A372282 -id:A372282 - cp's OEIS frontend

A372351 Odd bisection of A371094.

21, 21, 341, 45, 117, 69, 341, 93, 213, 117, 5461, 141, 309, 165, 725, 189, 405, 213, 1877, 237, 501, 261, 1109, 285, 597, 309, 5461, 333, 693, 357, 1493, 381, 789, 405, 3413, 429, 885, 453, 1877, 477, 981, 501, 87381, 525, 1077, 549, 2261, 573, 1173, 597, 4949, 621, 1269, 645, 2645, 669, 1365, 693, 11605, 717

Offset: 1

Antti Karttunen, Apr 28 2024

nonn

Row 2 of A372282.
Cf. A371094, and array A371100 (gives the same terms, in different order).
Cf. A372290 (the range of this sequence), A372291 (numbers that occur only once), A372292 (more than once), A372293 (odd numbers not occurring here).

Table[With[{e = IntegerExponent[6*n - 2, 2]}, (6*n - 2)*2^e + (4^e - 1)/3], {n, 100}] (* Paolo Xausa, Apr 29 2024 *)

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372351(n) = A371094(n+n-1);

def A372351(n): return ((m:=6*n-2)<<(e:=(~m & m-1).bit_length()))+((1<<(e<<1))-1)//3 # Chai Wah Wu, Apr 28 2024

a(n) = A371094(2*n-1).

A372356 Array read by upward antidiagonals: A(n, k) = A372354(1+n,k)-A372354(n,k), n,k >= 1.

Original entry on oeis.org

4, 8, 3, 16, 8, 6, 32, 16, 12, 3, 64, 32, 24, 5, 3, 128, 64, 48, 9, 7, 3, 256, 128, 96, 21, 13, 5, 5, 512, 256, 192, 44, 25, 13, 12, 3, 1024, 512, 384, 88, 53, 28, 24, 5, 3, 2048, 1024, 768, 176, 108, 56, 48, 8, 9, 2, 4096, 2048, 1536, 352, 216, 112, 96, 21, 20, 7, 8, 8192, 4096, 3072, 704, 432, 224, 192, 44, 40, 13, 16, 3

Offset: 1

Antti Karttunen, Apr 30 2024

			Array begins:
n\k|    1     2     3    4    5    6     7    8     9   10    11   12   13   14
---+-----------------------------------------------------------------------------
1  |    4,    3,    6,   3,   3,   3,    5,   3,    3,   2,    8,   3,   4,   3,
2  |    8,    8,   12,   5,   7,   5,   12,   5,    9,   7,   16,   4,   6,   5,
3  |   16,   16,   24,   9,  13,  13,   24,   8,   20,  13,   32,  13,  15,  11,
4  |   32,   32,   48,  21,  25,  28,   48,  21,   40,  25,   64,  28,  29,  21,
5  |   64,   64,   96,  44,  53,  56,   96,  44,   80,  53,  128,  56,  57,  40,
6  |  128,  128,  192,  88, 108, 112,  192,  88,  160, 108,  256, 112, 117,  81,
7  |  256,  256,  384, 176, 216, 224,  384, 176,  320, 216,  512, 224, 236, 161,
8  |  512,  512,  768, 352, 432, 448,  768, 352,  640, 432, 1024, 448, 472, 324,
9  | 1024, 1024, 1536, 704, 864, 896, 1536, 704, 1280, 864, 2048, 896, 944, 647,

Columnwise first differences of A372354.

Cf. A000523, A371094, A372282, A372357.

up_to = 78;
A000523(n) = logint(n,2);
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
A372356sq(n,k) = { my(x=A372282sq(n,k)); (A000523(A371094(x))-A000523(x)); };
A372356list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372356sq((a-(col-1)),col))); (v); };
v372356 = A372356list(up_to);
A372356(n) = v372356[n];

A372360 Array read by upward antidiagonals: A(n, k) = A000120(A372361(n, k)), n,k >= 1; Binary weights of terms of arrays A372359 and A372361.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3

Offset: 1

Antti Karttunen, May 01 2024

Entry A(n, k) at row n and column k tells how many bits needs to be flipped in the binary expansion of the (n-1)-th iterate of Reduced Collatz function R, when started from 2*k-1, to obtain the unique term of A086893 with the same binary length as that (n-1)-th iterate. That is, A(n, k) gives the Hamming distance between A372283(n, k) and A086893(1+A000523(A372283(n, k))).

