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A064520
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a(n) = + 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 - 10 + 11 + 12 + 13 + 14 + 15 - ... + (+-1)*n, where there is one plus, two minuses, three pluses, etc. (see A002024).
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3
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1, -1, -4, 0, 5, 11, 4, -4, -13, -23, -12, 0, 13, 27, 42, 26, 9, -9, -28, -48, -69, -47, -24, 0, 25, 51, 78, 106, 77, 47, 16, -16, -49, -83, -118, -154, -117, -79, -40, 0, 41, 83, 126, 170, 215, 169, 122, 74, 25, -25, -76, -128, -181, -235, -290, -234, -177, -119, -60, 0, 61, 123, 186, 250, 315, 381, 314, 246, 177
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OFFSET
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1,3
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COMMENTS
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|a(n)| takes its locally maximal values when n is a triangular number, the maximal values being given by A019298.
The maximal positive/negative values occur for n = 1, 3, 6, 10, 15, 21 ... the triangular numbers and are a(n) = 1, -4, 11, -23, 42, -69,106, 215, 381, 616 ... +- int(sqrt(n^3/2) + 0.22098 * sqrt(n)). a(n) = n for n = 5, 13, 25, 41, 61, 85, ... m*(m*2-2)+1 and the previous number is equal to 0. Positive numbers which do not occur in this sequence are 2, 3, 6, 7, 8, 10, 12, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, ...
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n} (-1)^(A002024(k)+1)*k.
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EXAMPLE
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a(9) = -13 because 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 = -13.
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MAPLE
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a := proc(n) option remember: if n=1 then RETURN(1) fi: a(n-1) + n*(-1)^( floor(1/2 + sqrt(2*n)+1)); end: for n from 1 to 150 do printf(`%d, `, a(n)) od:
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MATHEMATICA
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Accumulate[Flatten[Table[(-1)^(n+1) Range[(n(n-1))/2+1, (n(n+1))/2], {n, 15}]]] (* Harvey P. Dale, Apr 22 2015 *)
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PROG
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(PARI) t(n) = floor(1/2+sqrt(2*n)) for(n=1, 200, print1(sum(k=1, n, (-1)^(t(k)+1)*k), " "))
(PARI) t(n)= { floor(sqrt(2*n) + 1/2) } { for (n=1, 1000, a=sum(k=1, n, (-1)^(t(k) + 1)*k); write("b064520.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 17 2009
(Python)
from math import isqrt
def A064520(n): return sum(k if (isqrt(k<<3)+1>>1)&1 else -k for k in range(1, n+1)) # Chai Wah Wu, Oct 16 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 07 2001
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EXTENSIONS
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STATUS
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approved
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