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, 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

Links

Table of n, a(n) for n=0..71.

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

Adjacent sequences: A362308 A362309 A362310 * A362312 A362313 A362314

Keyword

nonn,tabf,easy

Author

Thomas Scheuerle, Apr 15 2023

Status

approved