A124745 Expansion of (1+x)/(1-x^2+x^3).

1, 1, 1, 0, 0, -1, 0, -1, 1, -1, 2, -2, 3, -4, 5, -7, 9, -12, 16, -21, 28, -37, 49, -65, 86, -114, 151, -200, 265, -351, 465, -616, 816, -1081, 1432, -1897, 2513, -3329, 4410, -5842, 7739, -10252, 13581, -17991, 23833, -31572, 41824, -55405, 73396, -97229, 128801

Offset: 0

Author

Paul Barry, Nov 06 2006

Keywords

• easy
• sign

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
• Index entries for linear recurrences with constant coefficients, signature (0,1,-1).

Crossrefs

Row sums of A124744.

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

Programs

Mathematica

LinearRecurrence[{0, 1, -1}, {1, 1, 1}, 100] (* Paolo Xausa, Aug 27 2024 *)

Formula

a(n) = Sum_{k=0..n} C(floor(k/2),n-k)*(-1)^(n-k) = (-1)^n*A078027(n).

a(n) = a(n-2) - a(n-3) with a(0) = a(1) = a(2) = 1. - Taras Goy, Mar 24 2019