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A047209
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Numbers that are congruent to {1, 4} mod 5.
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45
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1, 4, 6, 9, 11, 14, 16, 19, 21, 24, 26, 29, 31, 34, 36, 39, 41, 44, 46, 49, 51, 54, 56, 59, 61, 64, 66, 69, 71, 74, 76, 79, 81, 84, 86, 89, 91, 94, 96, 99, 101, 104, 106, 109, 111, 114, 116, 119, 121, 124, 126, 129, 131, 134, 136, 139, 141, 144, 146, 149, 151, 154
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OFFSET
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1,2
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COMMENTS
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Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 72 ).
Numbers n such that Kronecker(5,n)==mu(gcd(5,n)). - Jon Perry, Sep 17 2002
Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore (2*h*n+(h-4)*(-1)^n-h)/4)^2-1=0 (mod h); in our case, a(n)^2-1=0 (mod 5). - Bruno Berselli, Nov 17 2010
The sum of the alternating series (-1)^(n+1)/a(n) from n=1 to infinity is Pi/5*cot(Pi/5), that is 1/5*sqrt(1+2/sqrt(5))*Pi. [Jean-François Alcover, May 03 2013]
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LINKS
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_Reinhard Zumkeller_, Table of n, a(n) for n = 1..1000
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
William A. Stein, The modular forms database
Eric Weisstein's World of Mathematics, Determined by Spectrum
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FORMULA
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G.f.: (1+3x+x^2)/((1-x)(1-x^2)).
a(n) = floor((5n+3)/2).
a(1)=1, a(n)=5(n-1)-a(n-1). - Benoit Cloitre, Apr 12 2003
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = (10*n+(-1)^n-5)/4.
a(n)-a(n-1)-a(n-2)+a(n-3)=0 for n>3.
a(n) = a(n-2)+5 for n>2.
a(n) = 5*A000217(n-1)+1 - 2*sum(a(i), i=1..n-1) for n>1.
a(n)^2 = 5*A036666(n)+1 (cf. also Comments). (End)
a(n) = 5*floor(n/2)+(-1)^(n+1). [Gary Detlefs, Dec 29 2011]
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MATHEMATICA
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Select[Range[0, 200], MemberQ[{1, 4}, Mod[#, 5]] &] (* From Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)
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PROG
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(Haskell)
a047209 n = a047209_list !! (n-1)
a047209_list = 1 : 4 : (map (+ 5) a047209_list)
-- Reinhard Zumkeller, Jan 05 2011
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CROSSREFS
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Cf. A000566.
Cf. A005408 (n=1 or 3 mod 4), A007310 (n=1 or 5 mod 6).
Cf. A036666.
Cf. A003114, A203776.
Cf. A045468 (primes), A032527 (partial sums).
Cf. A047336, A047522, A056020, A090771, A175885, A091998, A175886, A175887.
Sequence in context: A190373 A010387 A010411 * A138812 A003259 A020935
Adjacent sequences: A047206 A047207 A047208 * A047210 A047211 A047212
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Edited by Michael Somos, Sep 22, 2002
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STATUS
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approved
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