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A340429
Array T(n, k) is the number x such that frac(x*phi) + frac(n*phi)*frac(k*phi) = 1 where phi is the golden ratio A001622 and frac(y) is the fractional part of y, read by antidiagonals.
3
1, 3, 3, 4, 8, 4, 6, 11, 11, 6, 8, 16, 15, 16, 8, 9, 21, 22, 22, 21, 9, 11, 24, 29, 32, 29, 24, 11, 12, 29, 33, 42, 42, 33, 29, 12, 14, 32, 40, 48, 55, 48, 40, 32, 14, 16, 37, 44, 58, 63, 63, 58, 44, 37, 16, 17, 42, 51, 64, 76, 72, 76, 64, 51, 42, 17
OFFSET
1,2
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).
FORMULA
T(n, k) = 2*n*k + A339765(n)*k/2 + A339765(k)*n/2.
T(n, k) = T(k, n), array is symmetric.
T(n, k) = 3*n*k - n*h(k) - k*h(n) where h(n) = ceiling(2*n / (sqrt(5) + 3)) = A189663(n + 1). - Peter Luschny, Mar 21 2024
EXAMPLE
Array begins:
1 3 4 6 8 ...
3 8 11 16 21 ...
4 11 15 22 29 ...
6 16 22 32 42 ...
8 21 29 42 55 ...
...
MAPLE
h := n -> ceil(2*n / (sqrt(5) + 3)):
T := (n, k) -> 3*n*k - n*h(k) - k*h(n):
seq(lprint(seq(T(n, k), k = 1..9)), n = 1..7); # Peter Luschny, Mar 21 2024
MATHEMATICA
A340429[n_, k_] := Floor[n * GoldenRatio] * k + Floor[k * GoldenRatio] * n - n * k;
Table[A340429[n - k + 1, k], {n, 15}, {k, n}] (* Paolo Xausa, Mar 21 2024 *)
PROG
(PARI) f(n) = 2*floor(n*(1+sqrt(5))/2) - 3*n; \\ A339765
T(n, k) = 2*n*k + f(n)*k/2 + f(k)*n/2;
CROSSREFS
Cf. A000201 (row 1), A003623 (row 2), A190509 (row 3), A371388 (main diagonal).
Sequence in context: A329216 A266616 A360846 * A147679 A339054 A137417
KEYWORD
nonn,tabl
AUTHOR
Michel Marcus, Jan 07 2021
STATUS
approved