A088512 -id:A088512 - cp's OEIS frontend

A285720 Number of ways to write n as a sum of two unordered squarefree numbers so that their addition in base-2 does not produce carries.

0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 11, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 7, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 0, 11, 0, 0, 0, 3, 0, 0, 0, 7, 0, 0, 0, 7, 0, 0, 0, 13, 0, 0, 0, 3, 0, 0, 0, 9, 0

Offset: 1

Antti Karttunen, May 02 2017

Antti Karttunen, Table of n, a(n) for n = 1..4095
Index entries for sequences related to binary expansion of n

Cf. A003987, A004198, A005117, A008683, A008966, A013928, A071068, A088512.

Table[Sum[Abs[MoebiusMu[i] MoebiusMu[n - i]] Boole[BitXor[i, n - i] == n], {i, Floor[n/2]}], {n, 120}] (* Michael De Vlieger, May 03 2017 *)

from sympy import mobius
def a003987(n, i): return i^(n - i) == n
def a(n): return sum([abs(mobius(i)*mobius(n - i))*(1*a003987(n, i)) for i in range(1, n//2 + 1)])
print([a(n) for n in range(1,121)]) # Indranil Ghosh, May 02 2017

(define (A285720 n) (let loop ((k (A013928 n)) (s 0)) (if (or (zero? k) (< (A005117 k) (- n (A005117 k)))) s (loop (- k 1) (+ s (if (and (= 1 (A008966 (- n (A005117 k)))) (zero? (A004198bi (A005117 k) (- n (A005117 k))))) 1 0)))))) ;; Where A004198bi implements bitwise-AND (A004198).

a(n) = Sum_{i=1..floor(n/2)} abs(mu(i)*mu(n-i))*[A003987(i,n-i) == n]. (Here [] is Iverson bracket, giving in this case 1 only if (i XOR (n-i)) is equal to n, and 0 otherwise. mu is Moebius mu function, A008683.)
a(n) <= A071068(n).
a(n) <= A088512(n).

A353292 a(n) is the number of positive integers k <= n that have at least one common 1-bit with n.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 5, 7, 1, 6, 7, 10, 9, 12, 13, 15, 1, 10, 11, 16, 13, 18, 19, 22, 17, 22, 23, 26, 25, 28, 29, 31, 1, 18, 19, 28, 21, 30, 31, 36, 25, 34, 35, 40, 37, 42, 43, 46, 33, 42, 43, 48, 45, 50, 51, 54, 49, 54, 55, 58, 57, 60, 61, 63, 1, 34, 35, 52, 37

Offset: 0

Rémy Sigrist, Apr 09 2022

See A353293 for the corresponding k's.

			For n = 10:
- we have:
      k   10 AND k
      --  --------
       1         0
       2         2
       3         2
       4         0
       5         0
       6         2
       7         2
       8         8
       9         8
      10        10
- so a(10) = #{2, 3, 6, 7, 8, 9, 10} = 7.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A062050, A080100, A088512, A115378, A353293.

a(n) = { my (h=hammingweight(n), w=#binary(n)); n-2^(w-1)+1 + (2^(h-1)-1)*2^(w-h) }

a(n) = n - A115378(n) for any n > 0.
a(n) = A062050(n) + A088512(n) * A080100(n) for any n > 0.
a(2^k) = 1 for any k >= 0.
a(2^k - 1) = 2^k - 1 for any k >= 0.

A262260 Number of triangles formed by the positions of odd numbers in the first n rows of Pascal's triangle, also known as Tartaglia's triangle.

