P[n],t,j),j=0..floor(n/3)) od; # yields sequence in triangular form

nn=15;Map[Select[#,#>0&]&,CoefficientList[Series[1/(1-2z-(u-1)z^3),{z,0,nn}],{z,u}]]//Grid (* Geoffrey Critzer, Dec 03 2013 *)

G.f.: G(t,z) = 1/[1-2z+(1-t)z^3]. Recurrence relation: T(n,k) = 2T(n-1,k) -T(n-3,k) +T(n-3,k-1) for n>=3.

cp's OEIS Frontend

A289207 a(n) = max(0, n-2).

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