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A042968
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a(n) = 1 + n + floor(n/3).
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53
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1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98, 99, 101, 102
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OFFSET
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0,2
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COMMENTS
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Positive integers not divisible by 4. - Michael Somos, Jun 17 2018
Equivalently, numbers whose square part is odd. Cf. A028982. - Peter Munn, Jul 14 2020
More generally the sequence of numbers not divisible by some fixed integer m >= 2 is given by a(n,m) = 1 + n + floor(n/(m-1)). - Benoit Cloitre, Jul 11 2009
Also a(n,m) = floor((m*n-1)/(m-1)) [with offset 1]. - Gary Detlefs, May 14 2011
Numbers not having more even than odd divisors: A048272(a(n)) >= 0. - Reinhard Zumkeller, Jan 21 2012
Extending the comments of Benoit Cloitre (Jul 11 2009) and Gary Detlefs (May 14 2011), the g.f. is A(m,x) = (1-x^m) / ((1-x^(m-1))*(1-x)^2) where m >= 2 is fixed. - Werner Schulte, Apr 26 2018
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
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FORMULA
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a(n) = a(n-1) + a(n-3) - a(n-4).
a(n) = a(n-3) + 4, with a(0) = 1.
G.f.: A(x) = (1+x) * (1+x^2) / ( (1+x+x^2)*(1-x)^2 ). - Michael Somos, Jan 12 2000
A064680(A064680(a(n))) = a(n). - Reinhard Zumkeller, Oct 19 2001
Nearest integer to (Sum_{k>n} 1/k^4)/(Sum_{k>n} 1/k^5). - Benoit Cloitre, Jun 12 2003
a(n) = n + 1 + floor(n/3). - Benoit Cloitre, Jul 11 2009
a(n) = floor((4*n+3)/3). - Gary Detlefs, May 14 2011
A214546(a(n)) >= 0 for n > 0. - Reinhard Zumkeller, Jul 20 2012
a(n) = 2*n - ceiling(2*n/3) + 1. - Arkadiusz Wesolowski, Sep 21 2012
Sum_{k=0..n} a(n) = A071619(n+1). - L. Edson Jeffery, Jul 30 2014
The g.f. A(x) satisfies x*A(x)^2 = (B(x)/x)^2 + (B(x)/x), where B(x) is the o.g.f. of A042965. - Peter Bala, Apr 12 2017
a(n) = (12*n + 6 + 3*cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017
Euler transform of length 4 sequence [2, 0, 1, -1]. - Michael Somos, Jun 17 2018
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Jun 17 2018
E.g.f.: (2/3)*exp(x)*(1 + 2*x) + (1/9)*exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)). - Stefano Spezia, Nov 16 2019
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EXAMPLE
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G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 6*x^4 + 7*x^5 + 9*x^6 + 10*x^7 + 11*x^8 + ... - Michael Somos, Jun 17 2018
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MAPLE
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seq(1+n+floor(n/3), n=0..80); # Muniru A Asiru, Feb 17 2019
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MATHEMATICA
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Select[Table[n, {n, 200}], Mod[#, 4] != 0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)
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PROG
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(PARI) {a(n) = 1 + n + n\3};
(Haskell)
a042968 = (`div` 3) . (subtract 1) . (* 4)
a042968_list = filter ((/= 0) . (`mod` 4)) [1..]
-- Reinhard Zumkeller, Sep 02 2012
(MAGMA) [n+1+Floor(n/3): n in [0..80]]; // Vincenzo Librandi, Aug 03 2015
(Sage) [1+n+floor(n/3) for n in (0..80)] # G. C. Greubel, Feb 17 2019
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CROSSREFS
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Cf. A001651, A001935, A028982, A070048, A042965, A064680.
Cf. A071619 (partial sums).
Sequence in context: A329974 A059557 A195291 * A337037 A048103 A276078
Adjacent sequences: A042965 A042966 A042967 * A042969 A042970 A042971
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Edited by Peter Munn, Nov 16 2019
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STATUS
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approved
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