A002571 From a definite integral. (Formerly M3802 N1553)

1, 5, 10, 30, 74, 199, 515, 1355, 3540, 9276, 24276, 63565, 166405, 435665, 1140574, 2986074, 7817630, 20466835, 53582855, 140281751, 367262376, 961505400, 2517253800, 6590256025, 17253514249, 45170286749, 118257345970

Offset

1,2

Comments

a(n) are the row sums of the elements of the Golden Triangle (A180662) with alternating signs. - Alexander Adamchuk, Oct 18 2010

Limit_{n->oo} A002570(n)/A002571(n) = 1/sqrt(5). - Sean A. Irvine, Apr 09 2014

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Links

Table of n, a(n) for n=1..27.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

L. R. Shenton, A determinantal expansion for a class of definite integral. Part 5. Recurrence relations, Proc. Edinburgh Math. Soc. (2) 10 (1957), 167-188.

L. R. Shenton and K. O. Bowman, Second order continued fractions and Fibonacci numbers, Far East Journal of Applied Mathematics, 20(1), 17-31, 2005.

Formula

Appears to have g.f. x/((1-3x+x^2)*(1+x)^2). - Ralf Stephan, Apr 14 2004

a(n) = (-1)^n*Sum_{i=1..n+1} (-1)^(i+1)*Fibonacci(i)*Fibonacci(i+1). - Alexander Adamchuk, Jun 16 2006

From Paul D. Hanna, Feb 20 2009: (Start)

Given g.f. A(x), then log(1+A(x)) = Sum_{n>=1} A000204(n)^2 * x^n/n where A000204 is the Lucas numbers.

a(n) = (1/n)*(A000204(n)^2 + Sum_{k=1..n-1} A000204(k)^2*a(n-k)) for n>1, with a(1) = 1. (End)

G.f.: -1 + 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A006206(n), where A006206(n) is the number of aperiodic binary necklaces of length n with no subsequence 00. - Paul D. Hanna, Jan 07 2012

a(n) = 8*a(n-2) - 8*a(n-4) + a(n-6) + 2(-1)^n, n>6. - Sean A. Irvine, Apr 09 2014

a(n) - a(n-2) = Fibonacci(n+1)^2. - Peter Bala, Aug 30 2015

Example

From Paul D. Hanna, Feb 20 2009: (Start)
G.f.: A(x) = x + 5*x^2 + 10*x^3 + 30*x^4 + 74*x^5 + 199*x^6 + ...
log(1+A(x)) = x + 3^2*x^2/2 + 4^2*x^3/3 + 7^2*x^4/4 + 11^2*x^5/5 + ... (End)
G.f.: A(x) = -1 + 1/((1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6) * (1-7*x^4+x^8) * (1-11*x^5-x^10)^2 * (1-18*x^6+x^12)^2 * (1-29*x^7-x^14)^4 * (1-47*x^8+x^16)^5 * (1-76*x^9-x^18)^8 * ...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A006206(n) * ...). - Paul D. Hanna, Jan 07 2012

Maple

A002571:=-(-1-4*z-z**2+z**3)/(z**2-3*z+1)/(1+z)**2; # conjectured (probably correctly) by Simon Plouffe in his 1992 dissertation

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, (fibonacci(m+1)+fibonacci(m-1))^2*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, Feb 20 2009

Crossrefs

Cf. A064831, A077916, A000045, A000204 (Lucas), A006206.

Cf. A001654, A180662 - The Golden Triangle. - Alexander Adamchuk, Oct 18 2010

Sequence in context: A156302 A156234 A048010 * A077916 A358728 A189315

Adjacent sequences: A002568 A002569 A002570 * A002572 A002573 A002574

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Max Alekseyev and Alexander Adamchuk, Oct 18 2010

Status

approved