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.

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. A001622, A101330, A189663, A339765.

Cf. A000201 (row 1), A003623 (row 2), A190509 (row 3), A371388 (main diagonal).

Sequence in context: A329216 A266616 A360846 * A147679 A339054 A137417

Adjacent sequences: A340426 A340427 A340428 * A340430 A340431 A340432

Keyword

nonn,tabl

Author

Michel Marcus, Jan 07 2021

Status

approved