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A134816 Padovan's spiral numbers.

%I A134816 #126 Mar 15 2025 16:51:05
%S A134816 1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351,465,
%T A134816 616,816,1081,1432,1897,2513,3329,4410,5842,7739,10252,13581,17991,
%U A134816 23833,31572,41824,55405,73396,97229,128801,170625,226030,299426,396655
%N A134816 Padovan's spiral numbers.
%C A134816 a(n) is the length of the edge of the n-th equilateral triangle in the Padovan triangle spiral.
%C A134816 Partial sums of A000931. - _Juri-Stepan Gerasimov_, Jul 17 2009
%C A134816 Rising diagonal sums of triangle A152198. - _John Molokach_, Jul 09 2013
%C A134816 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
%H A134816 Muniru A Asiru, <a href="/A134816/b134816.txt">Table of n, a(n) for n = 1..1500</a>
%H A134816 J.-L. Baril, <a href="https://doi.org/10.46298/dmtcs.2158">Avoiding patterns in irreducible permutations</a>, Discrete Mathematics and Theoretical Computer Science, Vol. 17, No. 3 (2016).
%H A134816 Eric Fernando Bravo, <a href="https://www.mathos.unios.hr/mc/index.php/mc/article/view/4733">On concatenations of Padovan and Perrin numbers</a>, Math. Commun. (2023) Vol 28, 105-119.
%H A134816 Alain Faisant, <a href="https://arxiv.org/abs/1905.07702">On the Padovan sequence</a>, arXiv:1905.07702 [math.NT], 2019.
%H A134816 Ed Harris, Pete McPartlan and Brady Haran, <a href="https://www.youtube.com/watch?v=PsGUEj4w9Cc">The Plastic Ratio</a>, Numberphile video (2019).
%H A134816 Mariana Nagy, Simon R. Cowell, and Valeriu Beiu, <a href="https://doi.org/10.3390/math12020201">On the Construction of 3D Fibonacci Spirals</a>, Mathematics (2024) Vol. 12, No. 2, 201.
%H A134816 Christian Richter, <a href="https://doi.org/10.1016/j.disc.2019.111745">Tilings of convex polygons by equilateral triangles of many different sizes</a>, Discrete Mathematics 343.3 (2020): 111745. (See Section 2.1.)
%H A134816 Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, <a href="https://doi.org/10.9734/ajpas/2024/v26i12691">A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers</a>, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
%H A134816 S. J. Tedford, <a href="https://www.fq.math.ca/Papers/57-4/tedford06222019.pdf">Combinatorial identities for the Padovan numbers</a>, Fib. Q., 57:4 (2019), 291-298.
%H A134816 Wikipedia, <a href="https://upload.wikimedia.org/wikipedia/commons/f/f5/Padovan_triangles.png">Padovan triangles</a>
%H A134816 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1).
%F A134816 a(n) = A000931(n+4).
%F A134816 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
%F A134816 a(1)=a(2)=a(3)=1, a(n) = a(n-2) + a(n-3) for n > 3. - _Robert FERREOL_, Oct 16 2017
%F A134816 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
%e A134816 a(6)=3 because 6+4=10 and A000931(10)=3.
%e A134816 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
%p A134816 a:=proc(n, p, q) option remember:
%p A134816 if n<=p then 1
%p A134816 elif n<=q then a(n-1, p, q)+a(n-p, p, q)
%p A134816 else add(a(n-k, p, q), k=p..q) fi end:
%p A134816 seq(a(n, 2, 3), n=0..100); # _Robert FERREOL_, Oct 16 2017
%t A134816 Drop[ CoefficientList[ Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, 52}], x], 5] (* _Robert G. Wilson v_, Sep 30 2009 *)
%t A134816 a[n_]=Round[Root[23#^3-5#-1&,1]Root[#^3-#-1&,1]^n ];a[Range[100]] (* OR *)
%t A134816 LinearRecurrence[{0, 1, 1}, {1, 1, 1}, 100] (* _Federico Provvedi_, Feb 12 2025 *)
%o A134816 (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
%o A134816 (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 */
%o A134816 (PARI) my(x='x+O('x^50)); Vec(x*(1+x)/(1-x^2-x^3)) \\ _Joerg Arndt_, Feb 07 2025
%Y A134816 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.
%Y A134816 Cf. A060006.
%K A134816 easy,nonn
%O A134816 1,4
%A A134816 _Omar E. Pol_, Nov 13 2007
%E A134816 More terms from _Robert G. Wilson v_, Sep 30 2009
%E A134816 First comment clarified by _Omar E. Pol_, Aug 12 2018