A014447 -id:A014447 - cp's OEIS frontend

A014730 Squares of odd Lucas numbers.

1, 9, 49, 121, 841, 2209, 15129, 39601, 271441, 710649, 4870849, 12752041, 87403801, 228826129, 1568397609, 4106118241, 28143753121, 73681302249, 505019158609, 1322157322201, 9062201101801, 23725150497409, 162614600673849, 425730551631121, 2918000611027441

Offset: 0

Mohammad K. Azarian

Harvey P. Dale, Table of n, a(n) for n = 0..1594
Index entries for linear recurrences with constant coefficients, signature (0,17,0,17,0,-1).

Cf. A014447, A081069.

Select[LucasL[Range[50]],OddQ]^2 (* Harvey P. Dale, Nov 13 2021 *)

Vec(-(x-1)*(x^4+10*x^3+42*x^2+10*x+1)/((x^2-4*x-1)*(x^2+1)*(x^2+4*x-1)) + O(x^100)) \\ Colin Barker, May 14 2014

a(n) = 17*a(n-2)+17*a(n-4)-a(n-6). - R. J. Mathar, Feb 10 2012

G.f.: -(x-1)*(x^4+10*x^3+42*x^2+10*x+1) / ((x^2-4*x-1)*(x^2+1)*(x^2+4*x-1)). - Colin Barker, May 14 2014

More terms from Colin Barker, May 14 2014

A181156 Odd Fibonacci numbers F which have a proper Fibonacci divisor G such that F/G is a Lucas number or a product of Lucas numbers.

Original entry on oeis.org

3, 21, 55, 377, 987, 6765, 17711, 121393, 317811, 2178309, 5702887, 39088169, 102334155, 701408733, 1836311903, 12586269025, 32951280099, 225851433717, 591286729879, 4052739537881, 10610209857723, 72723460248141, 190392490709135, 1304969544928657, 3416454622906707

Offset: 1

Vladimir Shevelev, Oct 07 2010

nonn

A conjectural statement that, for odd prime p, the ratio F_{p^2}/F_{p} is never a Lucas number or a product of some Lucas numbers, yields that
a) an odd Fibonacci number F is in the sequence iff for its maximal proper Fibonacci divisor G, we have: ind G does not equal sqrt(ind F) and F/G does not have a proper Fibonacci divisor > 3;
b) an odd Fibonacci number F is in the sequence iff its index has one of the forms: 6k+2 or 6k+4 (see A047235).

			F = 3 has the proper Fibonacci divisor G=1, and F/G = 3 is a Lucas number.
F = 317811 has the proper Fibonacci divisors 3, 13, and 377, and F/377 = 843 is a Lucas number.

Cf. A000045, A000032, A014437, A014447.

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A014730 Squares of odd Lucas numbers.

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A181156 Odd Fibonacci numbers F which have a proper Fibonacci divisor G such that F/G is a Lucas number or a product of Lucas numbers.

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