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A306562 a(n) = 1 + 2 - 3 - 4 + 5 + 6 + 7 - 8 - 9 - 10 - 11 + 12 + 13 + 14 + 15 + ... + (+-1)*n, where, after the 1st summand there is one plus, two minuses, three plusses, etc.

%I A306562 #38 Jun 08 2025 13:40:34
%S A306562 1,3,0,-4,1,7,14,6,-3,-13,-24,-12,1,15,30,46,29,11,-8,-28,-49,-71,-48,
%T A306562 -24,1,27,54,82,111,81,50,18,-15,-49,-84,-120,-157,-119,-80,-40,1,43,
%U A306562 86,130,175,221,174,126,77,27,-24,-76,-129,-183,-238,-294,-237,-179
%N A306562 a(n) = 1 + 2 - 3 - 4 + 5 + 6 + 7 - 8 - 9 - 10 - 11 + 12 + 13 + 14 + 15 + ... + (+-1)*n, where, after the 1st summand there is one plus, two minuses, three plusses, etc.
%H A306562 Alois P. Heinz, <a href="/A306562/b306562.txt">Table of n, a(n) for n = 0..10011</a> (first 1001 terms from Harvey P. Dale)
%F A306562 F(n) = ((-1)^(n+1)(2n+1)(2n^2+2n+5)+21)/16 gives local extrema 3, -4, 14, -24, 46, -71, 111, -157, ... (conjectured). - _Jean-François Alcover_, Jun 01 2019
%F A306562 For n > 0, a(n) = 1 + Sum_{k=1..n} (-1)^(A002024(k)+1)*(k+1). - _Jinyuan Wang_, Aug 06 2019
%F A306562 a(n) = 1 <=> n in { A046092 }. - _Alois P. Heinz_, Jun 08 2025
%e A306562 a(0) = 1                         =  1
%e A306562 a(1) = 1 + 2                     =  3
%e A306562 a(2) = 1 + 2 - 3                 =  0
%e A306562 a(3) = 1 + 2 - 3 - 4             = -4
%e A306562 a(4) = 1 + 2 - 3 - 4 + 5         =  1
%e A306562 a(5) = 1 + 2 - 3 - 4 + 5 + 6     =  7
%e A306562 a(6) = 1 + 2 - 3 - 4 + 5 + 6 + 7 = 14
%p A306562 a:= proc(n) option remember: `if`(n=0, 1,
%p A306562       a(n-1)+(n+1)*(-1)^floor(sqrt(2*n)-1/2))
%p A306562     end:
%p A306562 seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 26 2019
%t A306562 With[{nn=20},Accumulate[Flatten[Join[{1,2},Times@@@Partition[Riffle[TakeList[Range[3,3+(nn(nn+1))/2],Range[2,nn]],{-1,1}],2]]]]] (* _Harvey P. Dale_, Mar 24 2024 *)
%o A306562 (Python)
%o A306562 from math import isqrt
%o A306562 def A306562(n): return 1+sum((k if isqrt(k-1<<3)+1&2 else -k) for k in range(2,n+2)) # _Chai Wah Wu_, Jun 07 2025
%Y A306562 Cf. A000217, A002024, A046092, A064520.
%K A306562 sign,look
%O A306562 0,2
%A A306562 _Brandon J. Butierres_, Feb 23 2019
%E A306562 New name from _Michel Marcus_, Apr 11 2019