A330827 -id:A330827 - cp's OEIS frontend

A331216 a(n) is the number of ways to write n = u + v where the binary representations of u and of v have the same number of 0's and the same number of 1's.

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 3, 2, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 3, 4, 3, 2, 5, 2, 3, 6, 3, 2, 5, 2, 3, 4, 3, 0, 3, 2, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 3, 4, 3, 2, 5, 2, 3, 6, 3, 4, 7, 2, 7, 6, 5

Offset: 0

Rémy Sigrist, Jan 12 2020

In other words, a(n) is the number of ways to write n as the sum of two binary anagrams.

Leading zeros are ignored.

			For n = 22:
- we can write 22 as u + v in the following ways:
  u   v   bin(u)  bin(v)
  --  --  ------  ------
  10  12    1010    1100
  11  11    1011    1011
  12  10    1100    1010
- hence a(22) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Rémy Sigrist, PARI program for A331216
Rémy Sigrist, Scatterplot of (x, y) such that 0 <= x, y <= 2^10 and x and y are binary anagrams (a(n) corresponds to the number of pixels (x, y) such that x+y = n)

Cf. A330827 (ternary analog), A331218 (decimal analog).

PARI
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a(2*n) > 0.
a(2*n) >= a(n).
Apparently, a(3*2^k-1-x) = a(3*2^k-1+x) for any k >= 0 and x = -2^k..2^k.

A331218 a(n) is the numbers of ways to write n = u + v where the decimal representations of u and of v have the same number of digits d for d = 0..9.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 4, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 5, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 6, 1, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 0

Rémy Sigrist, Jan 12 2020

In other words, a(n) is the number of ways to write n as the sum of two anagrams.

Leading zeros are ignored.

			For n = 44:
- we have the following ways to write 44 as a sum of two anagrams:
  u   v
  --  --
  13  31
  22  22
  31  13
- hence a(44) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..20000
Rémy Sigrist, PARI program for A331218
Rémy Sigrist, Scatterplot of (x, y) such that 0 <= x, y <= 10^3 and x and y are decimal anagrams (a(n) corresponds to the number of pixels (x, y) such that x+y = n)

Cf. A330827 (ternary analog), A331216 (binary analog).

PARI
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See Links section.
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A330831 a(n) = (F_n^2 - 1)^2, where F_n is a Fermat prime, A019434.

Original entry on oeis.org

64, 576, 82944, 4362338304, 18447869990796263424

Offset: 1

Walter Kehowski, Jan 06 2020

nonn

Also the second element of the power-spectral basis of A330829.

The first element of the power-spectral basis of A330829 is A330830.

			a(0) = (3^2 - 1)^2 = 64.

Cf. A000215, A001146, A019434, A330827, A330828, A330829, A330830.

F := proc(n) return 2^(2^n)+1 end;
a := proc(n) if isprime(F(n)) then return (F(n)^2-1)^2 fi; end;
[seq(a(n),n=0..4)];

a(n) = (F(n)^2 - 1)^2, where F(n) = 2^(2^n)+1 is a Fermat prime.

cp's OEIS Frontend

A331216 a(n) is the number of ways to write n = u + v where the binary representations of u and of v have the same number of 0's and the same number of 1's.

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A331218 a(n) is the numbers of ways to write n = u + v where the decimal representations of u and of v have the same number of digits d for d = 0..9.

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PARI

A330831 a(n) = (F_n^2 - 1)^2, where F_n is a Fermat prime, A019434.

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