0,3

Nonnegative values of y satisfying x^2 - 21*y^2 = 4; values of x are in A003501. - W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pps. 231-242.

F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971.

A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=5, q=-1.

A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pps. 245-252.

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=7.

F. M. van Lamoen, Square wreaths around hexagons, Forum Geometricorum, 6 (2006) 311-325.

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

Index entries for sequences related to linear recurrences with constant coefficients

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.

Tanya Khovanova, Recursive Sequences

F. M. van Lamoen, Article in Forum Geometricorum

Index entries for sequences related to Chebyshev polynomials.

G.f.: x/(1-5*x+x^2). a(n)= S(2*n-1, sqrt(7))/sqrt(7) = S(n-1, 5); S(n, x)=U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.

(A003501)=sqrt{[21*(a(n))^2]+4}.

a(n)={[((5+sqrt(21))/2)^n]-[((5-sqrt(21))/2)^n]}/[sqrt(21)]. - Barry E. Williams, Aug 29 2000

a(n)=sum{k=0..n-1, binomial(n+k, 2k+1)2^k} - Paul Barry (pbarry(AT)wit.ie), Nov 30 2004

[A004253(n), a(n)] = [1,3; 1,4]^n * [1,0] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 19 2008

a(n+1)=sum{k=0..n, Gegenbauer_C(n-k,k+1,2)}. [From Paul Barry (pbarry(AT)wit.ie), Apr 21 2009]

A004254:=1/(1-5*z+z**2); [S. Plouffe in his 1992 dissertation.]

(PARI) a(n)=if(n<0, 0, subst(4*poltchebi(n+1)-10*poltchebi(n), x, 5/2)/21)

(PARI) a(n)=if(n<0, 0, imag((5+quadgen(84))^n)/2^(n-1))

sage: [lucas_number1(n, 5, 1) for n in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

Partial sums of A004253.

Cf. A000027, A001906, A001353, A003501, A030221. a(n) = sqrt((A003501(n)^2 - 4)/21).

First differences of a(n) are in A004253, partial sums in A089817.

Cf. A004253.

INVERT transformation yields A001109. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 11 2008]

Sequence in context: A140766 A026388 A057969 this_sequence A086347 A026707 A110190

Adjacent sequences: A004251 A004252 A004253 this_sequence A004255 A004256 A004257

easy,nonn

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

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 31 2000