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

1,4

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

References

Richter, Christian. "Tilings of convex polygons by equilateral triangles of many different sizes." Discrete Mathematics 343.3 (2020): 111745. (See Section 2.1.)

S. J. Tedford, Combinatorial identities for the Padovan numbers, Fib. Q., 57:4 (2019), 291-298.

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.

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

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] (* Or *) a[1] = a[2] = a[3] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; Array[ a, 48] (* Robert G. Wilson v, Sep 30 2009 *)
a[ n_] := If[ n >= 0, SeriesCoefficient[ (x + x^2) / (1 - x^2 - x^3), {x, 0, n}], SeriesCoefficient[ (x + x^2) / (1 + x - x^3), {x, 0, Abs@n}]];

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 */

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.

Sequence in context: A133034 A000931 A078027 * A228361 A182097 A290697

Adjacent sequences: A134813 A134814 A134815 * A134817 A134818 A134819

Keyword

easy,nonn

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