A151842 a(3n)=n, a(3n+1)=2n+1, a(3n+2)=n+1.

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

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

Table of n, a(n) for n=0..70.

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

From Ridouane Oudra, Nov 23 2024: (Start)

a(n) = 5*n/6 + n^2/2 - n^3/3 + (2*n^2 - n - 3/2)*floor(n/3) - (3*n + 3/2)*floor(n/3)^2.

a(n) = t(n+2)*t(n+3) - t(n)*t(n+1), where t(n) = floor(n/3) = A002264(n).

a(n) = A008133(n+2) - A008133(n). (End)

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.

Cf. A002264, A008133.

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