A199770 -id:A199770 - cp's OEIS frontend

A007461 Shifts left under AND-convolution with itself.

1, 1, 2, 1, 2, 4, 0, 5, 2, 4, 0, 10, 0, 12, 4, 13, 6, 12, 0, 18, 12, 20, 20, 36, 20, 36, 16, 44, 32, 60, 40, 73, 50, 56, 40, 58, 44, 52, 60, 84, 36, 112, 88, 108, 136, 132, 152, 178, 136, 232, 108, 260, 244, 256, 304, 288

Offset: 0

N. J. A. Sloane

a(A000225(n)) mod 2 = 1, a(A062289(n)) mod 2 = 0. [Reinhard Zumkeller, Apr 02 2012]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Cf. A007460, A199770.

import Data.Bits ((.&.))
a007461 n = a007461_list !! n
a007461_list = 1 : f [1,1] where
   f xs = x : f (x:xs) where
     x = sum $ zipWith (.&.) xs $ tail $ reverse xs :: Integer
-- Reinhard Zumkeller, Apr 02 2012

a:= proc(n) option remember; `if`(n=0, 1, add(
      Bits[And](a(i), a(n-1-i)), i=0..n-1))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 16 2018

a[0]=1; a[1]=1; a[n_] := a[n] = Sum[BitAnd[a[k], a[n-k-1]], {k, 0, n-1}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 07 2012 *)

A007462 Shifts left under XOR-convolution with itself.

Original entry on oeis.org

0, 1, 2, 4, 14, 38, 118, 338, 1006, 2990, 8974, 26862, 80510, 241390, 723934, 2171046, 6512910, 19536974, 58608782, 175821710, 527470318, 1582385678, 4747139342, 14241362318, 42724100334, 128172182990, 384516408110, 1153548740206, 3460645850030, 10381936700110

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Cf. A192484, A199770.

import Data.Bits (xor)
a007462 n = a007462_list !! n
a007462_list = 0 : 1 : f [1,0] where
   f xs = y : f (y : xs) where
     y = sum $ zipWith xor xs $ reverse xs :: Integer
-- Reinhard Zumkeller, Jul 15 2012

with(Bits):
a:= proc(n) option remember;
      `if`(n<2, n, add(Xor(a(i), a(n-1-i)), i=0..n-1))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 22 2012

a[0]=0; a[1]=1; a[n_] := a[n] = Sum[BitXor[a[k], a[n-k-1]], {k, 0, n-1}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Sep 07 2012, after Alois P. Heinz *)

a(n) ~ c * 3^n, where c = 0.151273188266276362886260408769663538575624024971525940842364624... . - Vaclav Kotesovec, Sep 10 2014

A192484 Shifts left under XOR-convolution: a(n) = Sum_{k=0..n-1} a(k) XOR a(n-k-1) for n>1 with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 6, 14, 38, 102, 294, 854, 2566, 7622, 22790, 68166, 204678, 613318, 1839750, 5518310, 16553798, 49656774, 148968774, 446888518, 1340652486, 4021929542, 12065804486, 36197270598, 108591619654, 325774522822, 977323956550

Offset: 0

Paul D. Hanna, Jul 02 2011

nonn

Limit a(n+1)/a(n) = 3.

			Given a(0)=1, a(1)=2, illustrate XOR convolution for the initial terms.
a(2) = 1 XOR 2 + 2 XOR 1 = 3 + 3 = 6;
a(3) = 1 XOR 6 + 2 XOR 2 + 6 XOR 1 = 7 + 0 + 7 = 14;
a(4) = 1 XOR 14 + 2 XOR 6 + 6 XOR 2 + 14 XOR 1 = 15 + 4 + 4 + 15 = 38; ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. variant: A007462.

Cf. A199770.

import Data.Bits (xor)
a192484 n = a192484_list !! n
a192484_list = 1 : 2 : f [2,1] where
   f xs = y : f (y : xs) where
     y = sum $ zipWith xor xs $ reverse xs :: Integer
-- Reinhard Zumkeller, Jul 15 2012

{a(n)=if(n<2,n+1,sum(k=0,n-1,bitxor(a(k),a(n-k-1))))}

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A007461 Shifts left under AND-convolution with itself.

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A007462 Shifts left under XOR-convolution with itself.

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A192484 Shifts left under XOR-convolution: a(n) = Sum_{k=0..n-1} a(k) XOR a(n-k-1) for n>1 with a(0)=1, a(1)=2.

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