Lucas multiplication table

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The entries of the $\scriptstyle n\,$th row of the Lucas multiplication table are defined as

$T(n,k):=L_{n}L_{k},\quad 1\leq k\leq n,\,n\geq 1,\,$

where $\scriptstyle L_{n}\,$ is the $\scriptstyle n\,$th Lucas number (A000032).

Lucas multiplication triangular table

In the table below, numbers that are 2 short of the square of a Lucas number are given a dashed black border, while those that are 2 over the square of a Lucas number are given a thick, solid border. The squares of Lucas numbers are given a thin, solid border. (The squares of the Lucas numbers are in A001254.)

																									A??????: Row sums
													1												1
												3		9											12
											4		12		16										32
										7		21		28		49									105
									11		33		44		77		121								286
								18		54		72		126		198		324							792
							29		87		116		203		319		522		841						2117
						47		141		188		329		517		846		1363		2209					5640
					76		228		304		532		836		1368		2204		3572		5776				14896
				123		369		492		861		1353		2214		3567		5781		9348		15129			39237

A180485 Lucas multiplication table as a triangle read by rows $T(n,k)=L(n)L(k)$, with $L(n)$ being a Lucas number (A000032).

{1, 3, 9, 4, 12, 16, 7, 21, 28, 49, 11, 33, 44, 77, 121, 18, 54, 72, 126, 198, 324, 29, 87, 116, 203, 319, 522, 841, 47, 141, 188, 329, 517, 846, 1363, 2209, 76, 228, 304, 532, ...}

An important difference from the Fibonacci multiplication table is that the near misses to the squares don't occur in as many columns.

Row sums of Lucas multiplication triangular table

The row sums

$\sum _{k=1}^{n}T(n,k)=L_{n}\sum _{k=1}^{n}L_{k},\,n\geq 1,\,$

give the sequence (A??????)

{1, 12, 32, 105, 286, 792, 2117, 5640, 14896, 39237, ...}

See also

• Lucas numbers
• Lucas polynomials

• Fibonacci numbers
• Fibonacci polynomials
• Fibonacci multiplication table