OFFSET
0,1
COMMENTS
Has been called the skiponacci sequence or skiponacci numbers. - N. J. A. Sloane, May 24 2013
For n >= 3, also the numbers of maximal independent vertex sets, maximal matchings, minimal edge covers, and minimal vertex covers in the n-cycle graph C_n. - Eric W. Weisstein, Mar 30 2017 and Aug 03 2017
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, Dec 13 2002
For n>=3, a(n) is the number of maximal independent sets in a cycle of order n. - Vincent Vatter, Oct 24 2006
Pisano period lengths are given in A104217. - R. J. Mathar, Aug 10 2012
From Roman Witula, Feb 01 2013: (Start)
Let r1, r2 and r3 denote the roots of x^3 - x - 1. Then the following identity holds: a(k*n) + (a(k))^n - (a(k) - r1^k)^n - (a(k) - r2^k)^n - (a(k) - r3^k)^n
= 0 for n = 0, 1, 2,
= 6 for n = 3,
= 12*a(k) for n = 4,
= 10*[2*(a(k))^2 - a(-k)] for n = 5,
= 30*a(k)*[(a(k))^2 - a(-k)] for n = 6,
= 7*[6*(a(k))^4 - 9*a(-k)*(a(k))^2 + 2*(a(-k))^2 - a(k)] for n = 7,
= 56*a(k)*[((a(k))^2 - a(-k))^2 - a(k)/2] for n = 8,
where a(-k) = -A078712(k) and the formula (5.40) from the paper of Witula and Slota is used. (End)
The parity sequence of a(n) is periodic with period 7 and has the form (1,0,0,1,0,1,1). Hence we get that a(n) and a(2*n) are congruent modulo 2. Similarly we deduce that a(n) and a(3*n) are congruent modulo 3. Is it true that a(n) and a(p*n) are congruent modulo p for every prime p? - Roman Witula, Feb 09 2013
The trinomial x^3 - x - 1 divides the polynomial x^(3*n) - a(n)*x^(2*n) + ((a(n)^2 - a(2*n))/2)*x^n - 1 for every n>=1. For example, for n=3 we obtain the factorization x^9 - 3*x^6 + 2*x^3 - 1 = (x^3 - x - 1)*(x^6 + x^4 - 2*x^3 + x^2 - x + 1). Sketch of the proof: Let p,s,t be roots of the Perrin polynomial x^3 - x - 1. Then we have (a(n))^2 = (p^n + s^n + t^n)^2 = a(2*n) + 2*a(n)*x^n -2*x^n + 2/x^n for every x = p,s,t, i.e., x^(3*n) - a(n)*x^(2*n) + ((a(n)^2 - a(2*n))/2)*x^n - 1 = 0 for every x = p,s,t, which finishes the proof. By discussion of the power(a(n))^3 = (p^n + s^n + t^n)^3 it can be deduced that the trinomial x^3 - x - 1 divides the polynomial 2*x^(4*n) - a(n)*x^(3*n) - a(2*n)*x^(2*n) + ((a(n)^3 - a(3*n) - 3)/3)*x^n - a(n) = 0. Co-author of these divisibility relations is also my young student Szymon Gorczyca (13 years old as of 2013). - Roman Witula, Feb 09 2013
The sum of powers of the real root and complex roots of x^3-x-1=0 as expressed as powers of the plastic number r, (see A060006). Let r0=1, r1=r, r2=1+r^(-1) and c0=2, c1=-r and c3 = r^(-5) then a(n) = r(n-2)+r(n-3) + c(n-2)+c(n-3). Example: a(5) = 1 + r^(-1) + 1 + r + 2 - r + r^(-5) = 4 + r^(-1) + r^(-5) = 5. - Richard Turk, Jul 14 2016
Also the number of minimal total dominating sets in the n-sun graph. - Eric W. Weisstein, Apr 27 2018
Named after the French engineer François Olivier Raoul Perrin (1841-1910). - Amiram Eldar, Jun 05 2021
a(p) = p*A127687(p) for p prime. - Robert FERREOL, Apr 09 2024
REFERENCES
Olivier Bordellès, Thèmes d'Arithmétique, Ellipses, 2006, Exercice 4.11, p. 127.0
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
Dmitry Fomin, On the properties of a certain recursive sequence, Mathematics and Informatics Quarterly, Vol. 3 (1993), pp. 50-53.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 70.
