0,2

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 23 ).

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 64 ).

n such that n and (n+1) have the same binary digital sum - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 05 2002

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]

A056753(a(n)) = 3. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 23 2009]

Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 16.

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))

William A. Stein, The modular forms database

Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")

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

G.f.: (5-x)/(1-x)^2 - Paul Barry (pbarry(AT)wit.ie), Feb 27 2003

(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

a(n) = A001969(n) + A000069(n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 04 2004

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

a(n)=A004766(n-1). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 26 2008]

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]

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

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

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]

(Other) sage: [i+1 for i in range(190) if gcd(i, 4) == 4] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 20 2009]

(Other) sage: [crt(1, n, 4, 3 ) for n in xrange(1, 49)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 07 2009]

a(n)= A093561(n+1, 1), (4, 1)-Pascal column.

Cf. A002943, A007395 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 11 2009]

A161700, A005408, A016921, A017281, A017533, A158057, A161705, A161709, A161714, A128470. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]

Adjacent sequences: A016810 A016811 A016812 this_sequence A016814 A016815 A016816

Sequence in context: A086408 A141135 A162502 this_sequence A004766 A145288 A057948

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).