A188652 First differences of A000463.

0, 1, 2, -1, 6, -5, 12, -11, 20, -19, 30, -29, 42, -41, 56, -55, 72, -71, 90, -89, 110, -109, 132, -131, 156, -155, 182, -181, 210, -209, 240, -239, 272, -271, 306, -305, 342, -341, 380, -379, 420, -419, 462, -461, 506, -505, 552, -551, 600, -599, 650, -649, 702, -701, 756, -755, 812, -811, 870, -869, 930, -929, 992, -991, 1056, -1055, 1122, -1121, 1190, -1189, 1260, -1259, 1332, -1331, 1406

Offset: 1

Author

Reinhard Zumkeller, Apr 13 2011

Keywords

• sign
• easy

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (-1,2,2,-1,-1).

Crossrefs

Cf. A188653 (first differences).

Cf. A000463, A002378, A165900.

Programs

Haskell

a188652 n = a188652_list !! (n-1)
a188652_list = zipWith (-) (tail a000463_list) a000463_list

Mathematica

Differences[Flatten[Array[{#,#^2}&,40]]] (* Harvey P. Dale, Aug 04 2012 *)

Formula

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

a(n) = A000463(n+1) - A000463(n).

a(2n-1) = A002378(n-1), a(2n) = - A165900(n).

G.f.: -x^2*(-1-3*x+x^2+x^3) / ( (x-1)^2*(1+x)^3 ). - R. J. Mathar, Apr 14 2011

a(n) = (2*n+3-(2*n^2-2*n-5)*(-1)^n)/8. - Luce ETIENNE, Dec 18 2014

E.g.f.: ((4 + x - x^2)*cosh(x) - (1 - x - x^2)*sinh(x) - 4)/4. - Stefano Spezia, Jul 08 2023

Sum_{n>=2} 1/a(n) = 1 + (sqrt(5)-5)*Pi*tan(sqrt(5)*Pi/2)/(5*(sqrt(5)-1)). - Amiram Eldar, May 11 2025