A130404 Partial sums of A093178.

1, 2, 3, 6, 7, 12, 13, 20, 21, 30, 31, 42, 43, 56, 57, 72, 73, 90, 91, 110, 111, 132, 133, 156, 157, 182, 183, 210, 211, 240, 241, 272, 273, 306, 307, 342, 343, 380, 381, 420, 421, 462, 463, 506, 507, 552, 553, 600, 601, 650, 651, 702, 703, 756, 757, 812, 813

Offset: 1

Author

Klaus Brockhaus, May 25 2007

Keywords

• nonn
• easy

Comments

Numbers n such that floor(n/2) is a positive triangular number. - Bruno Berselli, Sep 15 2014

Links

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

Crossrefs

Cf. A093178, A002061, A002378, A247375.

Programs

Magma

&cat[ [ n^2-n+1, n*(n+1) ]: n in [1..29] ];

Mathematica

Table[If[EvenQ[n], 1, n], {n, 0, 56}] // Accumulate (* Jean-François Alcover, Jun 10 2013 *)
Accumulate[Join[{1},Riffle[Range[1,85,2],1]]] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{1,2,3,6,7},90] (* Harvey P. Dale, Jun 01 2016 *)

PARI

{s=0; for(n=1, 57, s=s+if(n%2>0, 1, n-1); print1(s, ","))}

PARI

{for(n=1, 57, print1(if(n%2>0, (n^2+3)/4, (n^2+2*n)/4), ","))}

Formula

a(1) = 1; for n > 1, a(n) = a(n-1)+1 if n is odd, a(n) = a(n-1)+(n-1) if n is even.

a(n) = A002061((n+1)/2) = (n^2+3)/4 if n is odd, a(n) = A002378(n/2) = (n^2+2*n)/4 if n is even.

G.f.: x*(1+x-x^2+x^3)/((1-x)^3*(1+x)^2).

a(n) = A093178(n) - A093178(n-1).

a(1) = 1; a(n) = a(n-1) + n^(n mod 2) = (1/4)*(n^2 + 2n + 4 + (n mod 2)*(2n-1)). - Rolf Pleisch, Feb 04 2008

a(n) = (2*(n-1)*(n+2) + (2*n-3)*(-1)^n+7)/8. - Bruno Berselli, Mar 31 2011