Zeros occur in the same locations as where they occur in A372359, etc.

			Array begins:
n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
---+-------------------------------------------------------------------------
1  | 0, 0, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 2, 3, 1, 2, 2, 3, 1, 2, 3, 4, 2, 3,
2  | 0, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 3, 2, 3, 2, 3, 2, 0, 1, 3, 2, 2, 1, 2,
3  | 0, 0, 0, 1, 2, 0, 0, 3, 0, 2, 0, 0, 1, 2, 1, 2, 2, 0, 2, 2, 3, 1, 0, 5,
4  | 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 3, 0, 5, 1, 0, 1, 3, 2, 1, 0, 4,
5  | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 4, 2, 0, 0, 2, 5, 1, 0, 3,
6  | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 3, 1, 0, 0, 2, 4, 2, 0, 3,
7  | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 3, 0, 0, 0, 1, 3, 1, 0, 4,
8  | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 4, 0, 0, 0, 2, 3, 0, 0, 3,
9  | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 1, 4, 0, 0, 4,
10 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 3, 0, 0, 4,
11 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 4, 0, 0, 0, 0, 4, 0, 0, 5,
12 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 5, 0, 0, 0, 0, 4, 0, 0, 3,
13 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 3, 0, 0, 0, 0, 5, 0, 0, 6,
14 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 6, 0, 0, 0, 0, 3, 0, 0, 2,
15 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 6, 0, 0, 4,
16 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4,
17 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 4, 0, 0, 4,
18 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 4, 0, 0, 3,
19 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 3, 0, 0, 0, 0, 4, 0, 0, 4,
20 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 3, 0, 0, 6,
21 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0, 4, 0, 0, 4,
We have A372283(5, 14) = 71, and when we compare the binary expansion of 71 = 1000111_2 with the term of A086893 that has a binary expansion of the same length, which in this case is 85 = 1010101_2, we see that only the bits at positions 1 and 4 (indexed from the right hand end, with 0 being the least significant bit position at right) need to be toggled to obtain the 71 from 85 or vice versa, therefore A(5, 14) = 2.
We have A372283(6, 14) = 107 = 1101011_2, and when xored with A086893(7) = 85 = 1010101_2, we obtain A372361(6, 14) = 62 = 111110_2, with five 1-bits, therefore A(6, 14) = 5. I.e., five bits (all except the least and the most significant bit) need to be flipped to change 85 to 107 or vice versa.

Cf. A000120, A003987, A086893, A372282, A372283, A372287, A372354, A372358, A372360.
Binary weights of A372359 and A372361.
Cf. also A372288.

up_to = 105;
R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A372283sq(n,k) = if(1==n,2*k-1,R(A372283sq(n-1,k)));
A000523(n) = logint(n,2);
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3);
A372358(n) = bitxor(A086893(1+A000523(n)),n);
A372361sq(n,k) = A372358(A372283sq(n,k));
A372361list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372361sq((a-(col-1)),col))); (v); };
v372361 = A372361list(up_to);
A372361(n) = v372361[n];

A(n, k) = A000120(A372361(n, k)) = A000120(A372358(A372283(n, k))).

A(n, k) = A000120(A372359(n, k)) = A000120(A372358(A372282(n, k))).

A372560 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = A372443(k-1).

Original entry on oeis.org

27, 165, 41, 8021, 501, 31, 12408149, 48469, 189, 47, 19607957362005, 299193685, 4565, 285, 71, 32439509492992549521282389, 7552911875069269, 1758549, 6869, 429, 107, 58947232705679751034215288252890081792789279233365, 3195535888075328282939605996885, 173230347605, 2643285, 10325, 645, 161

Offset: 1

Antti Karttunen, May 08 2024

			Array begins:
n\k|        1          2        3        4        5         6           7
---+-----------------------------------------------------------------------
1  |       27,        41,      31,      47,      71,      107,        161,
2  |      165,       501,     189,     285,     429,      645,       1941,
3  |     8021,     48469,    4565,    6869,   10325,    31061,     374101,
4  | 12408149, 299193685, 1758549, 2643285, 7951701, 95769941, 9216283989,

