A025244 - OEIS

A025244

a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 1, 2, 2.

1, 1, 2, 2, 4, 10, 22, 52, 130, 326, 832, 2162, 5674, 15032, 40178, 108154, 292956, 797994, 2184430, 6006028, 16579138, 45929838, 127656504, 355863330, 994735442, 2787537904, 7829586914, 22038759218, 62158281844, 175636481738, 497142172806, 1409444977380

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OFFSET

1,3

LINKS

Robert Israel, Table of n, a(n) for n = 1..2113

FORMULA

G.f.: (1 + x + x^2 - sqrt(1 - 2*x - x^2 - 6*x^3 + x^4))/2. - Michael Somos, Jun 08 2000

Conjecture: n*a(n) +(-2*n+3)*a(n-1) +(-n+3)*a(n-2) +3*(-2*n+9)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Feb 25 2015

Conjecture verified (for n >= 6) using differential equation 2*x^4 - 3*x^2 - x - 1 + (-2*x^3 + 9*x^2 + x + 1)*g(x) + (x^4 - 6*x^3 - x^2 - 2*x + 1)*g'(x) = 0 satisfied by g.f. - Robert Israel, Sep 28 2025

MAPLE

f:= gfun:-rectoproc({n*a(n) +(-2*n+3)*a(n-1) +(-n+3)*a(n-2) +3*(-2*n+9)*a(n-3) +(n-6)*a(n-4)=0, a(1)=1, a(2)=1, a(3)=2, a(4)=2, a(5)=4}, a(n), remember):

map(f, [$1..40]); # Robert Israel, Sep 28 2025

PROG

(PARI) a(n)=polcoeff((x+x^2-sqrt(1-2*x-x^2-6*x^3+x^4+x*O(x^n)))/2, n)

CROSSREFS

Sequence in context: A308986 A100088 A217212 * A132824 A298898 A339830

Adjacent sequences: A025241 A025242 A025243 * A025245 A025246 A025247

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved