login
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.
3

%I #19 Sep 23 2021 01:26:53

%S 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,

%T 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

%N 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.

%C This triangle is motivated by sequence A256885 by _Wesley Ivan Hurt_, which is the sequence of row sums for n >= 1.

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

%C 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.

%H Reinhard Zumkeller, <a href="/A257232/b257232.txt">Rows n = 1..125 of triangle, flattened</a>

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

%F 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.

%F 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

%e The triangle T(n, k) begins:

%e n\k 1 2 3 4 5 6 7 8 9 10 11 ...

%e 1: 1

%e 2: 2 0

%e 3: 3 1 0

%e 4: 4 2 1 1

%e 5: 5 3 2 2 0

%e 6: 6 4 3 3 1 1

%e 7: 7 5 4 4 2 2 0

%e 8: 8 6 5 5 3 3 1 1

%e 9: 9 7 6 6 4 4 2 2 1

%e 10: 10 8 7 7 5 5 3 3 2 1

%e 11: 11 9 8 8 6 6 4 4 3 2 0

%e ...

%t Table[n - (k - 1) - Boole[PrimeQ@ k], {n, 11}, {k, n}] // Flatten (* _Michael De Vlieger_, Apr 19 2015 *)

%o (Haskell)

%o a257232 n k = a257232_tabl !! (n-1) !! (k-1)

%o a257232_row n = a257232_tabl !! (n-1)

%o a257232_tabl = iterate

%o (\xs@(x:_) -> map (+ 1) xs ++ [1 - a010051 (x + 1)]) [1]

%o -- _Reinhard Zumkeller_, Apr 21 2015

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

%Y Cf. A010051.

%K nonn,easy,tabl

%O 1,2

%A _Wolfdieter Lang_, Apr 19 2015