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A064520 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). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
|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, ...
LINKS
FORMULA
a(n) = Sum_{k=1..n} (-1)^(A002024(k)+1)*k.
EXAMPLE
a(9) = -13 because 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 = -13.
MAPLE
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:
MATHEMATICA
Accumulate[Flatten[Table[(-1)^(n+1) Range[(n(n-1))/2+1, (n(n+1))/2], {n, 15}]]] (* Harvey P. Dale, Apr 22 2015 *)
PROG
(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
CROSSREFS
Sequence in context: A372274 A355921 A164108 * A267313 A108174 A134530
KEYWORD
sign,look,easy
AUTHOR
Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 07 2001
EXTENSIONS
More terms from James A. Sellers, Jason Earls and Vladeta Jovovic, Oct 08 2001
STATUS
approved

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Last modified June 21 12:50 EDT 2024. Contains 373544 sequences. (Running on oeis4.)