A083392 Alternating partial sums of A000217.

0, -1, 2, -4, 6, -9, 12, -16, 20, -25, 30, -36, 42, -49, 56, -64, 72, -81, 90, -100, 110, -121, 132, -144, 156, -169, 182, -196, 210, -225, 240, -256, 272, -289, 306, -324, 342, -361, 380, -400, 420, -441, 462, -484, 506, -529, 552, -576, 600, -625, 650, -676, 702

Offset

0,3

Comments

Conjecture: for n > 0, a(n-1) is equal to the determinant of an n X n symmetric Toeplitz matrix M(n) whose first row consists of a single zero followed by successive positive integers repeated (A004526). - Stefano Spezia, Jan 10 2020

Links

William A. Tedeschi, Table of n, a(n) for n = 0..10000

Wikipedia, Toeplitz Matrix

Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Formula

a(n) = Sum_{i=0..n} (-1)^i*t(i) where t(i) = i*(i+1)/2.

From R. J. Mathar, Feb 09 2010: (Start)

a(n) = -2*a(n-1) + 2*a(n-3) + a(n-4) for n > 3.

G.f.: x/((x-1)*(1+x)^3). (End)

a(n) = (-1)^n * ((n^2+n)/2 - floor(n^2/4)). - William A. Tedeschi, Aug 24 2010

E.g.f.: (1/4)*((x - 3)*x*cosh(x) - (x^2 - 3*x + 1)*sinh(x)). - Stefano Spezia, Jan 11 2020

Negative of the Euler transform of length 2 sequence [-2, 3]. - Michael Somos, Apr 27 2020

Example

a(4) = t(0) - t(1) + t(2) - t(3) + t(4) = 0 - 1 + 3 - 6 + 10 = 6.
G.f. = - x + 2*x^2 - 4*x^3 + 6*x^4 - 9*x^5 + 12*x^6 - 16*x^7 + ... - Michael Somos, Apr 27 2020

Mathematica

LinearRecurrence[{-2, 0, 2, 1}, {0, -1, 2, -4}, 60] (* Harvey P. Dale, Mar 16 2016 *)

Prog

(PARI) t(n)=n*(n+1)/2;
for (n=0, 30, print1(sum(i=0, n, (-1)^i*t(i)), ", "))
(Magma) [(-1)^n*((n^2+n)/2 - Floor(n^2/4)): n in [0..50]]; // G. C. Greubel, Oct 29 2017

Crossrefs

Cf. A000217, A002620, A004526.

Sequence in context: A086378 A088900 A371972 * A076921 A002620 A087811

Adjacent sequences: A083389 A083390 A083391 * A083393 A083394 A083395

Keyword

sign,easy

Author

Jon Perry, Jun 11 2003

Extensions

More terms from David W. Wilson, Jun 14 2003

Status

approved