A133646 -id:A133646 - 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

A133647 A133566 * A000244.

Original entry on oeis.org

1, 3, 12, 27, 108, 243, 972, 2187, 8748, 19683, 78732, 177147, 708588, 1594323, 6377292, 14348907, 57395628, 129140163, 516560652, 1162261467, 4649045868, 10460353203, 41841412812, 94143178827, 376572715308, 847288609443, 3389154437772, 7625597484987, 30502389939948

Offset: 0

Gary W. Adamson, Sep 19 2007

A133646 is a companion case.

			a(3) = 27 = 3^3.
a(4) = 108 = 4 * 3^3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,9).

Cf. A000244, A133566, A133646.

Join[{1},Flatten[NestList[9#&,{3,12},20]]] (* or *) LinearRecurrence[{0,9},{1,3,12},40] (* Harvey P. Dale, Aug 01 2019 *)

A133566 * A000244, where A133566 = an infinite lower triangular matrix and A000244 = [3^0, 3^1, 3^2, ...]. For odd n, a(n) = 3^n. For even n, a(n) = 4 * 3^(n-1).
From Philippe Deléham, Apr 06 2012: (Start)
G.f.: (1+3*x+3*x^2)/(1-9*x^2).
a(n+2) = 9*a(n), a(0) = 1, a(1) = 3, a(2) = 12. (End)
From Amiram Eldar, Jun 02 2025: (Start)
Sum_{n>=0} 1/a(n) = 47/32.
Sum_{n>=0} (-1)^n/a(n) = 23/32. (End)

More terms from Philippe Deléham, Apr 06 2012

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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