A080079 -id:A080079 - cp's OEIS frontend

A328732 Irregular table read by rows; for any n >= 0, the n-th row contains the numbers of the form u - v with u + v = n and u AND v = 0 (where AND denotes the bitwise AND operator), in ascending order.

0, -1, 1, -2, 2, -3, -1, 1, 3, -4, 4, -5, -3, 3, 5, -6, -2, 2, 6, -7, -5, -3, -1, 1, 3, 5, 7, -8, 8, -9, -7, 7, 9, -10, -6, 6, 10, -11, -9, -7, -5, 5, 7, 9, 11, -12, -4, 4, 12, -13, -11, -5, -3, 3, 5, 11, 13, -14, -10, -6, -2, 2, 6, 10, 14

Offset: 0

Rémy Sigrist, Oct 26 2019

The n-th row:
- has A001316(n) terms,
- has -n as first term and n as last term,
- has least positive term T(n, A001316(n)/2+1) = A080079(n) (when n > 0).

			Table begins:
    0;
   -1,   1;
   -2,   2;
   -3,  -1,   1,   3;
   -4,   4;
   -5,  -3,   3,   5;
   -6,  -2,   2,   6;
   -7,  -5,  -3,  -1,   1,   3,   5,   7;
   -8,   8;
   -9,  -7,   7,   9;
  -10,  -6,   6,  10;
  -11,  -9,  -7,  -5,   5,   7,   9,  11;
  -12,  -4,   4,  12;
  -13, -11,  -5,  -3,   3,   5,  11,  13;
  ...

Rémy Sigrist, Table of n, a(n) for n = 0..6562 (Rows for n = 0..256)
Rémy Sigrist, Scatterplot of (n, T(n, k)) for n = 0..1024 and k = 1..A001316(n)

Cf. A001316, A080079.

row(n) = my (r=[0], b=Vecrev(binary(n))); for (k=0, #b-1, if (b[k+1], r=concat(apply(v -> [v-2^k,v+2^k], r)))); Set(r)

A348647 Irregular table read by rows; the n-th row contains the lengths of the runs of consecutive terms with the same parity in the n-th row of Pascal's triangle (A007318).

Original entry on oeis.org

1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 7, 1, 2, 6, 2, 1, 1, 1, 5, 1, 1, 1, 4, 4, 4, 1, 3, 1, 3, 1, 3, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 15, 1, 2, 14, 2, 1, 1, 1, 13, 1, 1, 1, 4, 12, 4, 1, 3, 1, 11, 1, 3, 1

Offset: 0

Rémy Sigrist, Oct 27 2021

For any n >= 0, the n-th row:
- is palindromic,
- has A038573(n+1) terms,
- has leading term A006519(n+1),
- has central term A080079(n+1).

			Triangle begins:
    1;
    2;
    1, 1, 1;
    4;
    1, 3, 1;
    2, 2, 2;
    1, 1, 1, 1, 1, 1, 1;
    8;
    1, 7, 1;
    2, 6, 2;
    1, 1, 1, 5, 1, 1, 1;
    4, 4, 4;
    1, 3, 1, 3, 1, 3, 1;
    2, 2, 2, 2, 2, 2, 2;
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
    16;
    ...

Rémy Sigrist, Table of n, a(n) for n = 0..6304 (rows for n = 0..254, flattened)
Index entries for triangles and arrays related to Pascal's triangle

Cf. A006519, A007318, A038573, A047999, A048896, A080079.

row(n) = { my (b=binomial(n)%2, r=[], p=1, w=1); for (k=2, #b, if (p==b[k], w++, r=concat(r, w); p=b[k]; w=1)); concat(r, w) }

Sum_{k = 1..A038573(n+1)} T(n, k) = n+1.
T(n, 1) = A006519(n+1).
T(n, A048896(n)) = A080079(n+1).
T(2^k-1, 1) = 2^k for any k >= 0.

A333906 For n >= 2, a(n) = Sum_{k=2..n} prevpower2(k) + nextpower2(k) - 2*k, where prevpower2(k) is the largest power of 2 < k, nextpower2(k) is the smallest power of 2 > k.

Original entry on oeis.org

1, 1, 3, 5, 5, 3, 7, 13, 17, 19, 19, 17, 13, 7, 15, 29, 41, 51, 59, 65, 69, 71, 71, 69, 65, 59, 51, 41, 29, 15, 31, 61, 89, 115, 139, 161, 181, 199, 215, 229, 241, 251, 259, 265, 269, 271, 271, 269, 265, 259, 251, 241, 229, 215, 199, 181, 161

Offset: 2

Ctibor O. Zizka, Apr 09 2020

nonn

Partial sums of b(k) = prevpower2(k) + nextpower2(k) - 2*k; b(k) = 0 for A007283.

			a(2) = (1 + 4 - 2*2) = 1;
a(3) = (1 + 4 - 2*2) + (2 + 4 - 2*3) = 1;
a(4) = (1 + 4 - 2*2) + (2 + 4 - 2*3) + (2 + 8 - 2*4) = 3.

Cf. A000079, A007283, A062050, A080079.

cp's OEIS Frontend

A328732 Irregular table read by rows; for any n >= 0, the n-th row contains the numbers of the form u - v with u + v = n and u AND v = 0 (where AND denotes the bitwise AND operator), in ascending order.

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A348647 Irregular table read by rows; the n-th row contains the lengths of the runs of consecutive terms with the same parity in the n-th row of Pascal's triangle (A007318).

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A333906 For n >= 2, a(n) = Sum_{k=2..n} prevpower2(k) + nextpower2(k) - 2*k, where prevpower2(k) is the largest power of 2 < k, nextpower2(k) is the smallest power of 2 > k.

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