0,2

Sequence also gives values of x satisfying 3*y^2 - x^2 = 2, the corresponding y being given by A001835(n+1). Moreover, quadruples(p, q, r, s) satisfying p^2 + q^2 + r^2 = s^2, where p=q, and r is either p+1 or p-1, are termed nearly isosceles Pythagorean, and are given by p={x + (-1)^n}/3, r=p-(-1)^n, s=y for n>1. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 19 2002

a(n) = L(n,-4)*(-1)^n, where L is defined as in A108299; see also A001835 for L(n,+4). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005

a(n)= A002531(1+2*n) - Anton Vrba (antonvrba(AT)yahoo.com), Feb 14 2007

361 written in base A001835(n+1)-1 is the square of a(n). E.g. a(12)=2672279, A001835(13)-1=1542840. We have 361_(1542840)=3*1542840+6*1542840+1=2672279^2 - Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 04 2007

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 375.

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq. (44) rhs, m=6.

P.-F. Teilhet, Reply to Query 2094, L'Interm\'{e}diaire des Math\'{e}maticiens, 10 (1903), 235-238.

F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.

T. D. Noe, Table of n, a(n) for n=0..200

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

G.f.: (1+x)/((1-4*x+x^2)). a(n)= S(2*n, sqrt(6)) = S(n, 4)+S(n-1, 4); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 4)= A001353(n).

For all members x of the sequence, 3*x^2 + 6 is a square. Lim. as n -> Inf. a(n)/a(n-1) = 2 + Sqrt(3). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 10 2002

a(n) = 1/2 ((1+sqrt(3))*(2+sqrt(3))^n + (1-sqrt(3))*(2-sqrt(3))^n). - Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 01 2002

a(n)=2*A001571(n)+1 - Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, -6)=a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002

With a=2+sqrt(3), b=2-sqrt(3): a(n)=(1/sqrt(2))(a^(n+1/2)-b^(n+1/2)). a(n)-a(n-1)=A003500(n). a(n)=sqrt(1+12*A061278(n)+12*A061278(n)^2). - Mario Catalani (mario.catalani(AT)unito.it), Apr 11 2003

a(n) = 2^(-n)*Sum{k>=0} binomial(2*n+1, 2*k)*3^k; see A091042 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 01 2004

a(n) = floor(sqrt(3)*A001835(n+1)). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 03 2004

a(n+1) - 2*a(n) = 3*A001835(n+1). Using the known relation A001835(n+1) = sqrt((a(n)^2 + 2)/3) it follows that a(n+1) - 2*a(n) = sqrt(3*(a(n)^2+2)). Therefore a(n+1)^2 + a(n)^2 - 4*a(n+1)*a(n) - 6 = 0. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Apr 18 2005

a(n)=Jacobi_P(n,1/2,-1/2,2)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry (pbarry(AT)wit.ie), Feb 03 2006

a(n)=((1+sqrt[3])^(2*n+1)+(1-sqrt[3])^(2*n+1))/2^(n+1) - Anton Vrba (antonvrba(AT)yahoo.com), Feb 14 2007

Equals binomial transform of A026150 starting (1, 4, 10, 28, 76,...) and double binomial transform of (1, 3, 3, 9, 9, 27, 27, 81, 81,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2007

A001834:=(1+z)/(1-4*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

a[0] = 1; a[1] = 5; a[n_] := a[n] = 4a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 25}] (from Robert G. Wilson v Apr 24 2004)

Table[Expand[((1+Sqrt[3])^(2*n+1)+(1+Sqrt[3])^(2*n+1))/2^(n+1)], {n, 0, 20}] - Anton Vrba (antonvrba(AT)yahoo.com), Feb 14 2007

Floretion Algebra Multiplication Program, FAMP Code: A001834 = (4/3)vesseq[ - .25'i + 1.25'j - .25'k - .25i' + 1.25j' - .25k' + 1.25'ii' + .25'jj' - .75'kk' + .75'ij' + .25'ik' + .75'ji' - .25'jk' + .25'ki' - .25'kj' + .25e], apart from initial term

A bisection of sequence A002531.

Cf. A001352, A001835.

Cf. A026150.

Adjacent sequences: A001831 A001832 A001833 this_sequence A001835 A001836 A001837

Sequence in context: A026590 A095073 A128349 this_sequence A099393 A083588 A086386

nonn,easy,nice

njas

More terms from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 07 2000