A135090 Array read by antidiagonals: T(n, k) = Knuth's Fibonacci (or circle) product of n and k ("n o k"), n >= 0, k >= 0.
0, 0, 0, 0, 3, 0, 0, 5, 5, 0, 0, 8, 8, 8, 0, 0, 11, 13, 13, 11, 0, 0, 13, 18, 21, 18, 13, 0, 0, 16, 21, 29, 29, 21, 16, 0, 0, 18, 26, 34, 40, 34, 26, 18, 0, 0, 21, 29, 42, 47, 47, 42, 29, 21, 0, 0, 24, 34, 47, 58, 55, 58, 47, 34, 24, 0, 0, 26, 39, 55, 65, 68, 68, 65, 55, 39, 26, 0, 0, 29, 42, 63, 76, 76, 84, 76, 76, 63, 42, 29, 0, 0, 32, 47
Offset: 0
Examples
Array begins: n\k | 0 1 2 3 4 5 6 7 8 9 ... ----+------------------------------------------------ 0 | 0 0 0 0 0 0 0 0 0 0 ... 1 | 0 3 5 8 11 13 16 18 21 24 ... 2 | 0 5 8 13 18 21 26 29 34 39 ... 3 | 0 8 13 21 29 34 42 47 55 63 ... 4 | 0 11 18 29 40 47 58 65 76 87 ... 5 | 0 13 21 34 47 55 68 76 89 102 ... 6 | 0 16 26 42 58 68 84 94 110 126 ... 7 | 0 18 29 47 65 76 94 105 123 141 ... 8 | 0 21 34 55 76 89 110 123 144 165 ... 9 | 0 24 39 63 87 102 126 141 165 189 ... ...
Links
- Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).
Programs
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Maple
h := n -> floor(2*(n + 1)/(sqrt(5) + 3)): # A060144(n+1) T := (n, k) -> 3*n*k - n*h(k) - k*h(n): seq(print(seq(T(n, k), k = 0..9)), n = 0..7); # Peter Luschny, Mar 21 2024
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Mathematica
A135090[n_, k_] := 3*n*k - n*Floor[(k + 1) / GoldenRatio^2] - k*Floor[(n + 1) / GoldenRatio^2]; Table[A135090[n-k, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Mar 21 2024 *)
Formula
T(n, k) = 3*n*k - n*h(k) - k*h(n) where h(n) = A060144(n + 1). - Peter Luschny, Mar 21 2024
Comments