A103424 -id:A103424 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

Cf. A053985, A063694, A063695, A088442.

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

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

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 *)

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

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

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

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

More terms from Reinhard Zumkeller, Dec 27 2003

A199573 Number of round trips of length n from any of the four vertices of the cycle graph C_4.

Original entry on oeis.org

1, 0, 2, 0, 8, 0, 32, 0, 128, 0, 512, 0, 2048, 0, 8192, 0, 32768, 0, 131072, 0, 524288, 0, 2097152, 0, 8388608, 0, 33554432, 0, 134217728, 0, 536870912, 0, 2147483648, 0, 8589934592, 0, 34359738368, 0, 137438953472, 0

Offset: 0

Wolfdieter Lang, Nov 08 2011

See the array w(N,L) and the triangle a(K,N) given in A199571.
Essentially the same as A103424.
This is A081294 and A000004 interleaved. - Omar E. Pol, Nov 09 2011

			a(4)=8 from the eight round trips of length 4 (starting from, say, vertex no. 1): 12121, 14141, 12141, 14121, 12321, 14341, 12341 and 14321.

R. J. Mathar, Counting Walks on Finite Graphs, Section 1.
Index entries for linear recurrences with constant coefficients, signature (0,4).

Cf. A078008 (N=3), A054877 (N=5), A199571.

CoefficientList[Series[(1 - 2 x^2)/(1 - (2 x)^2), {x, 0, 40}], x] (* or *) Riffle[Join[{1},NestList[4#&,2,20]],0] (* or *) LinearRecurrence[ {0,4},{1,0,2},80] (* Harvey P. Dale, Dec 04 2015 *)

a(n) =  2^(n-2)*(1+(-1)^n), n>=2, a(0)=1.
O.g.f.: (1-2*x^2)/(1-(2*x)^2).
E.g.f.: 1+(1 + 2*x^2/(U(0) - 2*x^2 + 1))*x^2 where U(k)=  4*k+5 + 2*x^2/(1 + (2*k+3)*(k+2)/U(k+1)) ; (continued fraction, 3rd kind, 2-step). - Sergei N. Gladkovskii, Oct 28 2012

A103425 a(n) = 3a(n-1) + a(n-2) - 3a(n-3).

Original entry on oeis.org

1, 3, 5, 15, 41, 123, 365, 1095, 3281, 9843, 29525, 88575, 265721, 797163, 2391485, 7174455, 21523361, 64570083, 193710245, 581130735, 1743392201, 5230176603, 15690529805, 47071589415, 141214768241, 423644304723, 1270932914165

Offset: 0

Paul Barry, Feb 05 2005

Binomial transform of A103424.

This is a (3, 1, -3) weighted tribonacci sequence, cf. A102001. The current sequence contains primes, including 3, 5, 41, 21523361. Is there an (a, b, c) weighted tribonacci sequence with a, b, c relatively prime which is prime-free? The general linear third-order recurrence equation x(n) = a*x(n-1) + b*x(n-2) + c*x(n-3) has a solution in terms of roots of a cubic polynomial, see Weisstein. - Jonathan Vos Post, Feb 05 2005

Eric Weisstein's World of Mathematics, Linear Recurrence Equation.
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

G.f.: (1-5x^2)/((1-x^2)(1-3x)).
E.g.f.: exp(x)(1+sinh(2x)).
a(n) = 1 + (3^n - (-1)^n)/2.

cp's OEIS Frontend

A004514 Generalized nim sum n + n in base 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Python

SageMath

Formula

Extensions

A199573 Number of round trips of length n from any of the four vertices of the cycle graph C_4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A103425 a(n) = 3a(n-1) + a(n-2) - 3a(n-3).

Views

Author

Keywords

Comments

Links

Formula

cp's OEIS Frontend

A004514 Generalized nim sum n + n in base 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Python

SageMath

Formula

Extensions

A199573 Number of round trips of length n from any of the four vertices of the cycle graph C_4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A103425 a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3).

Views

Author

Keywords

Comments

Links

Formula

A103425 a(n) = 3a(n-1) + a(n-2) - 3a(n-3).