Manfred Schroeder, Number Theory in Science and Communication, 3rd ed., Springer, 1997.
A. G. Shannon, P. G. Anderson and A. F. Horadam, Properties of Cordonnier, Perrin and Van der Laan numbers, International Journal of Mathematical Education in Science and Technology, Volume 37:7 (2006), 825-831. See Q_n.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..8172 (terms 0..1000 from T. D. Noe)
William Adams and Daniel Shanks, Strong primality tests that are not sufficient, Math. Comp., Vol. 39, No. 159 (1982), pp. 255-300.
Kouèssi Norbert Adédji, Japhet Odjoumani, and Alain Togbé, Padovan and Perrin numbers as products of two generalized Lucas numbers, Archivum Mathematicum, Vol. 59 (2023), No. 4, 315-337.
Bill Amend, "Foxtrot" cartoon, Oct 11, 2005 (Illustration of initial terms! From www.ucomics.com/foxtrot/.)
Mohamadou Bachabi and Alain S. Togbé, Products of Fermat or Mersenne numbers in some sequences, Math. Comm. (2024) Vol. 29, 265-285.
Herbert Batte, Taboka P. Chalebgwa and Mahadi Ddamulira, Perrin numbers that are concatenations of two distinct repdigits, arXiv:2105.08515 [math.NT], 2021.
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, Linear recurrence sequences with indices in arithmetic progression and their sums, arXiv:1505.06339 [math.NT], 2015.
Eric Fernando Bravo, On concatenations of Padovan and Perrin numbers, Math. Commun. (2023) Vol 28, 105-119.
Kevin S. Brown, Perrin's Sequence
J. Chick, Problem 81G, Math. Gazette, Vol. 81, No. 491 (1997), p. 304.
Tomislav Doslic and I. Zubac, Counting maximal matchings in linear polymers, Ars Mathematica Contemporanea, Vol. 11 (2016), pp. 255-276.
Robert Dougherty-Bliss, The Meta-C-finite Ansatz, arXiv preprint arXiv:2206.14852 [math.CO], 2022.
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 34.
Robert Dougherty-Bliss and Doron Zeilberger, Lots and Lots of Perrin-Type Primality Tests and Their Pseudo-Primes, arXiv:2307.16069 [math.NT], 2023.
E. B. Escott, Problem 151, Amer. Math. Monthly, Vol. 15, No. 11 (1908), p. 209.
Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly, Vol. 6 (1968), pp. 64-70.
Daniel C. Fielder, Errata:Special integer sequences controlled by three parameters, Fibonacci Quarterly, Vol. 6 (1968), pp. 64-70.
Zoltán Füredi, The number of maximal independent sets in connected graphs, J. Graph Theory, Vol. 11, No. 4 (1987), pp. 463-470.
Bill Gasarch, When does n divide a_n in this sequence? (2016).
A. Justin Gopinath and B. Nithya, Mathematical and Simulation Analysis of Contention Resolution Mechanism for IEEE 802.11 ah Networks, Computer Communications (2018) Vol. 124, 87-100.
Christian Holzbaur, Perrin pseudoprimes [Original link broke many years ago. This is a cached copy from the WayBack machine, dated Apr 24 2006]
Dmitry I. Ignatov, On the Maximal Independence Polynomial of the Covering Graph of the Hypercube up to n = 6, Int'l Conf. Formal Concept Analysis, 2023.
Stanislav Jakubec and Karol Nemoga, On a conjecture concerning sequences of the third order, Mathematica Slovaca, Vol. 36, No. 1 (1986), pp. 85-89.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 90.
Bir Kafle, Salah Eddine Rihane and Alain Togbé, A note on Mersenne Padovan and Perrin numbers, Notes on Num. Theory and Disc. Math., Vol. 27, No. 1 (2021), pp. 161-170.
