A101385 Array read by antidiagonals: T(n,k) = variant of Knuth's Fibonacci (or circle) product of n and k (A101330).

3, 8, 8, 21, 34, 21, 24, 144, 144, 24, 55, 152, 987, 152, 55, 58, 610, 1008, 1008, 610, 58, 63, 618, 6765, 1032, 6765, 618, 63, 144, 644, 6786, 6820, 6820, 6786, 644, 144, 147, 2584, 6909, 6844, 75025, 6844, 6909, 2584, 147, 152, 2592, 46368, 6972, 75080

Offset: 1

Author

N. J. A. Sloane, Jan 25 2005

Keywords

• nonn
• tabl
• easy

Comments

Let n = Sum_{i >= 2} eps(i) Fib_i and k = Sum_{j >= 2} eps(j) Fib_j be the Zeckendorf expansions of n and k, respectively (cf. A035517, A014417). The product of n and k is defined here to be Sum_{i,j} eps(i)*eps(j) Fib_{i*j} (= T(n,k)).

Examples

			Array begins:
3 8 21 24 55 ...
8 34 144 152 ...
21 144 987 ...
24 152 ...
55 ...
		

Links

• D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60.

Crossrefs

Cf. A101330, A035517, A014417. Main diagonal is A101633.

First 3 rows give A101643, A101644, A101645.

Programs

Mathematica

zeck[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[ fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr]]; kfpv[n_, m_] := Block[{y = Reverse[ IntegerDigits[ zeck[ n]]], z = Reverse[ IntegerDigits[ zeck[ m]]]}, Sum[ y[[i]]*z[[j]]*Fibonacci[(i + 1)(j + 1)], {i, Length[y]}, {j, Length[z]}]]; (* Robert G. Wilson v, Feb 09 2005 *)
Flatten[ Table[ kfpv[i, n - i], {n, 2, 12}, {i, n - 1, 1, -1}]] (* Robert G. Wilson v, Feb 09 2005 *)

Extensions

More terms from David Applegate, Jan 26 2005