A225883 a(n) = (-1)^n * (1 - 2^n).

0, 1, -3, 7, -15, 31, -63, 127, -255, 511, -1023, 2047, -4095, 8191, -16383, 32767, -65535, 131071, -262143, 524287, -1048575, 2097151, -4194303, 8388607, -16777215, 33554431, -67108863, 134217727, -268435455, 536870911, -1073741823, 2147483647, -4294967295

Offset: 0

Author

Michael Somos, May 19 2013

Keywords

• sign
• easy

Comments

A001047(n) = p(0) where p(x) is the unique degree-(n-1) polynomial such that p(k) = a(k) for k = 1, 2, ..., n.

a(n) = p(n) where p(x) is the unique degree-(n-1) polynomial such that p(k) = (-1)^k for k = 0, 1, ..., n-1.

Examples

			G.f. = x - 3*x^2 + 7*x^3 - 15*x^4 + 31*x^5 - 63*x^6 + 127*x^7 - 255*x^8 + 511*x^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28.
• Index entries for linear recurrences with constant coefficients, signature (-3,-2).

Crossrefs

Cf. A000225, A001003, A001047, A108520.

Programs

Magma

[(-1)^n*(1 - 2^n): n in [0..50]]; // G. C. Greubel, Aug 09 2018

Mathematica

a[ n_] := If[ n<0, 0, (-1)^n (1 - 2^n)];
LinearRecurrence[{-3,-2}, {0,1}, 50] (* G. C. Greubel, Aug 09 2018 *)

PARI

{a(n) = if( n<0, 0, (-1)^n * (1 - 2^n))};

PARI

{a(n) = if( n<0, 0, polcoeff( x / ((1 + x) * (1 + 2*x)) + x * O(x^n), n))};

Formula

G.f.: x / ((1 + x) * (1 + 2*x)) = 1 / (1+x) - 1 / (1 + 2*x).

E.g.f.: exp(-x) - exp(-2*x). a(n) = -2 * a(n-1) - (-1)^n if n>0.

a(n) = -(-1)^n * A000225(n). a(n) = -3 * a(n-1) - 2 * a(n-2) if n>1.

REVERT transform is A001003 omitting a(0)=0.

INVERT transform is A108520.

2^n = a(n+1)^2 - a(n) * a(n+2).

|a(n)| = A168604(n+2)= A000225(n).