Vedran Krcadinac, A new generalization of the golden ratio, Fibonacci Quart., Vol. 44, No. 4 (2006), pp. 335-340.
G. C. Kurtz, Daniel Shanks and H. C. Williams, Fast primality tests for numbers less than 50*10^9, Mathematics of Computation, Vol. 46, No. 174 (1986), pp. 691-701. [Studies primes in this sequence. - N. J. A. Sloane, Jul 28 2019]
I. E. Leonard and A. C. F. Liu, A familiar recurrence occurs again, Amer. Math. Monthly, Vol. 119, No. 4 (2012), 333-336.
J. M. Luck and A. Mehta, Universality in survivor distributions: Characterising the winners of competitive dynamics, Physical Review E, Vol. 92, No. 5 (2015), 052810; arXiv preprint, arXiv:1511.04340 [q-bio.QM], 2015.
Matthew Macauley, Jon McCammond and Henning S. Mortveit, Dynamics groups of asynchronous cellular automata, Journal of Algebraic Combinatorics, Vol 33, No 1 (2011), pp. 11-35.
Gregory Minton, Three approaches to a sequence problem, Math. Mag., Vol. 84, No. 1 (2011), pp. 33-37.
Gregory T. Minton, Linear recurrence sequences satisfying congruence conditions, Proc. Amer. Math. Soc., Vol. 142, No. 7 (2014), pp. 2337-2352. MR3195758.
B. H. Neumann and L. G. Wilson, Some Sequences like Fibonacci's, Fibonacci Quart., Vol. 17, No. 1 (1979), p. 83.
Mathilde Noual, Dynamics of Circuits and Intersecting Circuits, in Language and Automata Theory and Applications, Lecture Notes in Computer Science, 2012, Volume 7183/2012, 433-444, DOI; also on arXiv, arXiv 1011.3930 [cs.DM], 2010.
Ahmet Öteleş, Bipartite Graphs Associated with Pell, Mersenne and Perrin Numbers, An. Şt. Univ. Ovidius Constantą, (2019) Vol. 27, Issue 2, 109-120.
R. Perrin, Query 1484, L'Intermédiaire des Mathématiciens, Vol. 6 (1899), p. 76.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Salah Eddine Rihane, Chèfiath Awero Adegbindin and Alain Togbé, Fermat Padovan And Perrin Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.2.
Salah Eddine Rihane and Alain Togbé, Repdigits as products of consecutive Padovan or Perrin numbers, Arab. J. Math. (2021).
David E. Rush, Degree n Relatives of the Golden Ratio and Resultants of the Corresponding Polynomials, Fibonacci Quart., Vol. 50, No. 4 (2012), pp. 313-325. See p. 318.
J. O. Shallit and J. P. Yamron, On linear recurrences and divisibility by primes, Fibonacci Quart., Vol. 22, No. 4 (1984), p. 366.
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Ondrej Such, An Insufficient Condition for Primality, Problem 10268, Amer. Math. Monthly, Vol. 102, No. 6 (1995), pp. 557-558.
Ondrej Such, An Insufficient Condition for Primality, Problem 10268, Amer. Math. Monthly, Vol. 103, No. 10 (1996), p. 911.
Pagdame Tiebekabe and Kouèssi Norbert Adédji, On Padovan or Perrin numbers as products of three repdigits in base delta, 2023.
Razvan Tudoran, Problem 653, College Math. J., Vol. 31, No. 3 (2000), pp. 223-224.
Vincent Vatter, Social distancing, primes, and Perrin numbers, Math Horiz., Vol. 29, No. 1, pp. 5-7.
Stan Wagon, Letter to the Editor, Math. Mag., Vol. 84, No. 2 (2011), p. 119.
Michel Waldschmidt, Lectures on Multiple Zeta Values, IMSC 2011.
Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Maximal Independent Edge Set
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
Eric Weisstein's World of Mathematics, Minimal Edge Cover.
