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A176794
Triangle read by rows: T(n, k) = 2^n*(q^k - 1)*(q^(n - k) - 1) + 1, where q = 3.
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
0,5
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)
EXAMPLE
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;
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
PROG
(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
(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
CROSSREFS
Cf. A000012 (q=1), A176793 (q=2), this sequence (q=3), A176795 (q=4).
Cf. A144843.
Sequence in context: A218115 A144442 A157151 * A176244 A022180 A156581
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Apr 26 2010
EXTENSIONS
Edited by G. C. Greubel, Oct 03 2024
STATUS
approved