A352111 In the factorial base expansion of n, replace each place value, say k! with k > 0, by (-1)^(k-1) * k!.

0, 1, -2, -1, -4, -3, 6, 7, 4, 5, 2, 3, 12, 13, 10, 11, 8, 9, 18, 19, 16, 17, 14, 15, -24, -23, -26, -25, -28, -27, -18, -17, -20, -19, -22, -21, -12, -11, -14, -13, -16, -15, -6, -5, -8, -7, -10, -9, -48, -47, -50, -49, -52, -51, -42, -41, -44, -43, -46, -45

Offset: 0

Author

Rémy Sigrist, Mar 05 2022

Keywords

• sign
• base

Comments

This sequence maps natural numbers uniquely onto integers.

Examples

			For n = 42:
- 42 = 1 * 4! + 3 * 3! + 0 * 2! + 0 * 1!,
- so a(42) = 1 * -4! + 3 * 3! + 0 * -2! + 0 * 1! = -6.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..5039 (terms for n < 7!)
• Index entries for sequences related to factorial base representation

Crossrefs

See A053985, A065369, A073835, A352112 for similar sequences.

Cf. A000142, A108731.

Programs

PARI

a(n) = { my (f=1, v=0); for (r=2, oo, if (n, v+=f*(n%r); n\=r; f*=-r, return (v))) }