OFFSET
0,1
COMMENTS
A106729 is sum of two consecutive squares of Lucas numbers (A001254), for which L(n)^2 + L(n+1)^2 = 5*{F(n)^2 + F(n+1)^2} = 5*A001519(n). A106789 is sum of two consecutive squares of Lucas 3-step numbers (A001644). Sum of two consecutive squares of Lucas 4-step numbers can be expressed in terms of tetranacci numbers, but not quite as neatly.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,4,6,12,-4,-6,0,-2,0,1).
FORMULA
EXAMPLE
MATHEMATICA
LinearRecurrence[{2, 4, 6, 12, -4, -6, 0, -2, 0, 1}, {17, 10, 58, 274, 901, 3277, 12402, 46282, 171170, 635953}, 40] (* G. C. Greubel, Apr 23 2019 *)
PROG
(PARI) my(x='x+O('x^40)); Vec((17-24*x-30*x^2+16*x^3-143*x^4-21*x^5 +46*x^6-32*x^7+2*x^8+17*x^9)/(1-2*x-4*x^2-6*x^3-12*x^4+4*x^5+6*x^6+2*x^8 -x^10)) \\ G. C. Greubel, Apr 23 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (17-24*x-30*x^2+16*x^3-143*x^4-21*x^5 +46*x^6-32*x^7+2*x^8+17*x^9)/(1-2*x-4*x^2 -6*x^3-12*x^4+4*x^5+6*x^6+2*x^8 -x^10) )); // G. C. Greubel, Apr 23 2019
(Sage) ((17-24*x-30*x^2+16*x^3-143*x^4-21*x^5 +46*x^6-32*x^7+2*x^8+ 17*x^9)/(1-2*x-4*x^2-6*x^3-12*x^4+4*x^5+6*x^6+2*x^8 -x^10)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019
(GAP) a:=[17, 10, 58, 274, 901, 3277, 12402, 46282, 171170, 635953];; for n in [11..40] do a[n]:=2*a[n-1]+4*a[n-2]+6*a[n-3]+12*a[n-4]-4*a[n-5] -6*a[n-6]-2*a[n-8]+a[n-10]; od; a; # G. C. Greubel, Apr 23 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, May 16 2005
STATUS
approved