0,1

With the terms indexed as shown, has property that p prime => p divides a(p). The smallest composite n such that n divides a(n) is 521^2. For quotients a(p)/p, where p is prime, see A014981.

Asymptotically, a(n) ~ r^n, with r=1.3247179572447... the inverse of the real root of 1-x^2-x^3=0 (see A060006). If n>9 then a(n)=round(r^n). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Dec 13 2002

The recursion can be used to compute a(-n). The result is -A078712(n). - T. D. Noe (noe(AT)sspectra.com), Oct 10 2006

For n>=3, a(n) is the number of maximal independent sets in a cycle of order n. - Vince Vatter (vatter(AT)gmail.com), Oct 24 2006

M^n * [3, 0, 2] = [a(n), a(n+1), a(n+2)]; e.g., M^7 * [3, 0, 2] = [7, 10, 12]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2007

W. W. Adams and D. Shanks, Strong primality tests that are not sufficient, Math. Comp. 39 (1982), 255-300.

J. Chick, Problem 81G, Math. Gazette, vol. 81 p. 304, 1997.

E. B. Escott, Problem 151, Amer. Math. Monthly, 15 (1908), 209.

D. C. Fielder, Special integer sequences controlled by three parameters. Fibonacci Quart 6 1968 64-70.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.

Z. Furedi, The number of maximal independent sets in connected graphs, J. Graph Theory 11 (1987), 463-470.

D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.

R. Perrin, Query 1484, L'Interm\'{e}diaire des Math\'{e}maticiens, 6 (1899), 76.

M. Schroeder, Number Theory..., 3rd ed., Springer, 1997.

I. Stewart, Math. Rec., Scientific American, #6, 1996 p 103.

Ondrej Such, An Insufficient Condition for Primality, Problem 10268, Amer. Math. Monthly, 102 (1995), 557-558; 103 (1996), 911.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 70.

T. D. Noe, Table of n, a(n) for n=0..1000

Bill Amend, "Foxtrot" cartoon, Oct 11, 2005 (Illustration of initial terms! From www.ucomics.com/foxtrot/.)

K. S. Brown, Perrin's Sequence

C. Holzbaur, Perrin pseudoprimes

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

I. Stewart, Tales of a Neglected Number

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Willem's Fibonacci site, Perrin and Fibonacci

Wikipedia, Perrin pseudoprime

G.f.: (3 - x^2)/(1 - x^2 - x^3).

a(n)=r1^n+r2^n+r3^n where r1, r2, r3 are three roots of x^3-x-1=0.

a(n-1)+a(n)+a(n+1)=a(n+4), a(n)-a(n-1)=a(n-5). - Jon Perry (perry(AT)globalnet.co.uk), Jun 05 2003

a(n) = the trace of M^n where M = the 3 X 3 matrix [0 1 0 / 0 0 1 / 1 1 0]. 2. a(n) = 2*A000931(n+3) + A000931(n) E.g. a(10) = 17 = (3 + 7 + 7) = trace of M^10. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 01 2004

A001608:=-z*(2+3*z)/(-1+z**2+z**3); [S. Plouffe in his 1992 dissertation.]

(PARI) a(n)=if(n<0, 0, polsym(x^3-x-1, n)[n+1])

Cf. A000931.

Adjacent sequences: A001605 A001606 A001607 this_sequence A001609 A001610 A001611

Sequence in context: A119493 A032531 A112455 this_sequence A112974 A113069 A136163

nonn,easy,nice

njas

More terms from Jon Perry (perry(AT)globalnet.co.uk), Jun 05 2003

Additional comments from Mike Baker, Oct 11, 2005