A134816 Padovan's spiral numbers.

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426, 396655

Offset: 1

Author

Omar E. Pol, Nov 13 2007

Keywords

• easy
• nonn

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

Examples

			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
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..1500
• J.-L. Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol. 17, No. 3 (2016).
• Eric Fernando Bravo, On concatenations of Padovan and Perrin numbers, Math. Commun. (2023) Vol 28, 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).
• Mariana Nagy, Simon R. Cowell, and Valeriu Beiu, On the Construction of 3D Fibonacci Spirals, Mathematics (2024) Vol. 12, No. 2, 201.
• Christian Richter, Tilings of convex polygons by equilateral triangles of many different sizes, Discrete Mathematics 343.3 (2020): 111745. (See Section 2.1.)
• 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):152-70, (2024). See p. 154.
• S. 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).

Crossrefs

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

Cf. A060006.

Programs

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

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 *)
a[n_]=Round[Root[23#^3-5#-1&,1]Root[#^3-#-1&,1]^n ];a[Range[100]] (* OR *)
LinearRecurrence[{0, 1, 1}, {1, 1, 1}, 100] (* Federico Provvedi, Feb 12 2025 *)

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

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

Extensions

More terms from Robert G. Wilson v, Sep 30 2009

First comment clarified by Omar E. Pol, Aug 12 2018