Cf. A371094, A372282, A372443 (the top row), A372444 (the leftmost column), A372561.

up_to = 28;
A000265(n) = (n>>valuation(n,2));
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372560sq(n,k) = if(1==n,A372443(k-1),A371094(A372560sq(n-1,k)));
A372560list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372560sq((a-(col-1)),col))); (v); };
v372560 = A372560list(up_to);
A372560(n) = v372560[n];

A372561 Array read by upward antidiagonals: A(n, k) = A265745(A372560(n, k)) for n > 1, k >= 1.

Original entry on oeis.org

3, 5, 5, 5, 5, 3, 5, 5, 5, 3, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 10347, 6251, 2155, 1131, 619, 363, 235, 107, 43, 27, 11, 7, 5

Offset: 1

Antti Karttunen, May 08 2024

In general, it seems that for n>2, k>1, A(n, k) = A(n-1, k+1) = A(k, n), except on those two anomalous antidiagonals, first on the thirteenth antidiagonal, where for n=1..13, A(n,14-n) obtains values 5, 7, 11, 27, 43, 107, 235, 363, 619, 1131, 2155, 6251, 10347, and then on the 30th antidiagonal, where for n=1.., A(n,31-n) obtains values 5, 11, 15, 23, 39, 71, 135, 391, 647, 1671, 2695, 4743, 17031, 33415, 49799, 82567, 148103, 410247, etc. The corresponding antidiagonals in A372560 begin as:
  233, 933, 14933, 978670933, 64138178286933, 1183140560213014108063589658350933, ..., and:
  911, 58325, 933205, 238900565, 15656587449685, 67244531063362552157525, etc. I conjecture that for the former sequence of numbers x, from 933 onward, A372555(x) = 7, and for the latter sequence of numbers y, from 58325 onward, A372555(y) = 9, and that the array A372555(A372560(n, k)) is symmetric apart from its borders, i.e, that for n, k > 1, A372555(A372560(n, k)) = A372555(A372560(k, n)).

			Array begins:
n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21
---+----------------------------------------------------------------
1  | 3, 5, 3, 3, 5, 3, 5, 5, 3, 5, 5, 5, 5, 3, 5, 3, 7, 5, 7, 5, 5,
2  | 5, 5, 5, 5, 3, 5, 5, 3, 5, 5, 5, 7, 5, 5, 5, 7, 5, 7, 7, 5, 5,
3  | 5, 5, 5, 3, 5, 5, 3, 5, 5, 5, 11, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7,
4  | 5, 5, 3, 5, 5, 3, 5, 5, 5, 27, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9,
5  | 5, 3, 5, 5, 3, 5, 5, 5, 43, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7,
6  | 3, 5, 5, 3, 5, 5, 5, 107, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7,
7  | 5, 5, 3, 5, 5, 5, 235, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7,
8  | 5, 3, 5, 5, 5, 363, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9,
9  | 3, 5, 5, 5, 619, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7,
10 | 5, 5, 5, 1131, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 1671,
11 | 5, 5, 2155, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 2695, 3,
12 | 5, 6251, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 4743, 3, 5,
13 | 10347, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 17031, 3, 5, 3,
14 | 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 33415, 3, 5, 3, 5,
15 | 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 49799, 3, 5, 3, 5, 5,
etc.
From column 19 to column 41, the first 11 rows:
n\k|19 20 ........................................................... 40 41
---+-------------------------------------------------------------------------
1  | 7, 5, 5, 5, 7, 7, 5, 5, 5, 7, 7, 5,    3, 3, 3, 5, 5, 5, 5, 3, 3, 3, 1,
2  | 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 11,   3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1,
3  | 5, 5, 7, 9, 7, 7, 7, 9, 7, 15,   3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1,
4  | 5, 7, 9, 7, 7, 7, 9, 7, 23,   3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1,
5  | 7, 9, 7, 7, 7, 9, 7, 39,   3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1,
6  | 9, 7, 7, 7, 9, 7, 71,   3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1,
7  | 7, 7, 7, 9, 7, 135,  3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1,
8  | 7, 7, 9, 7, 391,  3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1,
9  | 7, 9, 7, 647,  3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1,
10 | 9, 7, 1671, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
11 | 7, 2695, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

Cf. A265745, A371094, A372282, A372288, A372555, A372560.

