A371094 -id:A371094 - cp's OEIS frontend

A372353 Array read by upward antidiagonals: A(n, k) = A372352(A372282(n, k)), n,k >= 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 24, 4, 0, 0, 0, 256, 32, 6, 0, 0, 0, 0, 6144, 16, 0, 0, 0, 0, 0, 16777216, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 896, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6144, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16777216, 0, 56, 4

Offset: 1

Antti Karttunen, Apr 29 2024

Zeros occur in the same locations where 1's occur in array A372287.

			Array begins:
n\k| 1  2  3    4         5   6  7    8  9        10 11  12                 13
---+---------------------------------------------------------------------------
1  | 0, 0, 0,   2,        4,  6, 0,   2, 4,        6, 0,  2,                 4,
2  | 0, 0, 0,  24,       32, 16, 0,   8, 0,       32, 0, 56,                96,
3  | 0, 0, 0, 256,     6144,  0, 0, 896, 0,     6144, 0,  0,              8192,
4  | 0, 0, 0,   0, 16777216,  0, 0,   0, 0, 16777216, 0,  0,         402653184,
5  | 0, 0, 0,   0,        0,  0, 0,   0, 0,        0, 0,  0, 72057594037927936,
6  | 0, 0, 0,   0,        0,  0, 0,   0, 0,        0, 0,  0,                 0,

Cf. A086893, A096773, A372352, A372282, A372287.

Cf. also A372285 and A372355 (columnwise first differences).

up_to = 91;
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
A372352(n) = { my(k); for(i=1,oo,k=A086893(i); if(k>n, return(n-A086893(i-1)))); };
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)));
A372353sq(n,k) = A372352(A372282sq(n,k));
A372353list(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] = A372353sq((a-(col-1)),col))); (v); };
v372353 = A372353list(up_to);
A372353(n) = v372353[n];

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];

A372359 Array read by upward antidiagonals: A(n, k) = A372358(A372282(n, k)), n,k >= 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 24, 4, 0, 0, 0, 256, 32, 6, 0, 0, 0, 0, 6144, 16, 0, 0, 0, 0, 0, 16777216, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 1408, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6144, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16777216, 0, 88, 12

Offset: 1

Antti Karttunen, May 01 2024

Zeros occur in the same locations as where they occur in A372353 and where 1's occur in array A372287.

			Array begins:
n\k| 1  2  3    4     5   6  7     8  9    10 11  12         13             14
---+----------------------------------------------------------------------------
1  | 0, 0, 0,   2,    4,  6, 0,    2, 4,    6, 0,  2,        12,            14,
2  | 0, 0, 0,  24,   32, 16, 0,    8, 0,   32, 0, 88,        96,           112,
3  | 0, 0, 0, 256, 6144,  0, 0, 1408, 0, 6144, 0,  0,      8192,          2560,
4  | 0, 0, 0,   0, 2^24,  0, 0,    0, 0, 2^24, 0,  0, 402653184,       6815744,
5  | 0, 0, 0,   0,    0,  0, 0,    0, 0,    0, 0,  0,      2^56, 4947802324992,
6  | 0, 0, 0,   0,    0,  0, 0,    0, 0,    0, 0,  0,         0,     31 * 2^79,
where 2^56 = 72057594037927936 and 31 * 2^79 = 18738350204026752207945728.

Cf. A003987, A086893, A371094, A372282, A372287, A372354, A372358, A372360 (binary weights).

Cf. also A372353.

up_to = 91;
A000523(n) = logint(n,2);
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
A372358(n) = bitxor(A086893(1+A000523(n)),n);
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)));
A372359sq(n,k) = A372358(A372282sq(n,k));
A372359list(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] = A372359sq((a-(col-1)),col))); (v); };
v372359 = A372359list(up_to);
A372359(n) = v372359[n];

A(n, k) = A372282(n,k) XOR A086893(1+A372354(n, k)), where XOR is bitwise-xor, A003987.

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];

A372289 a(n) = n*2^e + (4^e - 1)/3, where e is the 2-adic valuation of n.

Original entry on oeis.org

1, 5, 3, 21, 5, 13, 7, 85, 9, 21, 11, 53, 13, 29, 15, 341, 17, 37, 19, 85, 21, 45, 23, 213, 25, 53, 27, 117, 29, 61, 31, 1365, 33, 69, 35, 149, 37, 77, 39, 341, 41, 85, 43, 181, 45, 93, 47, 853, 49, 101, 51, 213, 53, 109, 55, 469, 57, 117, 59, 245, 61, 125, 63, 5461

Offset: 1

Antti Karttunen and Ali Sada, Apr 26 2024

nonn

Construction: take the binary expansion of n, and substitute "01" for all trailing 0-bits that follow after its odd part (= A000265(n)). See the examples.

			For n=4, "100" in binary, when we substitute 01's for the two trailing 0's, we obtain 21, "10101" in binary, therefore a(4) = 21.
For n=11, "1011" in binary, there are no trailing 0's, and thus no changes, therefore a(11) = 11.

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 08 2025
Index entries for sequences related to binary expansion of n

Cf. A000265, A005408 (odd bisection), A007814, A371094 [= a(3n+1)].

a := proc(n) padic[ordp](n, 2): n*2^% + ((2^%)^2 - 1)/3 end:
seq(a(n), n = 1..64);  # Peter Luschny, Apr 27 2024

a[n_]:=n*(2^IntegerExponent[n, 2]) + ((4^IntegerExponent[n, 2]) - 1)/3; Array[a, 75] (* Stefano Spezia, Apr 26 2024 *)

A372289(n) = { my(e=valuation(n,2)); n*2^e + (4^e-1)/3 }

def A372289(n): return (n<<(e:=(~n & n-1).bit_length()))+((1<<(e<<1))-1)//3 # Chai Wah Wu, Apr 26 2024

For n >= 0, a(2n+1) = 2n+1.

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];

A372454 a(n) = A372444(n) - A086893(1+A372449(n)).

Original entry on oeis.org

6, -48, 2560, -1572864, -3848290697216, 6649092007880460460883968, -18999521285301737936647902825311679255527123058688, 76895533293152762966220781422103876125697362804839499718093497881599910128103059800826635129716736

Offset: 0

Antti Karttunen, May 05 2024

sign

The difference between A372444(n) and the term of A086893 with the same binary length.

			The term of A086893 that has same binary length as A372444(0) = 27 is 21 [as 21 = 10101_2 in binary, and 27 = 11011_2 in binary], therefore a(0) = 27-21 = 6.
The term of A086893 that has same binary length as A372444(1) = 165 is 213, therefore a(1) = 165-213 = -48.

Antti Karttunen, Table of n, a(n) for n = 0..10

Cf. A000523, A086893, A372444, A372448, A372449, A372453.

A000523(n) = logint(n,2);
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372444(n) = { my(x=27); while(n, x=A371094(x); n--); (x); };
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3);
A372454(n) = { my(x=A372444(n)); (x - A086893(1+A000523(x))); };

A372454(n) = if(!n, A372453(n), (4^A372448(n-1))*A372453(n));

a(n) = A372444(n) - A086893(1+A000523(A372444(n))).

a(0) = A372453(0) = 6; and for n > 0, a(n) = 4^A372448(n-1) * A372453(n).

cp's OEIS Frontend

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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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A372359 Array read by upward antidiagonals: A(n, k) = A372358(A372282(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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A372454 a(n) = A372444(n) - A086893(1+A372449(n)).

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