OFFSET
1,4
COMMENTS
a(n) is the length of the edge of the n-th equilateral triangle in the Padovan triangle spiral.
Partial sums of A000931. - Juri-Stepan Gerasimov, Jul 17 2009
Rising diagonal sums of triangle A152198. - John Molokach, Jul 09 2013
a(n) is the number of pairs of rabbits living at month n with the following rules: a pair of rabbits born in month n begins to procreate in month n + 2, procreates again in month n + 3, and dies at the end of this month (each pair therefore gives birth to 2 pairs); the first pair is born in month 1. - Robert FERREOL, Oct 16 2017
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..1500
Jean-Luc Baril, Avoiding patterns in irreducible permutations, Disc. Math. Theor. Comp. Sci. 17(3) (2016).
Eric Fernando Bravo, On concatenations of Padovan and Perrin numbers, Math. Commun. 28 (2023), 105-119.
Alain Faisant, On the Padovan sequence, arXiv:1905.07702 [math.NT], 2019.
Ed Harris, Pete McPartlan and Brady Haran, The Plastic Ratio, Numberphile video (2019).
Hiroki Minamide, On the Adjacency spectra of alternating-oriented n-gonal staircase digraphs, arXiv:2603.03913 [math.CO], 2026. See p. 12.
Mariana Nagy, Simon R. Cowell, and Valeriu Beiu, On the Construction of 3D Fibonacci Spirals, Mathematics 12(2) (2024), Art. 201.
Christian Richter, Tilings of convex polygons by equilateral triangles of many different sizes, Disc. Math. 343(3) (2020), Art. 111745. (See Section 2.1.)
Antonín Slavík and Marie Vestenická, Through the Tiling Glass: Tribonacci Identities, Univ. Karlova (Czechia, 2026). See p. 12.
Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12) (2024), 152-170. See p. 154.
Steven J. Tedford, Combinatorial identities for the Padovan numbers, Fib. Q., 57(4) (2019), 291-298.
Wikipedia, Padovan triangles.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).
FORMULA
a(n) = A000931(n+4).
G.f.: x * (1 + x) / (1 - x^2 - x^3) = x / (1 - x / (1 - x^2 / (1 + x / (1 - x / (1 + x))))). - Michael Somos, Jan 03 2013
a(1)=a(2)=a(3)=1, a(n) = a(n-2) + a(n-3) for n > 3. - Robert FERREOL, Oct 16 2017
a(n) = round(x*rho^n), where the Silver constant rho = Limit_{n->oo} a(n+1)/a(n) = A060006, and x is the real solution of the cubic 23*x^3-5*x-1 = 0. - Federico Provvedi, Feb 12 2025
EXAMPLE
a(6)=3 because 6+4=10 and A000931(10)=3.
G.f. = x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + ... - Michael Somos, Jan 01 2019
MAPLE
a:=proc(n, p, q) option remember:
if n<=p then 1
elif n<=q then a(n-1, p, q)+a(n-p, p, q)
else add(a(n-k, p, q), k=p..q) fi end:
seq(a(n, 2, 3), n=0..100); # Robert FERREOL, Oct 16 2017
MATHEMATICA
Drop[ CoefficientList[ Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, 52}], x], 5] (* Robert G. Wilson v, Sep 30 2009 *)
(* Alternative: *)
a[n_]=Round[Root[23#^3-5#-1&, 1]Root[#^3-#-1&, 1]^n ]; a[Range[100]] (* Federico Provvedi, Feb 12 2025 *)
(* Alternative: *)
LinearRecurrence[{0, 1, 1}, {1, 1, 1}, 100] (* Federico Provvedi, Feb 12 2025 *)
PROG
(GAP) a:=[1, 1, 1];; for n in [4..50] do a[n]:=a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Aug 12 2018
(PARI) {a(n) = if( n>=0, polcoeff( (x + x^2) / (1 - x^2 - x^3) + x * O(x^n), n), polcoeff( (x + x^2) / (1 + x - x^3) + x * O(x^-n), -n))}; /* Michael Somos, Jan 01 2019 */
(PARI) my(x='x+O('x^50)); Vec(x*(1+x)/(1-x^2-x^3)) \\ Joerg Arndt, Feb 07 2025
CROSSREFS
KEYWORD
easy,nonn,changed
AUTHOR
Omar E. Pol, Nov 13 2007
EXTENSIONS
More terms from Robert G. Wilson v, Sep 30 2009
First comment clarified by Omar E. Pol, Aug 12 2018
STATUS
approved