A318303 -id:A318303 - cp's OEIS frontend

A317825 a(1) = 1, a(n) = 3*a(n/2) if n is even, a(n) = n - a(n-1) if n is odd.

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

Offset: 1

Altug Alkan and Antti Karttunen, Aug 22 2018

Sequence has an elegant fractal-like scatter plot, situated (approximately) symmetrically over X-axis.
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(n-1) 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 fractal-like 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

Antti Karttunen, Table of n, a(n) for n = 1..16383
Altug Alkan, A scatterplot of a(n) for n <= 2^15-1
Altug Alkan, A scatterplot of f_e(n) for n <= 2^15-1
Altug Alkan, A scatterplot of (A317825(n), abs(A318303(n)))
Rémy Sigrist, A colored scatterplot of (A317825(n), abs(A318303(n))) for n = 1..2^20-1 (where the color is function of n)

Cf. A318265, A318303.

[n eq 1 select 1 else IsEven(n) select 3*Self(n div 2) else n- Self(n-1): n in [1..80]]; // Vincenzo Librandi, Sep 03 2018

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

A317825(n) = if(1==n,n,if(!(n%2),3*A317825(n/2),n-A317825(n-1)));

aa = [0]
a,n = 0,0
while n < 16383:
    n = n+1
    if n%2 == 0:
        a = 3*aa[n//2]
    else:
        a = n-a
    aa = aa+[a]
    print(n,a) # A.H.M. Smeets, Sep 01 2018

From A.H.M. Smeets, Sep 01 2018: (Start)
Sum_{i = 1..2*n-1} 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)

A318488 a(0) = 0, a(n) = -5*a(n/3) if n is divisible by 3, otherwise a(n) = n + a(n-1).

Original entry on oeis.org

0, 1, 3, -5, -1, 4, -15, -8, 0, 25, 35, 46, 5, 18, 32, -20, -4, 13, 75, 94, 114, 40, 62, 85, 0, 25, 51, -125, -97, -68, -175, -144, -112, -230, -196, -161, -25, 12, 50, -90, -50, -9, -160, -117, -73, 100, 146, 193, 20, 69, 119, -65, -13, 40, -375, -320, -264, -470, -412, -353, -570, -509, -447, -200, -136, -71, -310

Offset: 0

Altug Alkan, Aug 27 2018

From a generalization of A318303 (compare the scatterplots in order to observe connection). In this case, A000244 is determinative for the boundaries of self-similar block structures of this sequence, i.e., n = 3^9 - 1 is a corresponding endpoint.

Altug Alkan, Table of n, a(n) for n = 0..19682
Rémy Sigrist, Colored scatterplot of a(n) for n = 0..3^10-1 (where the color is function of floor(n / 3^(A081604(n)-5)))

Cf. A081604, A318303.

[0] cat [n eq 1 select 1 else n mod 3 eq 0 select -5*Self(n div 3) else Self(n-1)+n: n in [1..70]]; // Vincenzo Librandi, Aug 28 2018

a[0]=0; a[n_] := a[n] = If[Mod[n, 3] == 0, -5 a[n/3], n + a[n - 1]]; Array[a, 70, 0] (* Giovanni Resta, Aug 27 2018 *)

a(n)=if(n==0, 0, if(n%3, n+a(n-1), -5*a(n/3)));

cp's OEIS Frontend

A317825 a(1) = 1, a(n) = 3*a(n/2) if n is even, a(n) = n - a(n-1) if n is odd.

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