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

1,2

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

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).

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A213203, A000217.

Sequence in context: A021332 A008344 A088230 * A330420 A072329 A068630

Adjacent sequences: A181479 A181480 A181481 * A181483 A181484 A181485

Keyword

nonn,easy

Author

Jon Perry, Oct 23 2010

Extensions

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

Status

approved