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A181482 The sum of the first n integers, with every third integer taken negative. 4
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 (list; graph; refs; listen; history; text; internal format)
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 to sequences with 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 *)

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

CROSSREFS

Cf. A213203, A000217.

Sequence in context: A021332 A008344 A088230 * A072329 A068630 A079406

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

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Last modified October 22 21:52 EDT 2014. Contains 248411 sequences.