A174013 Sequence whose Hankel transform is a (1,1) Somos-4 sequence.

1, 1, 3, 6, 12, 22, 37, 56, 73, 75, 44, -21, -39, 297, 1751, 5749, 14104, 27136, 38163, 22135, -80421, -369611, -934754, -1637758, -1559395, 2019629, 14766699, 44732254, 94865112, 138114302, 61077521

Offset

0,3

Comments

Continued fraction form of g.f. A(x) given by A(x)=1/(1-x(1+x)/(1-x/(1+x*A(x))).

Hankel transform is A174017. Diagonal sums of the Deleham array

[1,1,-1,1,1,-1,1,...] Delta [1,0,0,1,0,0,1,0,0,1,0,...], or A174014.

Links

Table of n, a(n) for n=0..30.

Formula

G.f.: (1-3x-x^2-sqrt(1-2x-x^2+6x^3+5x^4))/(2x(1-x-x^2));

G.f.: 1/(1-x(1+x)/(1-x/(1+x/(1-x(1+x)/(1-x/(1+x/(1-... continued fraction).

Conjecture: (n+1)*a(n) +n*a(n-1) +12*(-n+1)*a(n-2) +3*(3*n-8)*a(n-3) +6*(6*n-23)*a(n-4) +11*(-n+2)*a(n-5) +(-49*n+253)*a(n-6) +20*(-n+6)*a(n-7)=0. - R. J. Mathar, Jan 12 2013

Example

G.f. = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 22*x^5 + 37*x^6 + ... - Michael Somos, Jul 11 2024

Mathematica

a[ n_] := SeriesCoefficient[2*(1-x)/(1-3*x-x^2 + Sqrt[1-2*x-x^2+6*x^3+5*x^4]), {x, 0, n}]; (* Michael Somos, Jul 11 2024 *)

Prog

(PARI) {a(n) = polcoeff(2*(1-x)/(1-3*x-x^2 + sqrt(1-2*x-x^2+6*x^3+5*x^4 + x*O(x^n))), n)}; /* Michael Somos, Jul 11 2024 */

Crossrefs

Sequence in context: A333820 A139422 A062483 * A309266 A081056 A242448

Adjacent sequences: A174010 A174011 A174012 * A174014 A174015 A174016

Keyword

easy,sign

Author

Paul Barry, Mar 05 2010

Status

approved