

A317825


a(1) = 1, a(n) = 3*a(n/2) if n is even, a(n) = n  a(n1) if n is odd.


5



1, 3, 0, 9, 4, 0, 7, 27, 18, 12, 23, 0, 13, 21, 6, 81, 64, 54, 73, 36, 57, 69, 46, 0, 25, 39, 12, 63, 34, 18, 49, 243, 210, 192, 227, 162, 199, 219, 180, 108, 149, 171, 128, 207, 162, 138, 185, 0, 49, 75, 24, 117, 64, 36, 91, 189, 132, 102, 161, 54, 115, 147, 84, 729, 664, 630, 697, 576
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OFFSET

1,2


COMMENTS

Sequence has an elegant fractallike scatter plot, situated (approximately) symmetrically over Xaxis.
This sequence can also be generalized with some modifications. Let f_k(1) = 1. f_k(n) = floor(k*a(n/2)) if n is even, f_k(n) = n  f_k(n1) if n is odd. This sequence is a(n) = f_k(n) where k = 3. For example, if k is e (A001113), then recurrence also provides a curious fractallike structure that has some similarities with a(n). See Links section for their plots.
A scatterplot of (Sum_{i = 1..2*n} a(i))  n^2 gives a similar plot as for a(n).  A.H.M. Smeets, Sep 01 2018


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16383
Altug Alkan, A scatterplot of a(n) for n <= 2^151
Altug Alkan, A scatterplot of f_e(n) for n <= 2^151
Altug Alkan, A scatterplot of (A317825(n), abs(A318303(n)))
Rémy Sigrist, A colored scatterplot of (A317825(n), abs(A318303(n))) for n = 1..2^201 (where the color is function of n)


FORMULA

From A.H.M. Smeets, Sep 01 2018: (Start)
Sum_{i = 1..2*n1} a(i) = n^2 for n >= 0.
Sum_{i = 1..2*n} a(i) = 3*a(n) + n^2 for n >= 0, a(0) = 0.
Sum_{i = 1..36*2^n} a(i) = 162*A085350(n) for n >= 0.
Lim_{n > infinity} a(n)/n^2 = 0.
Lim_{n > infinity} (Sum_{i = 1..n} a(i))/n^2 = 1/4. (End)


MATHEMATICA

Nest[Append[#1, If[EvenQ[#2], 3 #1[[#2/2]], #2  #1[[1]] ]] & @@ {#, Length@ # + 1} &, {1}, 67] (* Michael De Vlieger, Aug 22 2018 *)


PROG

(PARI) A317825(n) = if(1==n, n, if(!(n%2), 3*A317825(n/2), nA317825(n1)));
(Python)
aa = [0]
a, n = 0, 0
while n < 16383:
....n = n+1
....if n%2 == 0:
........a = 3*aa[n//2]
....else:
........a = na
....aa = aa+[a]
....print(n, a) # A.H.M. Smeets, Sep 01 2018
(MAGMA) [n eq 1 select 1 else IsEven(n) select 3*Self(n div 2) else n Self(n1): n in [1..80]]; // Vincenzo Librandi, Sep 03 2018


CROSSREFS

Cf. A318265, A318303.
Sequence in context: A021101 A154202 A214699 * A002346 A021327 A297053
Adjacent sequences: A317822 A317823 A317824 * A317826 A317827 A317828


KEYWORD

sign,look


AUTHOR

Altug Alkan and Antti Karttunen, Aug 22 2018


STATUS

approved



