A086893 -id:A086893 - 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];

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.

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))).

A372361 Array read by upward antidiagonals: A(n, k) = A372358(A372283(n, k)), n,k >= 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 4, 0, 0, 0, 4, 2, 6, 0, 0, 0, 0, 6, 4, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 22, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 22, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 14

Offset: 1

Antti Karttunen, May 01 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
---+------------------------------------------------------------------------
1  | 0, 0, 0, 2, 4, 6, 0,  2, 4, 6, 0,  2, 12,  14, 8,  10, 20, 22, 16, 18,
2  | 0, 0, 0, 6, 2, 4, 0,  2, 0, 8, 0, 22,  6,  28, 6,  26, 12,  0,  2, 14,
3  | 0, 0, 0, 4, 6, 0, 0, 22, 0, 6, 0,  0,  8,  10, 4,  18,  6,  0,  6, 12,
4  | 0, 0, 0, 0, 4, 0, 0,  0, 0, 4, 0,  0,  6,  26, 0,  62,  8,  0,  4, 22,
5  | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  4,  18, 0, 116,  6,  0,  0, 48,
6  | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0,  62, 0,  44,  4,  0,  0,  6,
7  | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0, 116, 0,  14,  0,  0,  0,  8,
8  | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0,  44, 0,  92,  0,  0,  0,  6,
9  | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0,  14, 0,  50,  0,  0,  0,  4,
10 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0,  92, 0,  78,  0,  0,  0,  0,
11 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0,  50, 0,  60,  0,  0,  0,  0,
12 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0,  78, 0, 122,  0,  0,  0,  0,
13 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0,  60, 0,  82,  0,  0,  0,  0,
14 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0, 122, 0, 222,  0,  0,  0,  0,
15 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0,  82, 0, 260,  0,  0,  0,  0,
16 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0, 222, 0, 232,  0,  0,  0,  0,
17 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0, 260, 0, 114,  0,  0,  0,  0,
18 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0, 232, 0,  46,  0,  0,  0,  0,
19 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0, 114, 0,  44,  0,  0,  0,  0,
20 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0,  46, 0,  78,  0,  0,  0,  0,
21 | 0, 0, 0, 0, 0, 0, 0,  0, 0, 0, 0,  0,  0,  44, 0, 252,  0,  0,  0,  0,

Cf. A075677, A086893, A372283, A372358, A372360 (binary weights), A372446 (column 14).

Cf. also A372359.

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); \\ From A086893
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];

A372446 a(n) = A372358(A372443(n)).

Original entry on oeis.org

14, 28, 10, 26, 18, 62, 116, 44, 14, 92, 50, 78, 60, 122, 82, 222, 260, 232, 114, 46, 44, 78, 252, 106, 138, 410, 354, 774, 1064, 218, 2, 1366, 336, 276, 228, 16, 8, 2, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, May 03 2024

nonn

These are the bitmasks (or symmetric differences) obtained when the n-th iterate of 27 with Reduced Collatz-function R [= A372443(n), where R(n) = A000265(3*n+1)] is xored with that term of A086893 that has the same binary length. The binary expansions of the terms of A086893 are always of the form 10101...0101 (i.e., alternating 1's and 0's starting and ending with 1) when the binary length is odd, and of the form 110101...0101 (i.e., 1 followed by alternating 1's and 0's, and ending with 1) when n is even. Note that for all n >= 1, R(A086893(2n-1)) = 1, and R(A086893(2n)) = 5 (with R(5) = 1), so the first zero here, a(39) = 0 indicates that the iteration will soon have reached the terminal 1, and indeed, A372443(41) = 1.

Index entries for sequences related to 3x+1 (or Collatz) problem

Column 14 of A372361.

Cf. A000265, A075677, A086893, A372358, A372443.

A000265(n) = (n>>valuation(n,2));
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);
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A372446(n) = A372358(A372443(n));

A283642 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 678", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, 178956971, 357913941, 715827883, 1431655765, 2863311531, 5726623061, 11453246123

Offset: 0

Robert Price, Mar 12 2017

Initialized with a single black (ON) cell at stage zero.
Similar to A001045.
It is not difficult to prove that one has indeed a(n) = round(4*2^n/3) = A001045(n+2) for all n. The proof as well as the growth of the pattern is nearly identical to that of the toothpick sequence A139250. - M. F. Hasler, Feb 13 2020
The decimal representations of the n-th interval of elementary cellular automata rules 28 and 156 (see A266502 and A266508) generate this sequence. - Karl V. Keller, Jr., Sep 03 2021

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A283641, A266502, A266508, A086893, A001045.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 678; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]

print([(4*2**n + 1)//3 for n in range(50)]) # Karl V. Keller, Jr., Sep 03 2021

From Colin Barker, Mar 14 2017: (Start)
G.f.: (1 + 2*x) / ((1 + x)*(1 - 2*x)).
a(n) = (2^(n+2) - 1) / 3 for n even.
a(n) = (2^(n+2) + 1) / 3 for n odd.
a(n) = a(n-1) + 2*a(n-2) for n>1.
(End)
I.e., a(n) = A001045(n+2) = A154917(n+2) = A167167(n+2) = |A077925(n+1)| = A328284(n+5) = round(4*2^n/3), cf. comments. - M. F. Hasler, Feb 13 2020
E.g.f.: (4*exp(2*x) - exp(-x))/3. - Stefano Spezia, Feb 13 2020
a(n) = floor((4*2^n + 1)/3). - Karl V. Keller, Jr., Sep 03 2021

A087293 Numbers k such that ((prime(k)*prime(k+1))^2 + 1)/2 is prime.

