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A270439 Alternating sum of nonsquares (A000037). 0
2, -1, 4, -2, 5, -3, 7, -4, 8, -5, 9, -6, 11, -7, 12, -8, 13, -9, 14, -10, 16, -11, 17, -12, 18, -13, 19, -14, 20, -15, 22, -16, 23, -17, 24, -18, 25, -19, 26, -20, 27, -21, 29, -22, 30, -23, 31, -24, 32, -25, 33, -26, 34, -27, 35, -28, 37, -29, 38, -30, 39, -31, 40, -32, 41, -33, 42, -34, 43, -35, 44, -36, 46, -37, 47 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Interleaving of nontriangular numbers (A014132) and negative integers (A001478).

LINKS

Table of n, a(n) for n=1..75.

FORMULA

a(n) = Sum_{k = 1..n} (-1)^(k+1)*(k + floor(1/2 + sqrt(k))).

a(n) = Sum_{k = 1..n} (-1)^(k+1)*A000037(k).

a(2^k*m) = -2^(k-1) * m, k > 0.

a(2k - 1) = k + floor(1/2 + sqrt(2*k)), a(2k) = -k, k > 0.

a(2k - 1) = A014132(k), a(2k) = A001478(k).

EXAMPLE

a(1) = a(2*1-1) = 1 + floor(1/2 + sqrt(2*1)) = 2;

a(2) = a(2*1) = -1;

a(3) = a(2*2-1) = 2 + floor(1/2 + sqrt(2*2)) = 4;

a(4) = a(2*2) = -2;

a(5) = a(2*3-1) = 3 + floor(1/2 + sqrt(2*3)) = 5;

a(6) = a(2*3) = -3, etc.

or

a(1) = 2;

a(2) = 2 - 3 = -1;

a(3) = 2 - 3 + 5 = 4;

a(4) = 2 - 3 + 5 - 6 = -2;

a(5) = 2 - 3 + 5 - 6 + 7 = 5;

a(6) = 2 - 3 + 5 - 6 + 7 - 8 = -3, etc.

(2, 3, 5, 6, 7, 8, ... is the nonsquares).

MATHEMATICA

Table[Sum[(-1)^(k + 1) (k + Floor[1/2 + Sqrt[k]]), {k, n}], {n, 75}]

PROG

(PARI) a(n)=if(n%2, sqrtint(4*n-3)+n+2, -n)\2 \\ Charles R Greathouse IV, Aug 03 2016

CROSSREFS

Cf. A000037, A001478, A014132, A086849.

Sequence in context: A276133 A054269 A086450 * A106044 A124896 A008742

Adjacent sequences:  A270436 A270437 A270438 * A270440 A270441 A270442

KEYWORD

easy,sign

AUTHOR

Ilya Gutkovskiy, Jul 12 2016

STATUS

approved

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Last modified August 17 16:20 EDT 2018. Contains 313816 sequences. (Running on oeis4.)