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 A042965 Nonnegative integers not congruent to 2 mod 4. 58
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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(2n+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 A214546(a(n)) != 0. - Reinhard Zumkeller, Jul 20 2012 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 REFERENCES 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) P. Haukkanen, L. Toth, An analogue of Ramanujan's sum with respect to regular integers (mod r), Ramanujan J. 27 (2012), no. 1, 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). 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 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 EXAMPLE 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 + ... MAPLE A042965:=n->(12*n-12+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9: seq(A042965(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016 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 *) PROG (PARI) a(n)=(4*n-3)\3 \\ Charles R Greathouse IV, Jul 25 2011 (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 CROSSREFS Cf. A001045, A009531, A020883, A020884, A047209, A214546, A143978, A042968. Essentially the complement of A016825. See A267958 for these numbers multiplied by 4. Sequence in context: A183147 A074227 A122906 * A260003 A005848 A187885 Adjacent sequences:  A042962 A042963 A042964 * A042966 A042967 A042968 KEYWORD nonn,nice,easy AUTHOR 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' formula and in Pari program by Reinhard Zumkeller, Nov 09 2012 STATUS approved

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Last modified November 27 10:12 EST 2020. Contains 338679 sequences. (Running on oeis4.)