Original entry on oeis.org

2, 3, 6, 14, 27, 30, 35, 37, 44, 55, 101, 106, 118, 127, 137, 140, 154, 172, 184, 233, 248, 256, 260, 289, 383, 389, 425, 461, 463, 485, 500, 503, 513, 552, 584, 610, 617, 630, 642, 696, 706, 714, 737, 746, 819, 884, 926, 952, 964, 978, 1004, 1008, 1019, 1027

Offset: 1

Ray Chandler, Aug 31 2003

nonn

Resulting primes are in A087294.

			2 is in the sequence because ((prime(2)*prime(3))^2 + 1)/2 = ((3*5)^2 + 1)/2 = 113 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A086893, A087294.

Select[Range[1100],PrimeQ[((Prime[#]Prime[#+1])^2+1)/2]&] (* Harvey P. Dale, Mar 18 2022 *)

Offset changed to 1 by Jinyuan Wang, Aug 06 2021

A177954 Triangle read by rows, A070909 * Pascal's triangle.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 3, 5, 4, 1, 3, 6, 7, 4, 1, 4, 11, 17, 14, 6, 1, 4, 12, 22, 24, 16, 6, 1, 5, 19, 43, 59, 51, 27, 8, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1, 6, 29, 86, 164, 212, 188, 113, 44, 10, 1

Offset: 0

Gary W. Adamson, May 15 2010

Row sums = A086893: (1, 3, 5, 13, 21, 53, 85,...).

			First few rows of the triangle =
1;
2, 1;
2, 2, 1;
3, 5, 4, 1;
3, 6, 7, 4, 1;
4, 11, 17, 14, 6, 1;
4, 12, 22, 24, 16, 6, 1;
5, 19, 43, 59, 51, 27, 8, 1;
5, 20, 50, 80, 86, 62, 29, 8, 1;
6, 29, 86, 164, 212, 188, 113, 44, 10, 1;
6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1;
7, 41, 150, 365, 626, 776, 701, 458, 211, 65, 12, 1;
...

Cf. A070909, A086893

Triangle read by rows, A070909 * A007318

a(46) corrected by Georg Fischer, May 20 2022

A320642 Number of 1's in the base-(-2) expansion of -n.

Original entry on oeis.org

2, 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 5, 4, 3, 2, 4, 3, 5, 4, 6, 5, 4, 3, 5, 4, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 5, 4, 3, 2, 4, 3, 5, 4, 6, 5, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 6, 5, 4, 3, 5, 4, 3, 2, 4, 3, 5, 4, 6, 5, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 6, 5, 7, 6, 8, 7, 6

Offset: 1

Jianing Song, Oct 18 2018

Number of 1's in A212529(n).
Define f(n) as: f(0) = 0, f(-2*n) = f(n), f(-2*n+1) = f(n) + 1, then a(n) = f(-n), n >= 1. See A027615 for the other half of f.
For k > 1, the earliest occurrence of k is n = A086893(k-1).

			A212529(11) = 110101 which has four 1's, so a(11) = 4.
A212529(25) = 111011 which has five 1's, so a(25) = 5.
A212529(51) = 11011101 which has six 1's, so a(51) = 6.

Jianing Song, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Negabinary.
Eric Weisstein's World of Mathematics, Negadecimal.
Wikipedia, Negative base.

Cf. A000120, A005352, A027615, A086893, A212529.

b[n_] := b[n] = b[Quotient[n - 1, -2]] + Mod[n, 2]; b[0] = 0; a[n_] := b[-n]; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

b(n) = if(n==0, 0, b(n\(-2))+n%2)
a(n) = b(-n)

a(n) == -n (mod 3).

a(n) = A000120(A005352(n)). - Michel Marcus, Oct 23 2018

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

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A372361 Array read by upward antidiagonals: A(n, k) = A372358(A372283(n, k)), n,k >= 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A372446 a(n) = A372358(A372443(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A283642 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 678", based on the 5-celled von Neumann neighborhood.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A087293 Numbers k such that ((prime(k)*prime(k+1))^2 + 1)/2 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A177954 Triangle read by rows, A070909 * Pascal's triangle.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A320642 Number of 1's in the base-(-2) expansion of -n.

Views