A242300 a(n) = Sum_{0<=iA000032(k) is the k-th Lucas number.

0, 2, 11, 35, 105, 292, 796, 2130, 5655, 14927, 39281, 103160, 270600, 709282, 1858291, 4867275, 12746265, 33375932, 87388676, 228801650, 599034975, 1568333527, 4106014561, 10749789360, 28143481680, 73680863042, 192899442971, 505018008755, 1322155461705

Offset: 0

Author

J. M. Bergot, May 10 2014

Keywords

• nonn
• easy

Comments

This sequence does for Lucas numbers what A190173 does for Fibonacci numbers.

Examples

			For L(12) = a(13) the sum is (L(13)-1)^2 + L(11) + 1 = 520^2 + 200 = 270600 and for L(13) = a(14) the sum is (L(14)-1)^2 + l(12) - 4 = 842^2 + 322 - 4 = 709282.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-2,-6,4,2,-1).

Crossrefs

Cf. A000032, A190173.

Programs

PARI

concat(0, Vec(-x*(x^3+5*x^2-3*x-2)/((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)) + O(x^100))) \\ Colin Barker, May 13 2014

Sage

[(lucas_number2(i+1,1,-1)-1)^2+lucas_number2(i-1,1,-1)+(5*(-1)^i-3)/2 for i in [0..50]] # Tom Edgar, May 13 2014

Formula

The sums are (1) for L(2*k): (L(2*k+1)-1)^2 + L(2*k-1) + 1 and (2) for L(2*k+1): (L(2*k+2)-1)^2 + L(2*k) - 4.

G.f.: -x*(x^3+5*x^2-3*x-2) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)). - Colin Barker, May 12 2014

a(n) = (L(n+1)-1)^2 + L(n-1) + (5*(-1)^n-3)/2. - Colin Barker, May 13 2014

Extensions

Typo in a(18) fixed by Colin Barker, May 12 2014

More terms from Colin Barker, May 12 2014