A355654 -id:A355654 - cp's OEIS frontend

A355653 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) = q(p(n)').

0, 0, 1, 0, 1, 6, 3, 0, 1, 12, 57, 6, 3, 30, 7, 0, 1, 24, 225, 12, 57, 966, 115, 6, 3, 60, 505, 30, 7, 126, 15, 0, 1, 48, 897, 24, 225, 7686, 451, 12, 57, 1932, 31801, 966, 115, 3870, 231, 6, 3, 120, 2017, 60, 505, 16326, 1011, 30, 7, 252, 4089, 126, 15, 510

Offset: 0

Rémy Sigrist, Jul 12 2022

In other words, a(n) encodes the derivative of the polynomial encoded by n.

For n > 0, the runs in binary expansion of a(n) are (w*r_w, ..., 1*r_1).

			The first terms, alongside their binary expansions and corresponding polynomials, are:
  n   a(n)  bin(n)  bin(a(n))  p(n)               p(a(n))
  --  ----  ------  ---------  -----------------  ---------------
   0     0       0          0                  0                0
   1     0       1          0                  1                0
   2     1      10          1              x + 1                1
   3     0      11          0                  2                0
   4     1     100          1              x + 2                1
   5     6     101        110        x^2 + x + 1          2*x + 1
   6     3     110         11            2*x + 1                2
   7     0     111          0                  3                0
   8     1    1000          1              x + 3                1
   9    12    1001       1100      x^2 + 2*x + 1          2*x + 2
  10    57    1010     111001  x^3 + x^2 + x + 1  3*x^2 + 2*x + 1
  11     6    1011        110        x^2 + x + 2          2*x + 1
  12     3    1100         11            2*x + 2                2
  13    30    1101      11110      2*x^2 + x + 1          4*x + 1

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Binary plot of the sequence for n = 0..2^10
Index entries for sequences related to binary expansion of n

Cf. A000225, A005811, A101211, A355654.

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) = { fromruns(Vec(deriv(Pol(toruns(n))))) }

a^A005811(n)(n) = 0 (where a^k denotes the k-th iterate of a).
a(n) = 0 iff n belongs to A000225.
a(n) = 1 iff n = 2^k for some k > 0.
A005811(a(n)) = A005811(n)-1 for any n > 0.

A355664 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)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 9, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 35, 15, 35, 5, 0, 0, 6, 38, 48, 48, 38, 6, 0, 0, 7, 49, 51, 271, 51, 49, 7, 0, 0, 8, 56, 60, 284, 284, 60, 56, 8, 0, 0, 9, 135, 63, 387, 313, 387, 63, 135, 9, 0, 0, 10, 142, 192, 448, 398, 398, 448, 192, 142, 10, 0

Offset: 0

Rémy Sigrist, Jul 13 2022

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

			Array A(n, k) begins:
  n\k|  0   1    2    3     4     5     6     7      8      9     10     11
  ---+---------------------------------------------------------------------
    0|  0   0    0    0     0     0     0     0      0      0      0      0
    1|  0   1    2    3     4     5     6     7      8      9     10     11
    2|  0   2    9   12    35    38    49    56    135    142    153    156
    3|  0   3   12   15    48    51    60    63    192    195    204    207
    4|  0   4   35   48   271   284   387   448   2111   2172   2275   2288
    5|  0   5   38   51   284   313   398   455   2168   2289   2502   2531
    6|  0   6   49   60   387   398   481   504   3079   3102   3185   3196
    7|  0   7   56   63   448   455   504   511   3584   3591   3640   3647
    8|  0   8  135  192  2111  2168  3079  3584  33279  33784  34695  34752
    9|  0   9  142  195  2172  2289  3102  3591  33784  34785  36622  36739
   10|  0  10  153  204  2275  2502  3185  3640  34695  36622  39993  40476
   11|  0  11  156  207  2288  2531  3196  3647  34752  36739  40476  40719
   12|  0  12  195  240  3087  3132  3843  4032  49215  49404  50115  50160

Index entries for sequences related to binary expansion of n

Cf. A001196, A097254, A101211, A355663.

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)))) }

A(n, k) = A(k, n).
A(n, 0) = 0.
A(n, 1) = n.
A(n, 3) = A001196(n).
A(n, 7) = A097254(n+1).
A(n, n) = A355654(n).

cp's OEIS Frontend

A355653 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) = q(p(n)').

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