A176794 Triangle read by rows: T(n, k) = 2^n*(q^k - 1)*(q^(n - k) - 1) + 1, where q = 3.

1, 1, 1, 1, 17, 1, 1, 129, 129, 1, 1, 833, 1025, 833, 1, 1, 5121, 6657, 6657, 5121, 1, 1, 30977, 40961, 43265, 40961, 30977, 1, 1, 186369, 247809, 266241, 266241, 247809, 186369, 1, 1, 1119233, 1490945, 1610753, 1638401, 1610753, 1490945, 1119233, 1

Offset: 0

Author

Roger L. Bagula, Apr 26 2010

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      17,       1;
  1,     129,     129,       1;
  1,     833,    1025,     833,       1;
  1,    5121,    6657,    6657,    5121,       1;
  1,   30977,   40961,   43265,   40961,   30977,       1;
  1,  186369,  247809,  266241,  266241,  247809,  186369,       1;
  1, 1119233, 1490945, 1610753, 1638401, 1610753, 1490945, 1119233,     1;
		

Links

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

Crossrefs

Cf. A000012 (q=1), A176793 (q=2), this sequence (q=3), A176795 (q=4).

Cf. A144843.

Programs

Magma

f:= func< n,k,q | 1 + (q^k-1)*(q^(n-k)-1)*2^n >;
A176794:= func< n,k | f(n,k,3) >;
[A176794(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Oct 03 2024

Mathematica

T[n_, k_, q_] := 2^n*(q^k - 1)*(q^(n - k) - 1) + 1;
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten

SageMath

def f(n, k, q): return 1 + (q^k -1)*(q^(n-k) -1)*2^n
def A176794(n,k): return f(n,k,3)
flatten([[A176794(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Oct 03 2024

Formula

T(n, k) = 1 - (f(n+1, 2*k+1, q) - f(n+1, 1, q)) - (f(n+1, 2*n-2*k+1, q) - f(n+1, 2*n+1, q)), where f(n, k, q) = 2^(n-1) * q^((k-1)/2), and q = 3.

From G. C. Greubel, Oct 02 2024: (Start)

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

T(2*n, n) = 1 + 4^n*(3^n - 1)^2 = 1 + 16*A144843(n).

Sum_{k=0..n} T(n, k) = 2^n*(n + 2 + (n-2)*3^n) + (n+1).

Sum_{k=0..n} (-1)^k*T(n, k) = (1/4)*(1 + (-1)^n)*(2 + 2^n - 6^n). (End)

Extensions

Edited by G. C. Greubel, Oct 03 2024