A257232 Triangle T(n, k) with the natural numbers in columns with nonprime k and the nonnegative numbers in columns with prime k, for 1 <= k <= n.

1, 2, 0, 3, 1, 0, 4, 2, 1, 1, 5, 3, 2, 2, 0, 6, 4, 3, 3, 1, 1, 7, 5, 4, 4, 2, 2, 0, 8, 6, 5, 5, 3, 3, 1, 1, 9, 7, 6, 6, 4, 4, 2, 2, 1, 10, 8, 7, 7, 5, 5, 3, 3, 2, 1, 11, 9, 8, 8, 6, 6, 4, 4, 3, 2, 0

Offset: 1

Author

Wolfdieter Lang, Apr 19 2015

Keywords

• nonn
• easy
• tabl

Comments

This triangle is motivated by sequence A256885 by Wesley Ivan Hurt, which is the sequence of row sums for n >= 1.

Row n ends in a 0 if n is prime; otherwise it ends in 1.

The alternating row sums give 1, seven times 2, six times 3, six times 4, four times 5, twice 6, ..., and the multiplicity sequence 1, 7, 6, 6, 4, 2, ... is given in A257233.

Examples

			The triangle T(n, k) begins:
  n\k  1  2  3  4  5  6  7  8  9 10 11 ...
   1:  1
   2:  2  0
   3:  3  1  0
   4:  4  2  1  1
   5:  5  3  2  2  0
   6:  6  4  3  3  1  1
   7:  7  5  4  4  2  2  0
   8:  8  6  5  5  3  3  1  1
   9:  9  7  6  6  4  4  2  2  1
  10: 10  8  7  7  5  5  3  3  2  1
  11: 11  9  8  8  6  6  4  4  3  2  0
  ...
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Crossrefs

Cf. A256885 (row sums), A257233 (multiplicities for alternating row sums).

Cf. A010051.

Programs

Haskell

a257232 n k = a257232_tabl !! (n-1) !! (k-1)
a257232_row n = a257232_tabl !! (n-1)
a257232_tabl = iterate
               (\xs@(x:_) -> map (+ 1) xs ++ [1 - a010051 (x + 1)]) [1]
-- Reinhard Zumkeller, Apr 21 2015

Mathematica

Table[n - (k - 1) - Boole[PrimeQ@ k], {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Apr 19 2015 *)

Formula

T(n, k) = n - (k-1) - [isprime(k)], with [isprime(k)] = A010051(k), for 1 <= k <= n.

O.g.f. for column k (with leading zeros): x^k/(1-x)^2 if k is nonprime, otherwise x^(k+1)/(1-x)^2.

T(n+1,k) = T(n,k) + 1, 1 <= k <= n, T(n+1,n+1) = 1 - A010051(n+1). - Reinhard Zumkeller, Apr 21 2015