Fibonacci multiplication table

With $\scriptstyle F_{n}\,$ defined as the $\scriptstyle n\,$^th Fibonacci number (Cf. A000045,) the Fibonacci multiplication table entries are defined by the formula

$T(n,k)\equiv F_{n}F_{k}.\,$

Row sums of Fibonacci multiplication triangular table

The row sums give the sequence (Cf. A143212)

{1, 2, 8, 21, 60, 160, 429, 1134, 2992, 7865, 20648, 54144, 141897, 371722, 973560, 2549421, 6675460, 17478176, 45761045, 119808150, 313668576, 821205937, ...}

which is given by the formula

$\sum _{k=0}^{n}T(n,k)=F_{n}(F_{n+2}-1).\,$

In the table below, numbers that are 1 short of the square of a Fibonacci number are given a dashed black border, while those that are 1 over the square of a Fibonacci number are given a thick, solid border. The squares of Fibonacci numbers are given a thin, solid border. (The squares of the Fibonacci numbers are in A007598).

This suggests two equations for the squares of the Fibonacci numbers:

$(F_{n})^{2}=F_{n-1}F_{n+1}-(-1)^{n}=F_{n-2}F_{n+2}+(-1)^{n}\,$.

This can be generalized still further to:

$(F_{n})^{2}=F_{n-i}F_{n+i}+(F_{i})^{2}(-1)^{i+n}.\,$

																									A143212: Row sums
													1												1
												1		1											2
											2		2		4										8
										3		3		6		9									21
									5		5		10		15		25								60
								8		8		16		24		40		64							160
							13		13		26		39		65		104		169						429
						21		21		42		63		105		168		273		441					1134
					34		34		68		102		170		272		442		714		1156				2992
				55		55		110		165		275		440		715		1155		1870		3025			7865
			89		89		178		267		445		712		1157		1869		3026		4895		7921		20648
		144		144		288		432		720		1152		1872		3024		4896		7920		12816		20736	54144
	233		233		466		699		A022088: Fibonacci-type sequence with initial values 0, 5. (The first four positive values are in row 5).																141897
377		377		754		A022086: Fibonacci-type sequence with initial values 0, 3. (The first three positive values are in row 3).																			371722
610		610		A055389: Twice the Fibonacci numbers																					973560
987		A000045: The Fibonacci numbers																							2549421
A000045: The Fibonacci numbers																									6675460

The triangle read by rows is at A143211.

See also

• Fibonacci numbers
• Fibonacci polynomials