A176794 - OEIS

%I #9 Oct 04 2024 07:41:06

%S 1,1,1,1,17,1,1,129,129,1,1,833,1025,833,1,1,5121,6657,6657,5121,1,1,

%T 30977,40961,43265,40961,30977,1,1,186369,247809,266241,266241,247809,

%U 186369,1,1,1119233,1490945,1610753,1638401,1610753,1490945,1119233,1

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

%H G. C. Greubel, <a href="/A176794/b176794.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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.

%F From _G. C. Greubel_, Oct 02 2024: (Start)

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

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

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

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

%e Triangle begins as:

%e   1;

%e   1,       1;

%e   1,      17,       1;

%e   1,     129,     129,       1;

%e   1,     833,    1025,     833,       1;

%e   1,    5121,    6657,    6657,    5121,       1;

%e   1,   30977,   40961,   43265,   40961,   30977,       1;

%e   1,  186369,  247809,  266241,  266241,  247809,  186369,       1;

%e   1, 1119233, 1490945, 1610753, 1638401, 1610753, 1490945, 1119233,     1;

%t T[n_, k_, q_] := 2^n*(q^k - 1)*(q^(n - k) - 1) + 1;

%t Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten

%o (Magma)

%o f:= func< n,k,q | 1 + (q^k-1)*(q^(n-k)-1)*2^n >;

%o A176794:= func< n,k | f(n,k,3) >;

%o [A176794(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Oct 03 2024

%o (SageMath)

%o def f(n, k, q): return 1 + (q^k -1)*(q^(n-k) -1)*2^n

%o def A176794(n,k): return f(n,k,3)

%o flatten([[A176794(n, k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Oct 03 2024

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

%Y Cf. A144843.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Apr 26 2010

%E Edited by _G. C. Greubel_, Oct 03 2024