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A106791 Sum of two consecutive squares of Lucas 4-step numbers (A073817). 1

%I

%S 17,10,58,274,901,3277,12402,46282,171170,635953,2364489,8785386,

%T 32637202,121265666,450571589,1674090725,6220049810,23110593298,

%U 85867345570,319039636721,1185390110881,4404311472106,16364198176874

%N Sum of two consecutive squares of Lucas 4-step numbers (A073817).

%C 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.

%H G. C. Greubel, <a href="/A106791/b106791.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,6,12,-4,-6,0,-2,0,1).

%F a(n) = A073817(n)^2 + A073817(n+1)^2.

%F a(n) = 5*A073817(n)^2 + 4*A073817(n)*A073817(n-4) + A073817(n-4)^2.

%F G.f.: (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- 3*x-3*x^2+x^3+x^4)*(1+x+2*x^2+2*x^3-2*x^4+x^5-x^6)). - _Colin Barker_, Dec 17 2012

%e a(0) = A073817(0)^2 + A073817(1)^2 = 4^2 + 1^2 = 16 + 1 = 17.

%e a(1) = A073817(1)^2 + A073817(2)^2 = 1^2 + 3^2 = 1 + 9 = 10.

%e a(2) = A073817(2)^2 + A073817(3)^2 = 3^2 + 7^2 = 9 + 49 = 58.

%e a(3) = A073817(3)^2 + A073817(4)^2 = 7^2 + 15^2 = 49 + 225 = 274.

%e a(4) = A073817(4)^2 + A073817(5)^2 = 15^2 + 26^2 = 225 + 676 = 901 = 30^2 + 1.

%e a(5) = A073817(5)^2 + A073817(6)^2 = 26^2 + 51^2 = 676 + 2601 = 3277.

%t 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 *)

%o (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

%o (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

%o (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

%o (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

%Y Cf. A001254, A001519, A001644, A073817, A106729.

%K easy,nonn

%O 0,1

%A _Jonathan Vos Post_, May 16 2005

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Last modified July 17 02:10 EDT 2019. Contains 325092 sequences. (Running on oeis4.)