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

0,2

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Index entries for sequences related to Nim-sums

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

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 *)

Prog

(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
(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
(PARI) a(n) = if(n, bitand(n, 2<<bitor(1, logint(n, 2)) \ 3)) << 1; \\ 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

Crossrefs

Cf. A053985, A063694, A063695, A088442.

Sequence in context: A029593 A196504 A182550 * A278748 A342286 A211930

Adjacent sequences: A004511 A004512 A004513 * A004515 A004516 A004517

Keyword

base,easy,nonn

Author

N. J. A. Sloane

Extensions

More terms from Reinhard Zumkeller, Dec 27 2003

Status

approved