Eric Weisstein's World of Mathematics, Minimal Vertex Cover
Eric Weisstein's World of Mathematics, Perrin Pseudoprime
Eric Weisstein's World of Mathematics, Perrin Sequence
Eric Weisstein's World of Mathematics, Sun Graph
Eric Weisstein's World of Mathematics, Total Dominating Set
Willem's Fibonacci site, Perrin and Fibonacci.
Wikipedia, Perrin Number.
Roman Witula and Damian Slota, Conjugate sequences in Fibonacci-Lucas sense and some identities for sums of powers of their elements, Integers, Vol. 7 (2007), #A08.
Richard J. Yanco, Letter and Email to N. J. A. Sloane, 1994
Richard Yanco and Ansuman Bagchi, K-th order maximal independent sets in path and cycle graphs, Unpublished manuscript, 1994. (Annotated scanned copy)
Fatih Yilmaz and Durmus Bozkurt, Hessenberg matrices and the Pell and Perrin numbers, Journal of Number Theory, Volume 131, Issue 8 (August 2011), pp. 1390-1396. [The terms given in the paper contain a typo]
Index entries for linear recurrences with constant coefficients, signature (0,1,1).
FORMULA
G.f.: (3 - x^2)/(1 - x^2 - x^3). - Simon Plouffe in his 1992 dissertation
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, Jun 05 2003
From Gary W. Adamson, Feb 01 2004: (Start)
a(n) = trace(M^n) where M is the 3 X 3 matrix [0 1 0 / 0 0 1 / 1 1 0], the companion matrix of the characteristic polynomial of this sequence, P = X^3 - X - 1.
M^n * [3, 0, 2] = [a(n), a(n+1), a(n+2)]; e.g., M^7 * [3, 0, 2] = [7, 10, 12].
a(n) = 3*p(n) - p(n-2) = 2*p(n) + p(n-3), with p(n) := A000931(n+3), n >= 0. - Wolfdieter Lang, Jun 21 2010
From Francesco Daddi, Aug 03 2011: (Start)
a(0) + a(1) + a(2) + ... + a(n) = a(n+5) - 2.
a(0) + a(2) + a(4) + ... + a(2*n) = a(2*n+3).
a(1) + a(3) + a(5) + ... + a(2*n+1) = a(2*n+4) - 2. (End)
From Francesco Daddi, Aug 04 2011: (Start)
a(0) + a(3) + a(6) + a(9) + ... + a(3*n) = a(3*n+2) + 1.
a(0) + a(5) + a(10) + a(15) + ... + a(5*n) = a(5*n+1)+3.
a(0) + a(7) + a(14) + a(21) + ... + a(7*n) = (a(7*n) + a(7*n+1) + 3)/2. (End)
a(n) = n*Sum_{k=1..floor(n/2)} binomial(k,n-2*k)/k, n > 0, a(0)=3. - Vladimir Kruchinin, Oct 21 2011
(a(n)^3)/2 + a(3n) - 3*a(n)*a(2n)/2 - 3 = 0. - Richard Turk, Apr 26 2017
2*a(4n) - 2*a(n) - 2*a(n)*a(3n) - a(2n)^2 + a(n)^2*a(2n) = 0. - Richard Turk, May 02 2017
a(n)^4 + 6*a(4n) - 4*a(3n)*a(n) - 3*a(2n)^2 - 12a(n) = 0. - Richard Turk, May 02 2017
a(n+5)^2 + a(n+1)^2 - a(n)^2 = a(2*(n+5)) + a(2*(n+1)) - a(2*n). - Aleksander Bosek, Mar 04 2019
From Aleksander Bosek, Mar 18 2019: (Start)
a(n+12) = a(n) + 2*a(n+4) + a(n+11);
a(n+16) = a(n) + 4*a(n+9) + a(n+13);
a(n+18) = a(n) + 2*a(n+6) + 5*a(n+12);
a(n+21) = a(n) + 2*a(n+12) + 6*a(n+14);
a(n+27) = a(n) + 3*a(n+9) + 4*a(n+22). (End)
a(n) = Sum_{j=0..floor((n-g)/(2*g))} 2*n/(n-2*(g-2)*j-(g-2)) * Hypergeometric2F1([-(n-2g*j-g)/2, -(2j+1)], [1], 1), g = 3 and n an odd integer. - Richard Turk, Oct 14 2019
E.g.f.: exp(r1*x) + exp(r2*x) + exp(r3*x), where r1, r2, r3 are three roots of x^3 - x - 1 = 0. - Fabian Pereyra, Nov 02 2024
EXAMPLE
From Roman Witula, Feb 01 2013: (Start)
We note that if a + b + c = 0 then:
1) a^3 + b^3 + c^3 = 3*a*b*c,
2) a^4 + b^4 + c^4 = 2*((a^2 + b^2 + c^2)/2)^2,
3) (a^5 + b^5 + c^5)/5 = (a^3 + b^3 + c^3)/3 * (a^2 +
b^2 + c^2)/2,
4) (a^7 + b^7 + c^7)/7 = (a^5 + b^5 + c^5)/5 * (a^2 + b^2 + c^2)/2 = 2*(a^3 + b^3 + c^3)/3 * (a^4 + b^4 + c^4)/4,
5) (a^7 + b^7 + c^7)/7 * (a^3 + b^3 + c^3)/3 = ((a^5 + b^5 + c^5)/5)^2.
