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A014601
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Congruent to 0 or 3 mod 4.
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24
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0, 3, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 28, 31, 32, 35, 36, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 67, 68, 71, 72, 75, 76, 79, 80, 83, 84, 87, 88, 91, 92, 95, 96, 99, 100, 103, 104, 107, 108, 111, 112, 115, 116, 119, 120, 123, 124
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Discriminants of imaginary quadratic fields with D=0,1 mod 4, D<0 (sequence gives -D).
n such that Langford-Skolem problem has a solution - see A014552.
A014494(n) = A000217(a(n)); complement of A042963. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 04 2004
Also called skew amenable numbers; a number n is skew amenable if there exist a set {a(i)} of integers satisfying the relations n = sum_(1,n) a(i) = -product_(1,n) a(i). Thus we have 8=1+1+1+1+1+1-2+4=-(1*1*1*1*1*1*(-2)*4). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 07 2005
A139131(a(n)) = A078636(a(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 10 2008
Possible nonpositive discriminants of quadratic equation a*x^2+b*x+c or discrminants of binary quadratic forms a*x^2+b*x*y+c^y^2 - Artur Jasinski (grafix(AT)csl.pl), Apr 28 2008
Contribution from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Oct 29 2008: (Start)
Also, disregarding the 0 term, positive integers m such that, equivalently,
(i) +-1 +-2 +-... +-m is even for all choices of signs,
(ii) +-1 +-2 +-... +-m = 0 for some choices of signs,
(iii) for all -m <= k <= m, k = +-1 +-2 +-... +-(k-1) +-(k+1) +-(k+2) +-... +-m for at least one choice of signs. (End)
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REFERENCES
| S. F. Barger, Solution to problem 10454, "Amenable Numbers", Amer. Math. Monthly Vol. 105 No. 4 April 1998 MAA Washington DC.
H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, pp. 514-5.
A. Scholz and B. Schoeneberg, Einfuehrung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, p. 108.
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LINKS
| S. R. Finch, Class number theory
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FORMULA
| a(n) = (n+1)*2 + 1 - n mod 2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 21 2003
a(n) = Sum{k=1..n, 2 - (-1)^k} - William A. Tedeschi (fynmun(AT)hotmail.com), Mar 20 2008
a(n)=a(n-1)+a(n-2)-a(n-3). G.f.: x*(3+x)/( (1+x)* (x-1)^2 ). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 25 2009]
a(n) = 2*n + (n mod 2) [From Paolo Valzasina (p.valzasina(AT)gmail.com), Nov 24 2009]
a(n) = (4*n-(-1)^n+1)/2. [From Bruno Berselli (berselli.bruno(AT)yahoo.it), Oct 06 2010]
a(n)=4*n-a(n-1)-1 (with a(0)=0) [From Vincenzo Librandi, Dec 24 2010]
a(n) = - A042948(-n). G.f.: 2*x/(1-x)^2 + (1/(1-x) + 1/(1+x))*x/2. - Michael Somos, Dec 27 2010
a(n)=Sum_k>=0 {A030308(n,k)*b(k)} with b(0)=3 and b(k)=2^(k+1) for k>0.- From DELEHAM Philippe, Oct 17 2011.
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MATHEMATICA
| aa = {}; Do[Do[Do[d = b^2 - 4 a c; If[d <= 0, AppendTo[aa, -d]], {a, 0, 50}], {b, 0, 50}], {c, 0, 50}]; Union[aa] - Artur Jasinski (grafix(AT)csl.pl), Apr 28 2008
Select[Range[0, 124], Or[Mod[#, 4] == 0, Mod[#, 4] == 3] &] (From Ant King Nov 18 2010)
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PROG
| (MAGMA)[n: n in [0..200]|n mod 4 in {0, 3}][From V. Librandi, Dec 24 2010]
(PARI) a(n) = 2*n + n%2 /* Michael Somos, Dec 27 2010 */
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CROSSREFS
| Cf. A079896.
Sequence in context: A032788 A070874 A187582 * A154708 A026444 A003171
Adjacent sequences: A014598 A014599 A014600 * A014602 A014603 A014604
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KEYWORD
| nonn,easy
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AUTHOR
| Eric Rains (rains(AT)caltech.edu)
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