A359250 Irregular triangle read by rows where T(n,k) is the coefficient of y^k in polynomial P(n) defined by P(2n) = P(n) and P(2n+1) = y*P(n) + P(n+1) starting P(0) = 0, P(1) = 1.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 3, 1, 3, 3, 1, 2, 1, 3, 4, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 3, 3, 1, 2, 2, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 4, 1, 4, 4

Offset

0,7

Comments

Row n length is wt(n) = A000120(n), the binary weight of n, so that k ranges 0 <= k < wt(n).

Evaluated at y=1, the recurrence for P is per Stern's diatomic sequence so that row n has sum A002487(n).

Evaluated at y=2, the recurrence for P is per A116528 so that Sum T(n,k)*2^k = A116528(n), and similarly variations such as y=10 for A178243.

Array A178239(r,n) is P(n) evaluated at y=r, and in particular P(n) is the polynomial for the values in column n there.

Column k=1, when that value exists, is T(n,1) = A080791(n-1), the number of 0 bits in n-1, since expressing the recurrence in Q(m) = P(m+1) adds y*Q(m-1) at each 0 bit in m.

Reversing the bits of n is no change to P(n), so that P(n) = P(A030101(n)).

Links

Kevin Ryde, Table of n, a(n) for rows 0..2048, flattened

Kevin Ryde, PARI/GP Code and Notes

Thomas Scheuerle, Scatter plot a(n) for n = 1 to 114689. This is in T(n,k) the range of n = 1 to 2^14. It looks a bit like narrow and tall fir trees like you could see in the Schwarzwald region.

Formula

G.f.: Sum_{n>=0} P(n)*x^n = x * Product_{e>=0} 1 + x^(2^e) + y*x^(2^(e+1)).

Example

Triangle begins:
      k=0  1  2
  n=0:  (empty)
  n=1:  1,
  n=2:  1,
  n=3:  1, 1,
  n=4:  1,
  n=5:  1, 2,
  n=6:  1, 1,
  n=7:  1, 1, 1,
  n=8:  1,
  n=9:  1, 3,

Prog

(MATLAB)
function a = A359250( max_n )
    ac = cell(1, 1); ac{1} = [1];
    for n = 2:max_n
        m = floor(n/2);
        if 2*m == n
            ac{n} = ac{m};
        else
            ac{n} = [ac{m+1} zeros(1, (length(ac{m})+1)-length(ac{m+1}))] ...
            + [0 ac{m}];
        end
    end
    a = cell2mat(ac);
end % Thomas Scheuerle, Dec 28 2022
(PARI) \\ See links.

Crossrefs

Cf. A000120 (row lengths), A002487 (row sums).

Cf. A178239 (array), A030101 (bit reversal).

Evaluated at y=2..10: A116528, A342633, A342634, A342635, A342603, A342636, A342637, A342638, A178243.

Cf. A125184.

Sequence in context: A359888 A204112 A186027 * A322482 A231071 A209156

Adjacent sequences: A359247 A359248 A359249 * A359251 A359252 A359253

Keyword

nonn,tabf,look,easy

Author

Kevin Ryde, Dec 28 2022

Status

approved