ペラン数
↑ Füredi, Z. (1987). “The number of maximal independent sets in connected graphs”. Journal of Graph Theory 11 (4): 463–470. doi:10.1002/jgt.3190110403.
↑ Lucas, E. (1878). “Théorie des fonctions numériques simplement périodiques”. American Journal of Mathematics 1: 197–240. doi:10.2307/2369311.
↑ Perrin, R. (1899). “Query 1484”. L'Intermediaire Des Mathematiciens 6: 76.
1 2 Adams, William; Shanks, Daniel (1982). “Strong primality tests that are not sufficient”. Mathematics of Computation 39 (159): 255–300. doi:10.2307/2007637. MR0658231.
↑ オンライン整数列大辞典の数列 A013998
↑ There are infinitely many Perrin pseudoprimes, Jon Grantham, Journal of Number Theory (2010).
u = G ( n − 1 ) , v = G ( n ) , w = G ( n + 1 ) {\displaystyle u=G(n-1),v=G(n),w=G(n+1)} とすれば
23 G ( 2 n − 1 ) = 4 u 2 + 3 v 2 + 9 w 2 + 18 u v − 12 u w − 4 v w 23 G ( 2 n ) = 18 u w − 4 u v + 6 v w − 6 u 2 + 7 v 2 − 2 w 2 23 G ( 2 n + 1 ) = 9 u 2 + v 2 + 3 w 2 + 6 u v − 4 u w − 14 v w 23 G ( 3 n − 1 ) = ( − 4 u 3 + 2 v 3 − w 3 + 9 ( u v 2 + v w 2 + w u 2 ) + 3 v 2 w + 6 u v w ) 23 G ( 3 n ) = ( 3 u 3 + 2 v 3 + 3 w 3 − 3 ( u v 2 + u w 2 + v w 2 + v u 2 ) + 6 v 2 w + 18 u v w ) 23 G ( 3 n + 1 ) = ( v 3 − w 3 + 6 u v 2 + 9 u w 2 + 6 v w 2 + 9 v u 2 − 3 w u 2 + 6 w v 2 − 6 u v w ) {\displaystyle {\begin{matrix}23G(2n-1)&=&4u^{2}+3v^{2}+9w^{2}+18uv-12uw-4vw\\23G(2n)&=&18uw-4uv+6vw-6u^{2}+7v^{2}-2w^{2}\\23G(2n+1)&=&9u^{2}+v^{2}+3w^{2}+6uv-4uw-14vw\\23G(3n-1)&=&\left(-4u^{3}+2v^{3}-w^{3}+9(uv^{2}+vw^{2}+wu^{2})+3v^{2}w+6uvw\right)\\23G(3n)&=&\left(3u^{3}+2v^{3}+3w^{3}-3(uv^{2}+uw^{2}+vw^{2}+vu^{2})+6v^{2}w+18uvw\right)\\23G(3n+1)&=&\left(v^{3}-w^{3}+6uv^{2}+9uw^{2}+6vw^{2}+9vu^{2}-3wu^{2}+6wv^{2}-6uvw\right)\end{matrix}}}