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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.

%I A340429 #15 Mar 21 2024 08:31:39
%S A340429 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,
%T A340429 11,12,29,33,42,42,33,29,12,14,32,40,48,55,48,40,32,14,16,37,44,58,63,
%U A340429 63,58,44,37,16,17,42,51,64,76,72,76,64,51,42,17
%N 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.
%H A340429 Paolo Xausa, <a href="/A340429/b340429.txt">Table of n, a(n) for n = 1..11325</a> (first 150 antidiagonals, flattened).
%F A340429 T(n, k) = 2*n*k + A339765(n)*k/2 + A339765(k)*n/2.
%F A340429 T(n, k) = T(k, n), array is symmetric.
%F A340429 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
%e A340429 Array begins:
%e A340429   1  3  4  6  8 ...
%e A340429   3  8 11 16 21 ...
%e A340429   4 11 15 22 29 ...
%e A340429   6 16 22 32 42 ...
%e A340429   8 21 29 42 55 ...
%e A340429   ...
%p A340429 h := n -> ceil(2*n / (sqrt(5) + 3)):
%p A340429 T := (n, k) -> 3*n*k - n*h(k) - k*h(n):
%p A340429 seq(lprint(seq(T(n, k), k = 1..9)), n = 1..7);  # _Peter Luschny_, Mar 21 2024
%t A340429 A340429[n_, k_] := Floor[n * GoldenRatio] * k + Floor[k * GoldenRatio] * n - n * k;
%t A340429 Table[A340429[n - k + 1, k], {n, 15}, {k, n}] (* _Paolo Xausa_, Mar 21 2024 *)
%o A340429 (PARI) f(n) = 2*floor(n*(1+sqrt(5))/2) - 3*n; \\ A339765
%o A340429 T(n, k) = 2*n*k + f(n)*k/2 + f(k)*n/2;
%Y A340429 Cf. A001622, A101330, A189663, A339765.
%Y A340429 Cf. A000201 (row 1), A003623 (row 2), A190509 (row 3), A371388 (main diagonal).
%K A340429 nonn,tabl
%O A340429 1,2
%A A340429 _Michel Marcus_, Jan 07 2021