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

1,2

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.

Links

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

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

Example

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
  ...

Mathematica

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

Prog

(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

Crossrefs

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

Cf. A010051.

Sequence in context: A003988 A185914 A144257 * A321980 A208544 A208535

Adjacent sequences: A257229 A257230 A257231 * A257233 A257234 A257235

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Apr 19 2015

Status

approved