A181482 The sum of the first n integers, with every third integer taken negative.

1, 3, 0, 4, 9, 3, 10, 18, 9, 19, 30, 18, 31, 45, 30, 46, 63, 45, 64, 84, 63, 85, 108, 84, 109, 135, 108, 136, 165, 135, 166, 198, 165, 199, 234, 198, 235, 273, 234, 274, 315, 273, 316, 360, 315, 361, 408, 360, 409, 459, 408, 460, 513, 459, 514, 570, 513, 571, 630

Offset: 1

Author

Jon Perry, Oct 23 2010

Keywords

• nonn
• easy

Comments

The partial sum for the first 10^k terms are 76, 57256, 55722556, 55572225556, 55557222255556,..., i.e., the palindrome 5{k}2{k-1}5{k} plus 1+2*10^(2*k-1). - R. J. Cano, Mar 10 2013, edited by M. F. Hasler, Mar 25 2013

Examples

			a(7) = 1 + 2 - 3 + 4 + 5 - 6 + 7 = 10.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Wolfram Alpha, WA Query
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Cf. A213203, A000217.

Programs

Haskell

a181482 n = a181482_list !! (n-1)
a181482_list = scanl1 (+) $ zipWith (*) [1..] $ cycle [1, 1, -1]
-- Reinhard Zumkeller, Nov 23 2014

JavaScript

c = 0; for (i = 1; i < 100; i++) {c += Math.pow(-1, (i + 1) % 3)*i; document.write(c, ", ");} // Jon Perry, Feb 17 2013

JavaScript

c=0; for (i = 1; i < 100; i++) { c += (1 - (i + 1) % 3 % 2 * 2) * i; document.write(c + ", "); } // Jon Perry, Mar 03 2013

Magma

I:=[1,3,0,4,9,3,10]; [n le 7 select I[n] else Self(n-1)+2*Self(n-3)-2*Self(n-4)-Self(n-6)+Self(n-7): n in [1..60]]; // Vincenzo Librandi, Feb 17 2013

Mathematica

a[n_] := Sum[If[Mod[j, 3] == 0, -j, j], {j, 1, n}]; Table[a[i], {i, 1, 50, 1}] (* Jon Perry *)
tri[n_] := n (n + 1)/2; f[n_] := tri@ n - 6 tri@ Floor[n/3]; Array[f, 63] (* Robert G. Wilson v, Oct 24 2010 *)
CoefficientList[Series[-(1 + 2*x + 2*x^3 + x^4 - 3*x^2)/((1 + x + x^2)^2*(x - 1)^3), {x, 0,30}], x] (* Vincenzo Librandi, Feb 17 2013 *)
Table[Sum[k * (-1)^Boole[Mod[k, 3] == 0], {k, n}], {n, 60}] (* Alonso del Arte, Feb 24 2013 *)
With[{nn=20},Accumulate[Times@@@Partition[Riffle[Range[3nn],{1,1,-1}],2]]] (* Harvey P. Dale, Feb 09 2015 *)

PARI

a(n)=sum(k=1,n,k*((-1)^(k%3==0)) )  \\ R. J. Cano, Feb 26 2013

PARI

a(n)={my(y=n\3);n*(n+1)\2-3*y*(y+1)} \\ R. J. Cano, Feb 28 2013

Formula

From R. J. Mathar, Oct 23 2010: (Start)

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7).

G.f.: -x*(1+2*x+2*x^3+x^4-3*x^2) / ( (1+x+x^2)^2*(x-1)^3 ).

a(n) = 2*A061347(n+1)/9 +4/9 + n*(n+1)/6 + 2*b(n)/3 where b(3k+1) = 0, b(3k) = -3k - 1 and b(3k+2) = 3k + 3. (End)

a(n) = sum((i+1)*A131561(i), i=0..n-1) = A000217(n)-6*A000217(floor(n/3)). [Bruno Berselli, Dec 10 2010]

a(0) = 0, a(n) = a(n-1) + (-1)^((n + 1) mod 3)*n - Jon Perry, Feb 17 2013

a(n) = n*(n+1)/2-3*floor(n/3)*(floor(n/3)+1). - R. J. Cano, Mar 01 2013 [Same as Berselli's formula. - Ed.]

a(3k) = 3k(k-1)/2. - Jon Perry, Mar 01 2013

a(0) = 0, a(n) = a(n-1) + (1 - ((n+1) mod 3 mod 2) * 2) * n. - Jon Perry, Mar 03 2013

Extensions

More terms added by R. J. Mathar, Oct 23 2010