\\ Needs also program from A372560.
up_to = 91;
A001045(n) = (2^n - (-1)^n) / 3;
A130249(n) = (#binary(3*n+1)-1);
A265745(n) = { my(s=0); while(n,s++; n -= A001045(A130249(n))); (s); };
A372561sq(n,k) = A265745(A372560sq(n,k));
A372561list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372561sq((a-(col-1)),col))); (v); };
v372561 = A372561list(up_to);
A372561(n) = v372561[n];

A372357 Array read by upward antidiagonals: A(n, k) = A372356(1+n,k)-2*A372356(n,k), n,k >= 1.

Original entry on oeis.org

0, 0, 2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 1, 0, 0, 0, 3, -1, -1, 0, 0, 0, 2, -1, 3, 2, 0, 0, 0, 0, 3, 2, 0, -1, 0, 0, 0, 0, 2, 0, 0, -2, 3, 0, 0, 0, 0, 0, 0, 0, 5, 2, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 5, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 3, -1

Offset: 1

Antti Karttunen, Apr 30 2024

			Array begins:
n\k| 1  2  3   4   5   6  7   8  9  10 11  12  13  14  15  16  17 18  19  20  21
---+-----------------------------------------------------------------------------
1  | 0, 2, 0, -1,  1, -1, 2, -1, 3,  3, 0, -2, -2, -1, -1, -1,  2, 5,  1,  1,  1,
2  | 0, 0, 0, -1, -1,  3, 0, -2, 2, -1, 0,  5,  3,  1, -1, -2, -2, 2, -1,  0, -1,
3  | 0, 0, 0,  3, -1,  2, 0,  5, 0, -1, 0,  2, -1, -1,  3,  1,  3, 0, -1, -2, -2,
4  | 0, 0, 0,  2,  3,  0, 0,  2, 0,  3, 0,  0, -1, -2,  2, -1, -1, 0,  3,  4,  1,
5  | 0, 0, 0,  0,  2,  0, 0,  0, 0,  2, 0,  0,  3,  1,  0,  2, -1, 0,  2, -2, -1,
6  | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  2, -1,  0, -1,  3, 0,  0,  3,  2,
7  | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  2,  0, -1,  2, 0,  0, -1, -1,
8  | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -1,  0,  1,  0, 0,  0, -1, -1,
9  | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -1,  0, -2,  0, 0,  0,  3,  1,
10 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  1,  0,  2,  0, 0,  0,  2, -2,
11 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -2,  0, -1,  0, 0,  0,  0,  2,
12 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  2,  0, -2,  0, 0,  0,  0, -1,
13 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -1,  0,  1,  0, 0,  0,  0, -2,
14 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -2,  0, -1,  0, 0,  0,  0,  1,
15 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  1,  0,  3,  0, 0,  0,  0, -1,
16 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -1,  0, -2,  0, 0,  0,  0,  3,
17 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  3,  0,  1,  0, 0,  0,  0, -2,
18 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -2,  0,  0,  0, 0,  0,  0,  1,
19 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  1,  0, -2,  0, 0,  0,  0,  0,
20 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0,  0,  0,  2,  0, 0,  0,  0, -2,
21 | 0, 0, 0,  0,  0,  0, 0,  0, 0,  0, 0,  0,  0, -2,  0, -2,  0, 0,  0,  0,  2,

Cf. A000523, A371094, A372282, A372354, A372356.

up_to = 105;
A000523(n) = logint(n,2);
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
A372356sq(n,k) = { my(x=A372282sq(n,k)); (A000523(A371094(x))-A000523(x)); };
A372357sq(n,k) = (A372356sq(1+n,k)-2*A372356sq(n,k));
A372357sq(n,k) = { my(x=A372282sq(n,k), y=A371094(x), z=A371094(y), xl=A000523(x), yl=A000523(y), zl=A000523(z)); ((zl-yl) - 2*(yl-xl)); };
A372357sq(n,k) = { my(x=A372282sq(n,k), y=A371094(x), z=A371094(y), xl=A000523(x), yl=A000523(y), zl=A000523(z)); (zl - 3*yl + 2*xl); };
A372357list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372357sq((a-(col-1)),col))); (v); };
v372357 = A372357list(up_to);
A372357(n) = v372357[n];

A372355 Array read by upward antidiagonals: A(n,k) = A372285(1+n, k)-A372285(n, k), n,k >= 1.

