A113772 Determinant of the 2 X 2 matrices where the first column is consecutive triangular numbers and the second column is the corresponding consecutive Fibonacci numbers.

-2, 0, -2, 5, 15, 49, 120, 279, 605, 1265, 2562, 5070, 9842, 18810, 35480, 66181, 122265, 223991, 407340, 735945, 1321903, 2361985, 4200468, 7437900, 13118950, 23056164, 40386850, 70529189, 122820915, 213323245, 369611232, 638945835, 1102195697, 1897522865

Offset

2,1

Links

G. C. Greubel, Table of n, a(n) for n = 2..1000

Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Formula

a(n) = t(n-1)*f(n) - t(n)*f(n-1), where t(n) = n(n+1)/2 and f(n) = f(n-1) + f(n-2) with f(1)=f(2)=1.

From Chai Wah Wu, May 29 2016: (Start)

a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6) for n>7.

G.f.: x^2*(2 - 6*x + 2*x^2 - x^3)/(x^2 + x - 1)^3. (End)

Example

a(5) = t(4)*f(5)-t(5)*f(4) = 10*5-15*3 = 50-45 = 5.

Mathematica

LinearRecurrence[{3, 0, -5, 0, 3, 1}, {-2, 0, -2, 5, 15, 49}, 50] (* G. C. Greubel, May 29 2016 *)

Prog

(PARI) t(n) = n*(n+1)/2 \\ A000217
f(n) = if (n==1, 1, if (n==2, 1, f(n-1)+f(n-2))) \\ A000045
det(n) = t(n-1)*f(n)-t(n)*f(n-1) \\ Michel Marcus, Jun 19 2013

Crossrefs

Sequence in context: A243159 A339327 A258144 * A033716 A115978 A033751

Adjacent sequences: A113769 A113770 A113771 * A113773 A113774 A113775

Keyword

easy,sign

Author

Patrick J. Costello (pat.costello(AT)eku.edu), Jan 19 2006

Extensions

More terms from Michel Marcus, Jun 19 2013

Status

approved