0,1

Continued fraction for (e-1)/(e+1).

No solutions to a(n)=b^2-c^2 - Henry Bottomley (se16(AT)btinternet.com), Jan 13 2001

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 70 ).

Sequence gives n such that 8 is the largest power of 2 dividing A003629(k)^n-1 for any k - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002

n such that sum(d|n,(-1)^d)=A048272(n)=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 15 2002

Also n such that sum(d|n,phi(d)*mu(n/d))=A007431(n)=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 15 2002

Also n such that sum(d|n,d/AOOOO5(d)*mu(n/d))=0, n such that sum(d|n,AOOOO5(d)/d*mu(n/d))=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 19 2002

Solutions to phi[x]=phi[x/2]; primorial numbers are here. - Labos E. (labos(AT)ana.sote.hu), Dec 16 2002

Together with 1, numbers that are not the leg of a primitive Pythagorean triangle. - Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 25 2003

Numbers having equal numbers of odd and even divisors: A001227(a(n))=A000005(2*a(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 28 2003

Maximum number of electrons in an atomic subshell with orbital quantum number l is 4l+2.

For n>0: complement of A107750 and A023416(a(n)-1)=A023416(a(n))<>A023416(a(n)+1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 23 2005

Also the minimal value of sum([p(i)-p(i+1)]^2, i=1..n+2), where p(n+3)=p(1), as p ranges over all permutations of {1,2,...,n+2} (see the Mihai reference). Example: a(2)=10 because the values of the sum for the permutations of {1,2,3,4}are 10 (8 times), 12 (8 times) and 18 (8 times). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 30 2005

Except for a(n)=2, numbers having 4 as an anti-divisor. - Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be), Oct 02 2005

This is also the number of polyacenes in carbon nanotubes. See page 413 equation 12 of the paper by I. Lukovits and D. Janezic. - Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Aug 22 2006

A139391(a(n)) = A006370(a(n)) = A005408(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 17 2008

Twice odd numbers. [From Omar E. Pol (info(AT)polprimos.com), Aug 16 2008]

Terms of a(n) in A102261= 2, 6, 6, 10, 14, 10, 10, 14, 14, 22, 26 . [From Paul Curtz (bpcrtz(AT)free.fr), Sep 07 2008]

Also a(n) = (n-1) + n + (n+1) + (n+2), so a(n) and -a(n) are all the integers that are sums of four consecutive integers. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Mar 21 2009]

(e-1)/(e+1) = tanh(1/2) [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 09 2009]

A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.

J. R. Goldman, The Queen of Mathematics, 1998, p. 70.

I. Lukovits and D. Janezic, "Enumeration of conjugated circuits in nanotubes", J. Chem. Inf. Comput. Sci., vol. 44, 410-414 (2004).

V. Mihai, Problem 10725, Amer. Math. Monthly, 108 (March 2001), pp. 272-273.

H. Bass, Mathematics, Mathematicians and Mathematics Education, Bull. Amer. Math. Soc. (N.S.) 42 (2004), no. 4, 417-430.

Harry J. Smith, Table of n, a(n) for n=0,...,20000

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

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, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Square Number

G. Xiao, Contfrac

Index entries for continued fractions for constants

a(n)=2*A005408(n) - Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 28 2003

a(n) = A118413(n+1,2) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 27 2006

G.f.: 2* (1+x)/(1-x)^2. E.g.f.: 2*(1+2*x)*exp(x). a(n)= a(n-1) + 4. a(-1-n)= -a(n). - Michael Somos Apr 11 2007

a(n)=8*n-a(n-1)-8 (with a(1)=2) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 24 2009]

0.4621171572600097585023184... = 0 + 1/(2 + 1/(6 + 1/(10 + 1/(14 + ...)))) [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 09 2009]

For n=2, a(2)=8*2-2-8=6; n=3, a(3)=8*3-6-8=10; n=4, a(4)=8*4-10-8=14 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 24 2009]

with(finance):seq(add(cashflows([0, 0, 4], 0 ), k=1..n)+2, n=0..58); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2008

lst={}; Do[AppendTo[lst, 4*n+2], {n, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 29 2008]

(MAGMA) [4*n+2 : n in [0..100] ];

(PARI) {a(n)= 4*n+2}

(PARI) { allocatemem(932245000); default(realprecision, 180000); x=contfrac(tanh(1/2)); for (n=2, 20002, write("b016825.txt", n-2, " ", x[n])); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 09 2009]

(Other) sage: [i+2 for i in range(236) if gcd(i, 4) == 4] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 20 2009]

(Other) sage: [crt(2, n, 4, 3 ) for n in xrange(2, 61)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 07 2009]

Cf. A107687. First differences of A001105.

Cf. A160327 = Decimal expansion. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 09 2009]

Sequence in context: A068977 A111284 A130824 this_sequence A161718 A122905 A132417

Adjacent sequences: A016822 A016823 A016824 this_sequence A016826 A016827 A016828

nonn,easy,nice,cofr,new

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