A122761 Triangular read by rows: T(n, k) = 3^k * (1 + (n mod 2)).

1, 2, 6, 1, 3, 9, 2, 6, 18, 54, 1, 3, 9, 27, 81, 2, 6, 18, 54, 162, 486, 1, 3, 9, 27, 81, 243, 729, 2, 6, 18, 54, 162, 486, 1458, 4374, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049

Offset

0,2

References

Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, Dover, New York, 1978, pp. 57-58.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = 3^k * (1 + (n mod 2)).

From G. C. Greubel, Dec 30 2022: (Start)

T(n, k) = 3^k*(3 - (-1)^n)/2.

T(n, 0) = (3 - (-1)^n)/2.

T(n, n) = 3^n*(3 - (-1)^n)/2.

T(2*n, n) = A000244(n).

T(2*n+1, n+1) = 6*T(2*n, n).

T(2*n+1, n) = 2*T(2*n, n).

T(2*n+1, n-1) = 6*T(2*n, n).

Sum_{k=0..n} T(n, k) = (1/4)*(3 - (-1)^n)*(3^(n+1) - 1).

Sum_{k=0..n} (-1)^k*T(n, k) = (1/8)*(3 - (-1)^n)*(1 + (-1)^n*3^(n+1)).

Sum_{k=0..floor(n/2)} T(n-k, k) = (1/8)*(3*(6 - (-1)^binomial(n+1, 2))*3^floor(n/2) - (6 + (-1)^n)). (End)

Example

Triangle begins as:
  1;
  2, 6;
  1, 3,  9;
  2, 6, 18, 54;
  1, 3,  9, 27,  81;
  2, 6, 18, 54, 162, 486;
  1, 3,  9, 27,  81, 243, 729;

Mathematica

Table[3^k*(1+Mod[n, 2]), {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(Magma) [3^k*(3-(-1)^n)/2: k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 30 2022
(SageMath)
def A122761(n, k): return 3^k*(3-(-1)^n)/2
flatten([[A122761(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 30 2022

Crossrefs

Cf. A000244.

Sequence in context: A121601 A369346 A355929 * A100469 A360857 A124320

Adjacent sequences: A122758 A122759 A122760 * A122762 A122763 A122764

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Sep 21 2006

Extensions

Name and formula corrected by Jon Perry, Oct 15 2012

Edited by G. C. Greubel, Dec 30 2022

Status

approved