A219977 Expansion of 1/(1+x+x^2+x^3).
1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0
Offset: 0
Examples
G.f. = 1 - x + x^4 - x^5 + x^8 - x^9 + x^12 - x^13 + x^16 - x^17 + x^20 - x^21 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..5000
- Elena Barcucci, Antonio Bernini, Stefano Bilotta, Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.
- Stefano Bilotta, Variable-length Non-overlapping Codes, arXiv preprint arXiv:1605.03785, 2016
- Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
- Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1).
Programs
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Magma
m:=100; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1+x+x^2+x^3))); // Vincenzo Librandi, Apr 22 2015 -
Mathematica
CoefficientList[Series[1/(1+x+x^2+x^3),{x,0,80}],x] (* or *) PadRight[{},120,{1,-1,0,0}] LinearRecurrence[{-1,-1,-1},{1,-1,0},80] (* Harvey P. Dale, May 22 2021 *) -
PARI
{a(n) = [1, -1, 0, 0][n%4 + 1]} /* Michael Somos, Dec 12 2012 */ -
PARI
Vec(1/(1+x+x^2+x^3) + O(x^100)) \\ Michel Marcus, Jan 28 2016
Formula
G.f.: 1/(1 +x +x^2 +x^3).
Euler transform of length 4 sequence [ -1, 0, 0, 1]. - Michael Somos, Dec 12 2012
a(n) = a(n+4) = -a(1-n). |a(n)| = A133872(n). REVERT transform is A036765. INVERT transform is A077962. - Michael Somos, Dec 12 2012
A038505(n+2) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Dec 12 2012
From Wesley Ivan Hurt, Apr 22 2015: (Start)
a(n) +a(n-1) +a(n-2) +a(n-3) = 0.
a(n) = (-1)^n/2 +(-1)^(n/2 +1/4 -(-1)^n/4)/2. (End)