A042965 Nonnegative integers not congruent to 2 mod 4.

0, 1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence (starting at 3) gives values of AUB, sorted and duplicates removed. Values of AUBUC give same sequence. - David W. Wilson

These are the nonnegative integers that can be written as a difference of two squares, i.e., n = x^2 - y^2 for integers x,y. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 25 2002. Equivalently, nonnegative numbers represented by the quadratic form x^2-y^2 of discriminant 4. The primes in this sequence are all the odd primes. - N. J. A. Sloane, May 30 2014

Numbers n such that Kronecker(4,n) == mu(gcd(4,n)). - Jon Perry, Sep 17 2002

Count, sieving out numbers of the form 2*(2*n+1) (A016825, "nombres pair-impairs"). A generalized Chebyshev transform of the Jacobsthal numbers: apply the transform g(x) -> (1/(1+x^2)) g(x/(1+x^2)) to the g.f. of A001045(n+2). Partial sums of 1,2,1,1,2,1,.... - Paul Barry, Apr 26 2005

For n>1, equals union of A020883 and A020884. - Lekraj Beedassy, Sep 28 2004

The sequence 1,1,3,4,5,... is the image of A001045(n+1) under the mapping g(x) -> g(x/(1+x^2)). - Paul Barry, Jan 16 2005

With offset 0 starting (1, 3, 4,...) = INVERT transform of A009531 starting (1, 2, -1, -4, 1, 6,...) with offset 0.

Apparently these are the regular numbers modulo 4 [Haukkanan & Toth]. - R. J. Mathar, Oct 07 2011

Numbers of the form x*y in nonnegative integers x,y with x+y even. - Michael Somos, May 18 2013

Convolution of A106510 with A000027. - L. Edson Jeffery, Jan 24 2015

Numbers that are the sum of zero or more consecutive odd positive numbers. - Gionata Neri, Sep 01 2015

Numbers that are congruent to {0, 1, 3} mod 4. - Wesley Ivan Hurt, Jun 10 2016

Nonnegative integers of the form (2+(3*m-2)/4^j)/3, j,m >= 0. - L. Edson Jeffery, Jan 02 2017

This is { x^2 - y^2; x >= y >= 0 }; with the restriction x > y one gets the same set without zero; with the restriction x > 0 (i.e., differences of two nonzero squares) one gets the set without 1. An odd number 2n-1 = n^2 - (n-1)^2, a number 4n = (n+1)^2 - (n-1)^2. - M. F. Hasler, May 08 2018

Examples

			G.f. = x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 11*x^9 + 12*x^10 + ...
		

References

• R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D9.
• J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 83.

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Pentti Haukkanen and László Tóth, An analogue of Ramanujan's sum with respect to regular integers (mod r), Ramanujan J., Vol. 27, No. 1 (2012), pp. 71-88.
• Ron Knott, Pythagorean Triples and Online Calculators.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A001045, A006165, A009531, A020883, A020884, A047209, A047222, A214546, A143978, A042968.

Essentially the complement of A016825.

See A267958 for these numbers multiplied by 4.

Programs

Haskell

a042965 =  (`div` 3) . (subtract 3) . (* 4)
a042965_list = 0 : 1 : 3 : map (+ 4) a042965_list
-- Reinhard Zumkeller, Nov 09 2012

Magma

[n: n in [0..100] | not n mod 4 in [2]]; // Vincenzo Librandi, Sep 03 2015

Maple

a_list := proc(len) local rec; rec := proc(n) option remember;
ifelse(n <= 4, [0, 1, 3, 4][n], rec(n-1) + rec(n-3) - rec(n-4)) end:
seq(rec(n), n=1..len) end: a_list(76); # Peter Luschny, Aug 06 2022

Mathematica

nn=100; Complement[Range[0,nn], Range[2,nn,4]] (* Harvey P. Dale, May 21 2011 *)
f[n_]:=Floor[(4*n-3)/3]; Array[f,70] (* Robert G. Wilson v, Jun 26 2012 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 1, 3, 4}, 70] (* L. Edson Jeffery, Jan 21 2015 *)
Select[Range[0, 100], ! MemberQ[{2}, Mod[#, 4]] &] (* Vincenzo Librandi, Sep 03 2015 *)

PARI

a(n)=(4*n-3)\3 \\ Charles R Greathouse IV, Jul 25 2011

Python

def A042965(n): return (n<<2)//3-1 # Chai Wah Wu, Feb 10 2025

Formula

Recurrence: a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.

a(n) = n - 1 + (3n-3-sqrt(3)*(1-2*cos(2*Pi*(n-1)/3))*sin(2*Pi*(n-1)/3))/9. Partial sums of the period-3 sequence 0, 1, 1, 2, 1, 1, 2, 1, 1, 2, ... (A101825). - Ralf Stephan, May 19 2013

G.f.: A(x) = x^2*(1+x)^2/((1-x)^2*(1+x+x^2)); a(n)=Sum{k=0..floor(n/2)}, binomial(n-k-1, k)*A001045(n-2*k), n>0. - Paul Barry, Jan 16 2005, R. J. Mathar, Dec 09 2009

a(n) = floor((4*n-3)/3). - Gary Detlefs, May 14 2011

A214546(a(n)) != 0. - Reinhard Zumkeller, Jul 20 2012

From Michael Somos, May 18 2013: (Start)

Euler transform of length 3 sequence [3, -2, 1].

a(2-n) = -a(n). (End)

From Wesley Ivan Hurt, Jun 10 2016: (Start)

a(n) = (12*n-12+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9.

a(3k) = 4k-1, a(3k-1) = 4k-3, a(3k-2) = 4k-4. (End)

a(n) = round((4*n-4)/3). - Mats Granvik, Sep 24 2016

The g.f. A(x) satisfies (A(x)/x)^2 + A(x)/x = x*B(x)^2, where B(x) is the o.g.f. of A042968. - Peter Bala, Apr 12 2017

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

From Peter Bala, Aug 03 2022: (Start)

a(n) = a(floor(n/2)) + a(1 + ceiling(n/2)) for n >= 2, with a(2) = 1 and a(3) = 3.

a(2*n) = a(n) + a(n+1); a(2*n+1) = a(n) + a(n+2). Cf. A047222 and A006165. (End)

E.g.f.: (9 + 12*exp(x)*(x - 1) + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Apr 05 2023

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Peter Pein and Ralf Stephan, Jun 17 2007

Typos fixed in Gary Detlefs's formula and in PARI program by Reinhard Zumkeller, Nov 09 2012