Original entry on oeis.org

4, 8, 5, 16, 8, 6, 32, 16, 12, 3, 64, 32, 24, 5, 2, 128, 64, 48, 12, 7, 3, 256, 128, 96, 23, 13, 8, 7, 512, 256, 192, 44, 28, 15, 12, 1, 1024, 512, 384, 88, 55, 28, 24, 5, 6, 2048, 1024, 768, 176, 108, 56, 48, 13, 11, 3, 4096, 2048, 1536, 352, 216, 112, 96, 23, 20, 7, 8, 8192, 4096, 3072, 704, 432, 224, 192, 44, 40, 13, 16, 3

Offset: 1

Antti Karttunen, Apr 29 2024

			Array begins:
n\k|    1     2      3     4     5     6      7     8      9    10     11    12
---+----------------------------------------------------------------------------
1  |    4,    5,     6,    3,    2,    3,     7,    1,     6,    3,     8,    3,
2  |    8,    8,    12,    5,    7,    8,    12,    5,    11,    7,    16,    9,
3  |   16,   16,    24,   12,   13,   15,    24,   13,    20,   13,    32,   15,
4  |   32,   32,    48,   23,   28,   28,    48,   23,    40,   28,    64,   28,
5  |   64,   64,    96,   44,   55,   56,    96,   44,    80,   55,   128,   56,
6  |  128,  128,   192,   88,  108,  112,   192,   88,   160,  108,   256,  112,
7  |  256,  256,   384,  176,  216,  224,   384,  176,   320,  216,   512,  224,
8  |  512,  512,   768,  352,  432,  448,   768,  352,   640,  432,  1024,  448,
9  | 1024, 1024,  1536,  704,  864,  896,  1536,  704,  1280,  864,  2048,  896,
10 | 2048, 2048,  3072, 1408, 1728, 1792,  3072, 1408,  2560, 1728,  4096, 1792,
11 | 4096, 4096,  6144, 2816, 3456, 3584,  6144, 2816,  5120, 3456,  8192, 3584,
12 | 8192, 8192, 12288, 5632, 6912, 7168, 12288, 5632, 10240, 6912, 16384, 7168,

Columnwise first differences of A372285.
Cf. A086893, A372282, A372286.
Cf. also A372353.

up_to = 78;
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
A372286(n) = { my(u=A371094(n), k1); for(i=1,oo,if(A086893(i)>=n,k1=i-1; break)); for(i=k1,oo,if(A086893(i)>u,return(i-k1-1))); };
A372355sq(n,k) = (A372286(A372282sq(1+n,k))-A372286(A372282sq(n,k)));
A372355list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372355sq((a-(col-1)),col))); (v); };
v372355 = A372355list(up_to);
A372355(n) = v372355[n];

cp's OEIS Frontend

A372351 Odd bisection of A371094.

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A372356 Array read by upward antidiagonals: A(n, k) = A372354(1+n,k)-A372354(n,k), n,k >= 1.

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A372360 Array read by upward antidiagonals: A(n, k) = A000120(A372361(n, k)), n,k >= 1; Binary weights of terms of arrays A372359 and A372361.

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A372560 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = A372443(k-1).

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A372561 Array read by upward antidiagonals: A(n, k) = A265745(A372560(n, k)) for n > 1, k >= 1.

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A372357 Array read by upward antidiagonals: A(n, k) = A372356(1+n,k)-2*A372356(n,k), n,k >= 1.

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A372355 Array read by upward antidiagonals: A(n,k) = A372285(1+n, k)-A372285(n, k), n,k >= 1.

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