A054240 Bit-interleaved number addition table; like binary addition but carries shift 2 instead of 1; addition base sqrt(2).

0, 1, 1, 2, 4, 2, 3, 3, 3, 3, 4, 6, 8, 6, 4, 5, 5, 9, 9, 5, 5, 6, 16, 6, 12, 6, 16, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 18, 12, 18, 16, 18, 12, 18, 8, 9, 9, 13, 13, 17, 17, 13, 13, 9, 9, 10, 12, 10, 24, 18, 20, 18, 24, 10, 12, 10, 11, 11, 11, 11, 19, 19, 19, 19, 11, 11, 11, 11, 12, 14, 32, 14

Offset: 0

Author

Marc LeBrun, Feb 07 2000

Keywords

• nonn
• base
• tabl
• easy
• look

Examples

			T(3,1)=6 because (0*2 + 1*sqrt(2) + 1*1) + (0*2 + 0*sqrt(2) + 1*1) = (1*2 + 1*sqrt(2) + 0*1) (i.e., base sqrt(2) addition).
		

Links

• Reinhard Zumkeller, Antidiagonals n=0..125 of square array, flattened

Crossrefs

Cf. A000695, A054239, A057300, A062880, A352909 (pairs (i,j) such that A(i,j) = i+j).

Cf. A201651 (triangle read by rows).

Programs

Haskell

import Data.Bits (xor, (.&.), shift)
a054240 :: Integer -> Integer -> Integer
a054240 x 0 = x
a054240 x y = a054240 (x `xor` y) (shift (x .&. y) 2)
a054240_adiag n =  map (\k -> a054240 (n - k) k) [0..n]
a054240_square = map a054240_adiag [0..]
-- Reinhard Zumkeller, Dec 03 2011

Formula

From Peter Munn, Dec 10 2019: (Start)

A(m,0) = A(0,m) = m.

A(n,k) = A(k,n).

A(n, A(m,k)) = A(A(n,m), k).

A(m,m) = 4*m.

A(2*n, 2*k) = 2*A(n,k).

A(A000695(n), A000695(k)) = A000695(n+k).

A(A000695(n), 2*A000695(k)) = A000695(n) + 2*A000695(k).

A(A000695(n) + 2*A000695(m), k) = A(A000695(n), k) + A(2*A000695(m), k) - k.

A(A057300(n), A057300(k)) = A057300(A(n,k)).

(End)