A100230 Main diagonal of triangle A100229.

1, 2, 10, 35, 118, 392, 1297, 4286, 14158, 46763, 154450, 510116, 1684801, 5564522, 18378370, 60699635, 200477278, 662131472, 2186871697, 7222746566, 23855111398, 78788080763, 260219353690, 859446141836, 2838557779201

Offset: 0

Author

Paul D. Hanna, Nov 29 2004

Keywords

• nonn

Comments

Let F(x) = Product_{n >= 1} (1 + x^(4*n + 1))/(1 - x^(4*n + 3)). Let alpha = (1/2)*(3 - sqrt(13)). This sequence occurs as partial numerators in the simple continued fraction expansion of the real number F(alpha) = 1.34372 29374 22358 27049 ... = 1 + 1/(2 + 1/(1 + 1/(10 + 1/(35 + 1/(1 + 1/(118 + 1/(392 + 1/(1 + ...)))))))). - Peter Bala, Oct 17 2019

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-2,-1).

Crossrefs

Cf. A100228, A100229.

Equals A006497(n) - 1.

Programs

Mathematica

LinearRecurrence[{4,-2,-1},{1,2,10},30] (* Harvey P. Dale, May 06 2012 *)

PARI

a(n)=if(n==0,1,n*polcoeff(log((1-x)/(1-3*x-x^2)+x*O(x^n)),n))

Formula

a(n) = 3*a(n-1) + a(n-2) + 3 for n>1, with a(0)=1, a(1)=2.

G.f.: Sum_{n>=1} a(n)*x^n/n = log((1-x)/(1-3*x-x^2)).

a(0)=1, a(1)=2, a(2)=10, a(n)=4*a(n-1)-2*a(n-2)-a(n-3). [Harvey P. Dale, May 06 2012]