Hence, by the Binet formula for a(n) we obtain the relations: a(3) = 3, a(4) = 2*(a(2)/2)^2 = 2, a(5)/5 = a(3)/3 * a(2)/2, i.e., a(5) = 5, and similarly that a(7) = 7. (End)
MAPLE
A001608 :=proc(n) option remember; if n=0 then 3 elif n=1 then 0 elif n=2 then 2 else procname(n-2)+procname(n-3); fi; end proc;
[seq(A001608(n), n=0..50)]; # N. J. A. Sloane, May 24 2013
MATHEMATICA
LinearRecurrence[{0, 1, 1}, {3, 0, 2}, 50] (* Harvey P. Dale, Jun 26 2011 *)
per = Solve[x^3 - x - 1 == 0, x]; f[n_] := Floor @ Re[N[ per[[1, -1, -1]]^n + per[[2, -1, -1]]^n + per[[3, -1, -1]]^n]]; Array[f, 46, 0] (* Robert G. Wilson v, Jun 29 2010 *)
a[n_] := n*Sum[Binomial[k, n-2*k]/k, {k, 1, n/2}]; a[0]=3; Table[a[n] , {n, 0, 45}] (* Jean-François Alcover, Oct 04 2012, after Vladimir Kruchinin *)
CoefficientList[Series[(3 - x^2)/(1 - x^2 - x^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 03 2015 *)
Table[RootSum[-1 - # + #^3 &, #^n &], {n, 0, 20}] (* Eric W. Weisstein, Mar 30 2017 *)
RootSum[-1 - # + #^3 &, #^Range[0, 20] &] (* Eric W. Weisstein, Dec 30 2017 *)
PROG
(PARI) a(n)=if(n<0, 0, polsym(x^3-x-1, n)[n+1])
(PARI) A001608_list(n) = polsym(x^3-x-1, n) \\ Joerg Arndt, Mar 10 2019
(Haskell)
a001608 n = a000931_list !! n
a001608_list = 3 : 0 : 2 : zipWith (+) a001608_list (tail a001608_list)
-- Reinhard Zumkeller, Feb 10 2011
(Python)
A001608_list, a, b, c = [3, 0, 2], 3, 0, 2
for _ in range(100):
a, b, c = b, c, a+b
A001608_list.append(c) # Chai Wah Wu, Jan 27 2015
(GAP) a:=[3, 0, 2];; for n in [4..20] do a[n]:=a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Jul 12 2018
(Magma) I:=[3, 0, 2]; [n le 3 select I[n] else Self(n-2) +Self(n-3): n in [1..50]]; // G. C. Greubel, Mar 18 2019
(Sage) ((3-x^2)/(1-x^2-x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Mar 18 2019
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Additional comments from Mike Baker, Oct 11 2005
Definition edited by Chai Wah Wu, Jan 27 2015
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021
STATUS
approved