A014601 Numbers congruent to 0 or 3 mod 4.

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

Offset: 0

Author

Eric Rains (rains(AT)caltech.edu)

Keywords

• nonn
• easy

Comments

Discriminants of orders in imaginary quadratic fields (negated). [Comment corrected by Christopher E. Thompson, Dec 11 2016]

Numbers such that Langford-Skolem problem has a solution - see A014552.

Complement of A042963. - Reinhard Zumkeller, Oct 04 2004

Also called skew amenable numbers; a number k is skew amenable if there exist a set {a(i)} of integers satisfying the relations k = Sum_{i=1..k} a(i) = -Product_{i=1..k} a(i). Thus we have 8 = 1 + 1 + 1 + 1 + 1 + 1 - 2 + 4 = -(1*1*1*1*1*1*(-2)*4). - Lekraj Beedassy, Jan 07 2005

Possible nonpositive discriminants of quadratic equation a*x^2 + b*x + c or discriminants of binary quadratic forms a*x^2 + b*x*y + c*y^2. - Artur Jasinski, Apr 28 2008

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. - Rick L. Shepherd, Oct 29 2008

A145768(a(n)) is even. - Reinhard Zumkeller, Jun 05 2012

Multiples of 4 interleaved with 1 less than multiples of 4. - Wesley Ivan Hurt, Nov 08 2013

((2*k+0) + (2*k+1) + ... + (2*k+m-1) + (2*k+m)) is even if and only if m = a(n) for some n where k is any nonnegative integer. - Gionata Neri, Jul 24 2015

Numbers whose binary reflected Gray code (A014550) ends with 0. - Amiram Eldar, May 17 2021

Examples

			G.f. = 3*x + 4*x^2 + 7*x^3 + 8*x^4 + 11*x^5 + 12*x^6 + 15*x^7 + 16*x^8 + ...
		

References

• H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, pp. 514-5.
• A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, p. 108.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• S. F. Barger, Solution to problem 10454: Amenable Numbers, Amer. Math. Monthly, Vol. 105, No. 4 (April 1998), p. 368.
• Steven R. Finch, Class number theory [Cached copy, with permission of the author]
• Heiko Harborth, Solution of Steinhaus's problem with plus and minus signs, Journal of Combinatorial Theory, Series A, Volume 12, Issue 2 (March 1972), Pages 253-259.
• Mickaël Launay, Les routes de Numland, L’énigme maths du "Monde" n°2. In French.
• Michael Penn, The most beautiful arrangement of numbers, YouTube video, 2024.
• Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A004676, A014550, A042948, A079896.

Cf. A274406. - Bruno Berselli, Jun 26 2016

Programs

Haskell

a014601 n = a014601_list !! n
a014601_list = [x | x <- [0..], mod x 4 `elem` [0, 3]]
-- Reinhard Zumkeller, Jun 05 2012

Magma

[n: n in [0..200]|n mod 4 in {0,3}]; // Vincenzo Librandi, Dec 24 2010

Maple

A014601:=n->3*n-2*floor(n/2); seq(A014601(k), k=0..100); # Wesley Ivan Hurt, Nov 08 2013

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, Apr 28 2008 *)
Select[Range[0, 124], Or[Mod[#, 4] == 0, Mod[#, 4] == 3] &] (* Ant King, Nov 18 2010 *)
CoefficientList[Series[2 x/(1 - x)^2 + (1/(1 - x) + 1/(1 + x)) x/2, {x, 0, 100}], x] (* Vincenzo Librandi, May 18 2014 *)
a[ n_] := 2 n + Mod[n, 2]; (* Michael Somos, Jul 24 2015 *)

PARI

{a(n) = 2*n + n%2}; /* Michael Somos, Dec 27 2010 */

Formula

a(n) = (n + 1)*2 + 1 - n mod 2. - Reinhard Zumkeller, Apr 21 2003

A014494(n) = A000217(a(n)). - Reinhard Zumkeller, Oct 04 2004

a(n) = Sum_{k=1..n} (2 - (-1)^k). - William A. Tedeschi, Mar 20 2008

A139131(a(n)) = A078636(a(n)). - Reinhard Zumkeller, Apr 10 2008

From R. J. Mathar, Sep 25 2009: (Start)

a(n) = a(n-1) + a(n-2) - a(n-3) for n > 2.

G.f.: x*(3+x)/((1+x)*(x-1)^2). (End)

a(n) = 2*n + (n mod 2). - Paolo Valzasina (p.valzasina(AT)gmail.com), Nov 24 2009

a(n) = (4*n - (-1)^n + 1)/2. - Bruno Berselli, Oct 06 2010

a(n) = 4*n - a(n-1) - 1 (with a(0) = 0). - Vincenzo Librandi, Dec 24 2010

a(n) = -A042948(-n) for all n in Z. - Michael Somos, Dec 27 2010

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. - Philippe Deléham, Oct 17 2011

a(n) = ceiling((4/3)*ceiling(3*n/2)). - Clark Kimberling, Jul 04 2012

a(n) = 3n - 2*floor(n/2). - Wesley Ivan Hurt, Nov 08 2013

a(n) = A042948(n+1) - 1 for all n in Z. - Michael Somos, Jul 24 2015

a(n) + a(n+1) = A004767(n) for all n in Z. - Michael Somos, Jul 24 2015

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2)/4 - Pi/8. - Amiram Eldar, Dec 05 2021

E.g.f.: ((4*x + 1)*exp(x) - exp(-x))/2. - David Lovler, Aug 04 2022