A234575 Triangle T(n,k) read by rows: T(n,k) = floor(n/k) + n mod k, with 1<=k<=n.

1, 2, 1, 3, 2, 1, 4, 2, 2, 1, 5, 3, 3, 2, 1, 6, 3, 2, 3, 2, 1, 7, 4, 3, 4, 3, 2, 1, 8, 4, 4, 2, 4, 3, 2, 1, 9, 5, 3, 3, 5, 4, 3, 2, 1, 10, 5, 4, 4, 2, 5, 4, 3, 2, 1, 11, 6, 5, 5, 3, 6, 5, 4, 3, 2, 1, 12, 6, 4, 3, 4, 2, 6, 5, 4, 3, 2, 1, 13, 7, 5, 4, 5, 3, 7, 6, 5

Offset

1,2

Links

Antti Karttunen, Rows n = 1..144 of triangular table, flattened

Formula

T(n,k) = A048158(n,k) + A010766(n,k). - Reinhard Zumkeller, Apr 29 2015

G.f. of the k-th column: x^k*((Sum_{i=0..k-1} x^i) - (k-1)*x^k)/((1 - x)^2*Sum_{i=0..k-1} x^i). - Stefano Spezia, May 08 2024

Example

Triangle begins:
   1
   2  1
   3  2  1
   4  2  2  1
   5  3  3  2  1
   6  3  2  3  2  1
   7  4  3  4  3  2  1
   8  4  4  2  4  3  2  1
   9  5  3  3  5  4  3  2  1
  10  5  4  4  2  5  4  3  2  1
  11  6  5  5  3  6  5  4  3  2  1
  12  6  4  3  4  2  6  5  4  3  2  1
  13  7  5  4  5  3  7  6  5  4  3  2  1
  14  7  6  5  6  4  2  7  6  5  4  3  2  1
  15  8  5  6  3  5  3  8  7  6  5  4  3  2  1

Mathematica

With[{rows=10}, Table[Floor[n/k]+Mod[n, k], {n, rows}, {k, n}]] (* Paolo Xausa, Sep 26 2023 *)

Prog

(Python)
for n in range(1, 19):
  for k in range(1, n+1):
    c = n//k + n%k
    print('%2d' % c, end=' ')
print()
(Scheme)
;; MIT/GNU Scheme
(define (A234575bi n k) (+ (floor->exact (/ n k)) (modulo n k)))
(define (A234575 n) (A234575bi (A002024 n) (A002260 n)))
;; Antti Karttunen, Dec 29 2013
(Haskell)
a234575 n k = a234575_tabl !! (n-1) !! (k-1)
a234575_row n = a234575_tabl !! (n-1)
a234575_tabl = zipWith (zipWith (+)) a048158_tabl a010766_tabl
-- Reinhard Zumkeller, Apr 29 2015

Crossrefs

Cf. A048158, A010766.

Sequence in context: A082404 A334725 A120885 * A294733 A275724 A193278

Adjacent sequences: A234572 A234573 A234574 * A234576 A234577 A234578

Keyword

nonn,easy,tabl,changed

Author

Alex Ratushnyak, Dec 28 2013

Status

approved