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A362311
Triangle read by rows (row length 2*n+1). Row n lists the integer solutions for x in the equation x - 2^n = x/y (x and y are integers).
2
2, 1, 3, 4, 2, 3, 5, 6, 8, 4, 6, 7, 9, 10, 12, 16, 8, 12, 14, 15, 17, 18, 20, 24, 32, 16, 24, 28, 30, 31, 33, 34, 36, 40, 48, 64, 32, 48, 56, 60, 62, 63, 65, 66, 68, 72, 80, 96, 128, 64, 96, 112, 120, 124, 126, 127, 129, 130, 132, 136, 144, 160, 192, 256, 128, 192, 224, 240, 248, 252, 254, 255
OFFSET
0,1
FORMULA
T(n, k) = 2^n - 2^(n-k-1), if k < n.
T(n, k) = 2^n + 2^(k-n), if k >= n.
T(n, 0..n-2) = 2*T(n-1, 0..n-2), for n > 1.
T(n, n-1) = 2^n - 1, for n > 0.
T(n, n) = 2^n + 1, for n > 0.
T(n, n+1..2*n) = 2*T(n-1, n-1..2*(n-1)), for n > 0.
EXAMPLE
Triangle begins:
k=0 1 2 3 4 5 6 7 8
n=0: 2
n=1: 1, 3, 4
n=2: 2, 3, 5, 6, 8
n=3: 4, 6, 7, 9,10,12,16
n=4: 8,12,14,15,17,18,20,24,32
...
Corresponding values for y in the equation:
k=0 1 2 3 4 5 6 7 8
n=0: 2
n=1: -1, 3, 2
n=2: -1,-3, 5, 3, 2
n=3: -1,-3,-7, 9, 5,3,2
n=4: -1,-3,-7,-15,17,9,5,3,2
...
PROG
(PARI) T(n, k) = if(k >= n, 2^n + 2^(k-n), 2^n - 2^(n-k-1));
(MATLAB)
function a = A362311( max_row )
r = 2; a = [];
for n = 1:max_row
a = [a r];
r = [2*r(1:n-1) 2^n-1 2^n+1 2*r(end-n+1:end)];
end
end
CROSSREFS
Cf. A036289 (row sums).
Cf. A362310.
Sequence in context: A088606 A140073 A333400 * A131389 A131394 A130585
KEYWORD
nonn,tabf,easy
AUTHOR
Thomas Scheuerle, Apr 15 2023
STATUS
approved