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 A151842 a(3n)=n, a(3n+1)=2n+1, a(3n+2)=n+1. 2
 0, 1, 1, 1, 3, 2, 2, 5, 3, 3, 7, 4, 4, 9, 5, 5, 11, 6, 6, 13, 7, 7, 15, 8, 8, 17, 9, 9, 19, 10, 10, 21, 11, 11, 23, 12, 12, 25, 13, 13, 27, 14, 14, 29, 15, 15, 31, 16, 16, 33, 17, 17, 35, 18, 18, 37, 19, 19, 39, 20, 20, 41, 21, 21, 43, 22, 22, 45, 23, 23, 47 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Take a list of numbers (like 0,1,2,3,4,5,...) and then pair them up like this: (0,1)(1,2),(2,3),(3,4)... Then sum each pair, and insert the sum between the numbers, like this: (0,1,1), (1,3,2), (2,5,3), ... Finally, remove the parentheses: 0,1,1,1,3,2,2,5,3,... This mirrors the pattern used to make a dragon curve fractal. You take two points, then find one to insert between them. In the next iteration, you take those three points and find two numbers to insert between them. (Rather than summing the two numbers, a different function is used to find a point relative to two other points.) LINKS Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1). FORMULA From R. J. Mathar, Jul 14 2009: (Start) G.f.: x*(1+x)*(1+x^2)/((x-1)^2*(1+x+x^2)^2). a(n) = 2*a(n-3) - a(n-6). (End) Expansion of x * (1 - x^4) / ((1 - x) * (1 - x^3)^2) in powers of x. - Michael Somos, Aug 12 2009 Euler transform of length 4 sequence [ 1, 0, 2, -1]. - Michael Somos, Aug 12 2009 -a(n) = a(-1-n). - Michael Somos, Nov 11 2013 EXAMPLE G.f. = x + x^2 + x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 5*x^7 + 3*x^8 + 3*x^9 + ... - Michael Somos, Aug 12 2009 MATHEMATICA CoefficientList[Series[x (1 + x) (1 + x^2) / ((x - 1)^2 (1 + x + x^2)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Feb 14 2015 *) PROG (Python) def pairup(x): return [x[i:i+2] for i in range(len(x)-1)] def combine(vals): return sum(vals) def expand(L, fn): return [(x[0], fn(x), x[1]) for x in pairup(L)] L = list(range(20)) print(expand(L, combine)) (PARI) {a(n) = kronecker(9, n) + (n\3) * [1, 2, 1][n%3 + 1]} /* Michael Somos, Aug 12 2009 */ (Magma) I:=[0, 1, 1, 1, 3, 2]; [n le 6 select I[n] else 2*Self(n-3)-Self(n-6): n in [1..80]]; // Vincenzo Librandi, Feb 14 2015 CROSSREFS See A076118 for a version with signs. Sequence in context: A092743 A139420 A092895 * A076118 A309045 A210956 Adjacent sequences: A151839 A151840 A151841 * A151843 A151844 A151845 KEYWORD nonn,easy AUTHOR Shane Geiger (shane.geiger(AT)gmail.com), Jul 14 2009 EXTENSIONS More terms from Vincenzo Librandi, Feb 14 2015 STATUS approved

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Last modified December 5 08:51 EST 2022. Contains 358585 sequences. (Running on oeis4.)