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

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

Author

Alex Ratushnyak, Dec 28 2013

Keywords

• nonn
• easy
• tabl

Examples

			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
		

Links

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

Crossrefs

Cf. A048158, A010766.

Programs

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

Mathematica

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

Python

for n in range(1, 19):
  for k in range(1, n+1):
    c = n//k + n%k
    print('%2d' % c, end=' ')
  print()

Python

def T(n, k) -> int: return n - (k - 1) * (n // k)
for n in range(1,16): print([T(n, k) for k in range(1,n+1)]) # Peter Luschny, Jun 01 2025

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

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

T(n, k) = n - (k - 1) * floor(n/k). - Peter Luschny, Jun 01 2025