A128918 a(n) = n*(n+1)/2 if n is odd, otherwise (n-1)*n/2 + 1.

1, 1, 2, 6, 7, 15, 16, 28, 29, 45, 46, 66, 67, 91, 92, 120, 121, 153, 154, 190, 191, 231, 232, 276, 277, 325, 326, 378, 379, 435, 436, 496, 497, 561, 562, 630, 631, 703, 704, 780, 781, 861, 862, 946, 947, 1035, 1036, 1128, 1129, 1225, 1226, 1326, 1327, 1431, 1432, 1540

Offset: 0

Author

N. J. A. Sloane, Sep 26 2007

Keywords

• nonn
• easy

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000384, A130883, A131179.

Programs

Haskell

a128918 n = (n + m - 1) * n' + m * n - m + 1  where (n', m) = divMod n 2
-- Reinhard Zumkeller, Oct 12 2013

Maple

A128918:=n->`if`((n mod 2) = 1, n*(n+1)/2, (n-1)*n/2+1): seq(A128918(n), n=0..100); # Wesley Ivan Hurt, Feb 03 2017

Mathematica

Table[If[OddQ[n],(n(n+1))/2,(n(n-1))/2+1],{n,0,60}] (* or *)
LinearRecurrence[{1,2,-2,-1,1},{1,1,2,6,7},60] (* Harvey P. Dale, Mar 31 2012 *)
CoefficientList[ Series[(-4x^3 + x^2 -1)/((x -1)^3 (x + 1)^2), {x, 0, 55}], x] (* Robert G. Wilson v, Jan 20 2018 *)

PARI

a(n)=if(n%2,n*(n+1),(n-1)*n+2)/2 \\ Charles R Greathouse IV, Oct 16 2015

PARI

Vec((1 - x^2 + 4*x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40)) \\ Colin Barker, Jan 20 2018

Python

def A128918(n): return n*(n-1)//2 + 1 + (n-1)*(n%2) # Chai Wah Wu, May 24 2022

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), with a(0)=1, a(1)=1, a(2)=2, a(3)=6, a(4)=7. - Harvey P. Dale, Mar 31 2012

a(n) = (1/2)*(-1)^n*(n+(-1)^n*((n-2)*n+2)-2). - Harvey P. Dale, Mar 31 2012

a(2*n) = A130883(n); a(2*n+1) = A000384(n+1). - Reinhard Zumkeller, Oct 12 2013

G.f.: (1 - x^2 + 4*x^3) / ((1 - x)^3*(1 + x)^2). - Colin Barker, Jan 20 2018