A372356 -id:A372356 - cp's OEIS frontend

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

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];

A372282 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1.

Original entry on oeis.org

1, 21, 3, 5461, 21, 5, 357913941, 5461, 341, 7, 1537228672809129301, 357913941, 1398101, 45, 9, 28356863910078205288614550619314017621, 1537228672809129301, 23456248059221, 1109, 117, 11, 9649340769776349618630915417390658987772498722136713669954798667326094136661, 28356863910078205288614550619314017621, 6602346876188694799461995861, 873813, 11605, 69, 13

Offset: 1

Antti Karttunen, Apr 28 2024

			Array begins:
n\k|    1     2        3     4      5     6        7     8      9     10
---+----------------------------------------------------------------------
1  |    1,    3,       5,    7,     9,   11,      13,   15,    17,    19,
2  |   21,   21,     341,   45,   117,   69,     341,   93,   213,   117,
3  | 5461, 5461, 1398101, 1109, 11605, 3413, 1398101, 2261, 87381, 11605,

Cf. A005408 (row 1), A372351 (row 2, bisection of A371094), A372444 (column 14).
Arrays derived from this one:
  A372283,
  A372285 the number of terms of A086893 in the interval [A(n, k), A(1+n, k)],
  A372287 the column index of A(n, k) in array A257852,
  A372288 the sum of digits of A(n, k) in "Jacobsthal greedy base",
  A372353 differences between A(n,k) and the largest term of A086893 <= A(n,k),
  A372354 floor(log_2(.)) of terms, A372356 (and their columnwise first differences),
  A372359 terms xored with binary words of the same length, either of the form 10101...0101 or 110101...0101, depending on whether the binary length is odd or even.
Cf. also arrays A371096, A371102 that give subsets of columns of this array, and array A371100 that gives the terms of the row 2 in different order.

up_to = 28;
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)));
A372282list(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] = A372282sq((a-(col-1)),col))); (v); };
v372282 = A372282list(up_to);
A372282(n) = v372282[n];

A372354 Array read by upward antidiagonals: A(n, k) = A000523(A372282(n, k)), n,k >= 1, where A000523(x) is one less than the number of bits in the binary expansion of x.

Original entry on oeis.org

0, 4, 1, 12, 4, 2, 28, 12, 8, 2, 60, 28, 20, 5, 3, 124, 60, 44, 10, 6, 3, 252, 124, 92, 19, 13, 6, 3, 508, 252, 188, 40, 26, 11, 8, 3, 1020, 508, 380, 84, 51, 24, 20, 6, 4, 2044, 1020, 764, 172, 104, 52, 44, 11, 7, 4, 4092, 2044, 1532, 348, 212, 108, 92, 19, 16, 6, 4, 8188, 4092, 3068, 700, 428, 220, 188, 40, 36, 13, 12, 4

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  |    0,    1,    2,   2,   3,   3,    3,   3,    4,   4,    4,   4,   4,   4,
2  |    4,    4,    8,   5,   6,   6,    8,   6,    7,   6,   12,   7,   8,   7,
3  |   12,   12,   20,  10,  13,  11,   20,  11,   16,  13,   28,  11,  14,  12,
4  |   28,   28,   44,  19,  26,  24,   44,  19,   36,  26,   60,  24,  29,  23,
5  |   60,   60,   92,  40,  51,  52,   92,  40,   76,  51,  124,  52,  58,  44,
6  |  124,  124,  188,  84, 104, 108,  188,  84,  156, 104,  252, 108, 115,  84,
7  |  252,  252,  380, 172, 212, 220,  380, 172,  316, 212,  508, 220, 232, 165,
8  |  508,  508,  764, 348, 428, 444,  764, 348,  636, 428, 1020, 444, 468, 326,
9  | 1020, 1020, 1532, 700, 860, 892, 1532, 700, 1276, 860, 2044, 892, 940, 650,

Cf. A000523, A371094, A372282, A372356 (columnwise first differences), A372357.

Row 1 is 0 followed by A113473.

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)));
A372354sq(n,k) = A000523(A372282sq(n,k));
A372354list(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] = A372354sq((a-(col-1)),col))); (v); };
v372354 = A372354list(up_to);
A372354(n) = v372354[n];

A372451 a(n) = A372449(1+n) - A372449(n).

Original entry on oeis.org

3, 5, 11, 21, 40, 81, 161, 324, 647, 1293, 2587, 5172, 10346, 20691, 41380, 82761, 165521, 331045, 662088, 1324177, 2648354, 5296706, 10593414, 21186826, 42373653, 84747305, 169494608, 338989217, 677978434, 1355956869, 2711913737, 5423827472, 10847654948, 21695309895, 43390619792, 86781239585, 173562479173, 347124958345

Offset: 0

Antti Karttunen, May 05 2024

nonn

a(n) tells how many bits the length of the binary expansion grows when we go from A372444(n) to A372444(1+n).

Antti Karttunen, Table of n, a(n) for n = 0..1001

First differences of A372449.
Column 14 of A372356.
Cf. A000523, A372443, A372444.

A372451(n) = (A372449(1+n)-A372449(n)); \\ Needs also code from A372449.

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

cp's OEIS Frontend

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

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A372282 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1.

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A372354 Array read by upward antidiagonals: A(n, k) = A000523(A372282(n, k)), n,k >= 1, where A000523(x) is one less than the number of bits in the binary expansion of x.

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A372451 a(n) = A372449(1+n) - A372449(n).

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