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 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 (list; graph; refs; listen; history; text; internal format)
 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). 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) 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 = a = a = 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

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Last modified October 2 23:51 EDT 2022. Contains 357230 sequences. (Running on oeis4.)