

A134816


Padovan's spiral numbers.


20



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,4


COMMENTS

a(n) is the length of the edge of the nth equilateral triangle in the Padovan triangle spiral.
Partial sums of A000931.  JuriStepan 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
J.L. Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol. 17, No. 3 (2016).
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(0)=a(1)=a(2)=1, a(n) = a(n2) + a(n3) for n >= 3.  Robert FERREOL, Oct 16 2017


EXAMPLE

a(6)=3 because 6+4=10 and A000931(10)=3.


MAPLE

a:=proc(n, p, q) option remember:
if n<=p then 1
elif n<=q then a(n1, p, q)+a(np, p, q)
else add(a(nk, 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 *)


PROG

(GAP) a:=[1, 1, 1];; for n in [4..50] do a[n]:=a[n2]+a[n3]; od; a; # Muniru A Asiru, Aug 12 2018


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



