A141169 -id:A141169 - cp's OEIS frontend

A119282 Alternating sum of the first n Fibonacci numbers.

0, -1, 0, -2, 1, -4, 4, -9, 12, -22, 33, -56, 88, -145, 232, -378, 609, -988, 1596, -2585, 4180, -6766, 10945, -17712, 28656, -46369, 75024, -121394, 196417, -317812, 514228, -832041, 1346268, -2178310, 3524577, -5702888, 9227464, -14930353, 24157816, -39088170, 63245985, -102334156, 165580140, -267914297, 433494436, -701408734, 1134903169, -1836311904, 2971215072, -4807526977, 7778742048

Offset: 0

Stuart Clary, May 13 2006

Apart from signs, same as A008346.
Natural bilateral extension (brackets mark index 0): ..., 88, 54, 33, 20, 12, 7, 4, 2, 1, 0, [0], -1, 0, -2, 1, -4, 4, -9, 12, -22, 3, ... This is A000071-reversed followed by A119282.
Alternating sums of rows of the triangle in A141169. - Reinhard Zumkeller, Mar 22 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
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 (0,2,-1)

Cf. A000071, A008346, A119283, A119284, A119285, A119286, A119287.

Cf. A128696, A128698.

[0] cat [(&+[(-1)^k*Fibonacci(k):k in [1..n]]): n in [1..30]]; // G. C. Greubel, Jan 17 2018

FoldList[#1 - Fibonacci@ #2 &, -Range@ 50] (* Michael De Vlieger, Jan 27 2016 *)
Accumulate[Table[(-1)^n Fibonacci[n], {n, 0, 49}]] (* Alonso del Arte, Apr 25 2017 *)

a(n) = sum(k=1, n, (-1)^k*fibonacci(k)); \\ Michel Marcus, Jan 27 2016

Let F(n) be the Fibonacci number A000045(n).
a(n) = Sum_{k = 1..n} (-1)^k F(k).
Closed form: a(n) = (-1)^n F(n-1) - 1 = (-1)^n A008346(n-1).
Recurrence: a(n) - 2 a(n-2) + a(n-3)= 0.
G.f.: A(x) = -x/(1 - 2 x^2 + x^3) = -x/((1 - x)(1 + x - x^2)).
Another recurrence: a(n) = a(n-2) - a(n-1) - 1. - Rick L. Shepherd, Aug 12 2009

A094570 Triangle T(n,k) read by rows: T(n,k) = F(k) + F(n-k) where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 5, 4, 3, 3, 4, 5, 8, 6, 4, 4, 4, 6, 8, 13, 9, 6, 5, 5, 6, 9, 13, 21, 14, 9, 7, 6, 7, 9, 14, 21, 34, 22, 14, 10, 8, 8, 10, 14, 22, 34, 55, 35, 22, 15, 11, 10, 11, 15, 22, 35, 55, 89, 56, 35, 23, 16, 13, 13, 16, 23, 35, 56, 89, 144, 90, 56, 36, 24, 18, 16, 18, 24, 36, 56, 90, 144

Offset: 0

Clark Kimberling, May 12 2004

			Triangle begins:
0;
1, 1;
1, 2, 1;
2, 2, 2, 2;
3, 3, 2, 3, 3;
5, 4, 3, 3, 4, 5;
8, 6, 4, 4, 4, 6, 8;
13, 9, 6, 5, 5, 6, 9, 13;
21, 14, 9, 7, 6, 7, 9, 14, 21;

Reinhard Zumkeller, Rows n=0..125 of triangle, flattened

Cf. A000045, A000071, A006355, A141169.

row(n) = vector(n+1, k, k--; fibonacci(k)+fibonacci(n-k)); \\ Michel Marcus, Mar 22 2021

Row n: F(0)+F(n), F(1)+F(n-1), F(2)+F(n-2), ..., F(n-1)+F(1), F(n)+F(0).
From Reinhard Zumkeller, Mar 21 2011: (Start)
T(n,0) = T(n,n) = A000045(n).
T(2*n,n) = A006355(n+1).
T(n,k) = A141169(n,k) + A141169(n,n-k). (End)
Sum(T(n,k), 0<=k<=n) = 2*A000071(n+2) = 2*A000045(n+2) - 2. - Philippe Deléham, Apr 07 2013

cp's OEIS Frontend

A119282 Alternating sum of the first n Fibonacci numbers.

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