A004514 Generalized nim sum n + n in base 4.

0, 2, 0, 2, 8, 10, 8, 10, 0, 2, 0, 2, 8, 10, 8, 10, 32, 34, 32, 34, 40, 42, 40, 42, 32, 34, 32, 34, 40, 42, 40, 42, 0, 2, 0, 2, 8, 10, 8, 10, 0, 2, 0, 2, 8, 10, 8, 10, 32, 34, 32, 34, 40, 42, 40, 42, 32, 34, 32, 34, 40, 42, 40, 42, 128, 130, 128, 130, 136, 138, 136, 138, 128

Offset: 0

Author

N. J. A. Sloane

Keywords

• base
• easy
• nonn

Comments

In the base 4 expansion of 2*n + 1, change 1 to 0 and 3 to 2. - Paolo Xausa, Feb 27 2025

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to Nim-sums

Crossrefs

Cf. A053985, A063694, A063695, A088442.

Programs

Haskell

a004514 = a063695 . (+ 1) . (* 2)  -- Reinhard Zumkeller, Sep 26 2015

Magma

function A063694(n)
  if n le 1 then return n;
  else return 4*A063694(Floor(n/4)) + ((n mod 4) mod 2);
  end if; return A063694;
end function;
A004514:= func< n | 2*A063694(n) >;
[A004514(n): n in [0..90]]; // G. C. Greubel, Dec 05 2022

Mathematica

A004514[n_]:= A004514[n]= If[n==0, 0, 2*n -2*A004514[Floor[n/2]]];
Table[A004514[n], {n,0,90}] (* G. C. Greubel, Dec 05 2022 *)
A004514[n_] := FromDigits[ReplaceAll[IntegerDigits[2*n + 1, 4], {1 -> 0, 3 -> 2}], 4];
Array[A004514, 100, 0] (* Paolo Xausa, Feb 27 2025 *)

PARI

a(n) = if(n, bitand(n, 2<Kevin Ryde, Dec 10 2022

Python

def A004514(n): return (n&((1<<(m:=n.bit_length())+(m&1))-1)//3)<<1 # Chai Wah Wu, Jan 30 2023

SageMath

def A063694(n):
    if (n<2): return n
    else: return 4*A063694(floor(n/4)) + ((n%4)%2)
def A004514(n): return 2*A063694(n)
[A004514(n) for n in range(91)] # G. C. Greubel, Dec 05 2022

Formula

Generalized nim sum m + n in base q: write m and n in base q and add mod q with no carries, e.g., 5 + 8 in base 3 = "21" + "22" = "10" = 1.

From Vladeta Jovovic, Feb 23 2003: (Start)

a(n) = 2*(n - a(floor(n/2))).

a(n) = 2*A063694(n). (End)

a(n) = A088442(n) - 1. - Chris Groer (cgroer(AT)math.uga.edu), Nov 10 2003

a(n) = n + A053985(n). - Reinhard Zumkeller, Dec 27 2003

a(n) = A063695(2*n+1). - Reinhard Zumkeller, Sep 26 2015

a(n) = Sum_{k>=0} A030308(n,k)*A103424(k+1). - Philippe Deléham, Jan 12 2023

Extensions

More terms from Reinhard Zumkeller, Dec 27 2003