Search: id:A016813
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%I A016813
%S A016813 1,5,9,13,17,21,25,29,33,37,41,45,49,53,57,61,65,69,73,
%T A016813 77,81,85,89,93,97,101,105,109,113,117,121,125,129,133,
%U A016813 137,141,145,149,153,157,161,165,169,173,177,181,185,189
%N A016813 4n+1.
%C A016813 Apart from initial term(s), dimension of the space of weight 2n cusp 
               forms for Gamma_0( 23 ).
%C A016813 Apart from initial term(s), dimension of the space of weight 2n cuspidal 
               newforms for Gamma_0( 64 ).
%C A016813 n such that n and (n+1) have the same binary digital sum - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Jun 05 2002
%C A016813 If A=[A002943] 4*n.^2+2*n (n>0, 6,20,42,. ,.,); Y=[A007395] 2 (2, 2, 
               2,..,); X=[A016813] 4*n+1 (n>0, 5,9,13,17, ,. .,), we have, for all 
               terms, Pell's equation X^2-A*Y^2=1. Example: 5^2-6 *2^2=1; 9^2-20*2^2=1; 
               13^2-42*2^2=1. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Mar 11 2009]
%C A016813 A056753(a(n)) = 3. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 23 2009]
%D A016813 Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269
%D A016813 L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 
               16.
%H A016813 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A016813 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A016813 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/dimskg0n.gp">
               Dimensions of the spaces S_k(Gamma_0(N))</a>
%H A016813 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/dimskg0new.gp">
               Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>
%H A016813 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/">The 
               modular forms database</a>
%H A016813 Konrad Knopp, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;
               c=umhistmath;idno=ACM1954.0001.001">Theorie und Anwendung der unendlichen 
               Reihen</a>, Berlin, J. Springer, 1922. (Original german edition of 
               "Theory and Application of Infinite Series")
%F A016813 sum(n=1, inf, (-1)^n/a(n))=1/4/sqrt(2)*(Pi+2ln(sqrt(2)+1)) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Apr 05 2002
%F A016813 G.f.: (5-x)/(1-x)^2 - Paul Barry (pbarry(AT)wit.ie), Feb 27 2003
%F A016813 (1 + 5x + 9x^2 + 13x^3...) = (1 + 2x + 3x^2...) / (1 - 3x + 9x^2 -27x^3...) 
               - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2003
%F A016813 a(n) = A001969(n) + A000069(n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Feb 04 2004
%F A016813 1 - 1/5 + 1/9 - 1/13 +... = (1/(4*sqrt(2))*(Pi + 2*ln(sqrt(2) + 1) [Jolley] 
               - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 20 2006
%F A016813 a(n)=A004766(n-1). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 26 2008]
%F A016813 a(n)=2*a(n-1)-a(n-2); a(0)=1, a(1)=5. a(n)=4+a(n-1). [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Nov 03 2008]
%F A016813 a(n)=8*n-a(n-1)-10 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 24 2009]
%e A016813 For n=2, a(2)=8*2-1-10=5; n=3, a(3)=8*3-5-10=9; n=4, a(4)=8*4-9-10=13 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 24 2009]
%p A016813 a[1]:=1:for n from 2 to 100 do a[n]:=a[n-1]+4 od: seq(a[n], n=1..48); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 16 2008
%p A016813 with(finance):seq(add(cashflows([0,0,4], 0 ),k=1..n)+1,n=0..50); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2008
%t A016813 f[n_]:=4*n+1; lst={};Do[a=f[n];AppendTo[lst,a],{n,0,6!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Jun 25 2009]
%o A016813 (Other) sage: [i+1 for i in range(190) if gcd(i,4) == 4] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 20 2009]
%o A016813 (Other) sage: [crt(1, n, 4,3 ) for n in xrange(1, 49)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jul 07 2009]
%Y A016813 a(n)= A093561(n+1, 1), (4, 1)-Pascal column.
%Y A016813 Cf. A002943, A007395 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Mar 11 2009]
%Y A016813 A161700, A005408, A016921, A017281, A017533, A158057, A161705, A161709, 
               A161714, A128470. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 17 2009]
%Y A016813 Sequence in context: A086408 A141135 A162502 this_sequence A004766 A145288 
               A057948
%Y A016813 Adjacent sequences: A016810 A016811 A016812 this_sequence A016814 A016815 
               A016816
%K A016813 nonn,easy,new
%O A016813 0,2
%A A016813 N. J. A. Sloane (njas(AT)research.att.com).

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