A355663 Square array A(n, k), n, k >= 0, read by antidiagonals; for any number n with runs in binary expansion (r_w, ..., r_0), let p(n) be the polynomial of a single indeterminate x where the coefficient of x^e is r_e for e = 0..w and otherwise 0, and let q be the inverse of p; A(n, k) = q(p(n) + p(k)).

0, 1, 1, 2, 3, 2, 3, 4, 4, 3, 4, 7, 12, 7, 4, 5, 8, 8, 8, 8, 5, 6, 11, 24, 15, 24, 11, 6, 7, 12, 19, 16, 16, 19, 12, 7, 8, 15, 28, 23, 48, 23, 28, 15, 8, 9, 16, 16, 24, 39, 39, 24, 16, 16, 9, 10, 19, 48, 31, 56, 51, 56, 31, 48, 19, 10, 11, 20, 35, 32, 32, 35, 35, 32, 32, 35, 20, 11

Offset

0,4

Comments

In other words, A(n, k) encodes the sum of the polynomials encoded by n and k.

Links

Table of n, a(n) for n=0..77.

Index entries for sequences related to binary expansion of n

Formula

A(n, k) = A(k, n).

A(n, 0) = n.

A(n, 1) = A014601(n) for any n > 0.

A(n, n) = A001196(n).

Example

Array A(n, k) begins:
  n\k|   0   1   2   3    4    5    6   7    8    9   10   11   12
  ---+------------------------------------------------------------
    0|   0   1   2   3    4    5    6   7    8    9   10   11   12
    1|   1   3   4   7    8   11   12  15   16   19   20   23   24
    2|   2   4  12   8   24   19   28  16   48   35   44   39   56
    3|   3   7   8  15   16   23   24  31   32   39   40   47   48
    4|   4   8  24  16   48   39   56  32   96   71   88   79  112
    5|   5  11  19  23   39   51   35  47   79   99   76  103   71
    6|   6  12  28  24   56   35   60  48  112   67   92   71  120
    7|   7  15  16  31   32   47   48  63   64   79   80   95   96
    8|   8  16  48  32   96   79  112  64  192  143  176  159  224
    9|   9  19  35  39   71   99   67  79  143  195  156  199  135
   10|  10  20  44  40   88   76   92  80  176  156  204  152  184
   11|  11  23  39  47   79  103   71  95  159  199  152  207  143
   12|  12  24  56  48  112   71  120  96  224  135  184  143  240

Prog

(PARI) toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
A(n, k) = { fromruns(Vec(Pol(toruns(n)) + Pol(toruns(k)))) }

Crossrefs

Cf. A001196, A014601, A101211, A355664.

Sequence in context: A054237 A360495 A129600 * A081388 A231070 A230252

Adjacent sequences: A355660 A355661 A355662 * A355664 A355665 A355666

Keyword

nonn,base,tabl

Author

Rémy Sigrist, Jul 13 2022

Status

approved