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A286236
Square array A(n,k) = P(A000010(k), (n+k-1)/k) if k divides (n+k-1), 0 otherwise, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.
4
1, 1, 2, 3, 0, 4, 3, 0, 2, 7, 10, 0, 0, 0, 11, 3, 0, 0, 5, 4, 16, 21, 0, 0, 0, 0, 0, 22, 10, 0, 0, 0, 5, 0, 7, 29, 21, 0, 0, 0, 0, 0, 8, 0, 37, 10, 0, 0, 0, 0, 14, 0, 0, 11, 46, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 10, 0, 0, 0, 0, 0, 5, 0, 8, 12, 16, 67, 78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79, 21, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 22, 92, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 17, 0, 106
OFFSET
1,3
COMMENTS
This is transpose of A286237, see comments there.
FORMULA
T(n,k) = A113998(n,k) * A286234(n,k).
EXAMPLE
The top left 12 X 12 corner of the array:
1, 1, 3, 3, 10, 3, 21, 10, 21, 10, 55, 10
2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
7, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0
11, 4, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0
16, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0
22, 7, 8, 0, 0, 5, 0, 0, 0, 0, 0, 0
29, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0
37, 11, 0, 8, 0, 0, 0, 14, 0, 0, 0, 0
46, 0, 12, 0, 0, 0, 0, 0, 27, 0, 0, 0
56, 16, 0, 0, 19, 0, 0, 0, 0, 14, 0, 0
67, 0, 0, 0, 0, 0, 0, 0, 0, 0, 65, 0
The first 15 rows when viewed as a triangle:
1,
1, 2,
3, 0, 4,
3, 0, 2, 7,
10, 0, 0, 0, 11,
3, 0, 0, 5, 4, 16,
21, 0, 0, 0, 0, 0, 22,
10, 0, 0, 0, 5, 0, 7, 29,
21, 0, 0, 0, 0, 0, 8, 0, 37,
10, 0, 0, 0, 0, 14, 0, 0, 11, 46,
55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56,
10, 0, 0, 0, 0, 0, 5, 0, 8, 12, 16, 67,
78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79,
21, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 22, 92,
36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 17, 0, 106
PROG
(Scheme)
(define (A286236 n) (A286236bi (A002260 n) (A004736 n)))
(define (A286236bi row col) (if (not (zero? (modulo (+ row col -1) col))) 0 (let ((a (A000010 col)) (b (/ (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Alternatively, with triangular indexing:
(define (A286236 n) (A286236tr (A002024 n) (A002260 n)))
(define (A286236tr n k) (A286236bi k (+ 1 (- n k))))
(Python)
from sympy import totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def t(n, k): return 0 if n%k!=0 else T(totient(k), n/k)
for n in range(1, 21): print [t(n, k) for k in range(1, n + 1)][::-1] # Indranil Ghosh, May 10 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, May 05 2017
STATUS
approved