Original entry on oeis.org

0, 1, 1, 4, 4, 6, 6, 13, 13, 15, 15, 21, 21, 25, 25, 40, 40, 42, 42, 48, 48, 52, 52, 66, 66, 70, 70, 82, 82, 90, 90, 121, 121, 123, 123, 129, 129, 133, 133, 147, 147, 151, 151, 163, 163, 171, 171, 201, 201, 205, 205, 217, 217, 225, 225, 253, 253, 261, 261, 285, 285, 301, 301, 364, 364

Offset: 0

Emmanuele Villa, Nov 09 2016

Named Tartaglia's triangle after the Italian mathematician Niccolò Fontana Tartaglia (1500-1577). - Amiram Eldar, Jun 11 2021

			Taking Pascal's triangle, removing the even terms and replacing each odd term with a dot, will give you this illustration (the circles are connected with lines to show the sub-triangles):
                                        triangle counts
                                        ---------------
   row                                     new total
   ===                                     === =====
    0                  o                    0    0
                      / \
    1                o---o                  1    1
                    /     \
    2              o       o                0    1
                  / \     / \
    3            o---o---o---o              3    4
                /             \
    4          o               o            0    4
              / \             / \
    5        o---o           o---o          2    6
            /     \         /     \
    6      o       o       o       o        0    6
          / \     / \     / \     / \
    7    o---o---o---o---o---o---o---o      7   13
        /                             \
    8  o                               o    0   13
.
.
Formula example:
given a(46) = 171, a(47) is computed as follows:
A = A001316(46) = 16
B = A001316(44) = 8
C = A001316(44) - 1 = 7
D = A001316((47+1-32)/8) - 1 = 1
a(47) = 171 + 16 + 8 + 7 - 1 = 201
.
.
You can find results for a(n), A, B, C and D in the links section for the first 500 rows.

Riccardo Perego and Emmanuele Villa, Tartaglia's Triangle Odd Distribution (in Italian), 2012.
Jon E. Schoenfield, Plot of terms through n=2^14.
Emmanuele Villa, A,B,C,D coefficients for the first 500 rows.
Emmanuele Villa, C# Program that calculates the first 50 rows.
Wikipedia, Pascal's Triangle.

Cf. A000120, A001316, A001317, A006943, A088512.

Empirical formula:
a(0)=0; a(1)=1; for n>1, a(n) = a(n-1) + A + B + C - D
where
A = A001316(n-1) if n = 2x+1, 0 otherwise
B = A001316(n-3) if n = 4x+1, 0 otherwise
C = B-1 if n = 8x+1, 0 otherwise
D = A088512(n+1) = A001316((n+1-m)/8)-1 if n = 8x+1, 0 otherwise, where m is the highest power of 2 less than n.

A335979 Number of partitions of n into exactly two parts with no decimal carries.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 2, 4, 7, 9, 12, 14, 17, 19, 22, 24, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 3, 6, 10, 13, 17, 20, 24, 27, 31, 34, 3, 7, 11, 15, 19, 23

Offset: 0

Jason Zimba, Jul 02 2020

a(m) = a(n) if m and n have the same nonzero digits, irrespective of order. For example, a(6044005) = a(45604).

			a(31) = 3 because there are three partitions of 31 into exactly two parts with no decimal carries: 30 + 1, 21 + 10, and 20 + 11.
a(100) = 0 because every partition of 100 into exactly two parts has at least one decimal carry.

Cf. A088512 (analogous sequence for base 2), A089898.

Ceiling[(1/2) Times @@ (IntegerDigits[n, 10] + 1)] - 1

If n has digits n_1, n_2, ..., n_k and all digits n_i are even, then a(n) = (1/2)(n_1 + 1)(n_2 + 1)...(n_k + 1) - 1/2. Otherwise, a(n) = (1/2)(n_1 + 1)(n_2 + 1)...(n_k + 1) - 1. Equivalently, a(n) = ceiling((1/2)(n_1 + 1)(n_2 + 1)...(n_k + 1)) - 1 for all n.

a(n) = ceiling((1/2)*A089898(n)) - 1.

cp's OEIS Frontend

A285720 Number of ways to write n as a sum of two unordered squarefree numbers so that their addition in base-2 does not produce carries.

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A353292 a(n) is the number of positive integers k <= n that have at least one common 1-bit with n.

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A262260 Number of triangles formed by the positions of odd numbers in the first n rows of Pascal's triangle, also known as Tartaglia's triangle.

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Examples

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Crossrefs

Formula

A335979 Number of partitions of n into exactly two parts with no decimal carries.

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Mathematica

Formula