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

Author

Altug Alkan and Antti Karttunen, Aug 22 2018

Keywords

• sign
• look

Comments

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

Links

• 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)

Crossrefs

Cf. A318265, A318303.

Programs

Magma

[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

Mathematica

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

PARI

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

Python

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

Formula

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)