

A134816


Padovan's spiral numbers.


21



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
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OFFSET

1,4


COMMENTS

a(n) is the length of the edge of the nth equilateral triangle in the Padovan triangle spiral.
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), 291298.


LINKS

Ed Harris, Pete McPartlan and Brady Haran, The Plastic Ratio, Numberphile video (2019)


FORMULA

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(n2) + a(n3) 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(n1, p, q)+a(np, p, q)
else add(a(nk, p, q), k=p..q) fi end:


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[n2]+a[n3]; 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



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



