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A047209 Numbers that are congruent to {1, 4} mod 5. 45
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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]

LINKS

_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

FORMULA

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]

MATHEMATICA

Select[Range[0, 200], MemberQ[{1, 4}, Mod[#, 5]] &] (* From Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)

PROG

(Haskell)

a047209 n = a047209_list !! (n-1)

a047209_list = 1 : 4 : (map (+ 5) a047209_list)

-- Reinhard Zumkeller, Jan 05 2011

CROSSREFS

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

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Edited by Michael Somos, Sep 22, 2002

STATUS

approved

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Last modified June 20 05:36 EDT 2013. Contains 226419 sequences.