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

17, 10, 58, 274, 901, 3277, 12402, 46282, 171170, 635953, 2364489, 8785386, 32637202, 121265666, 450571589, 1674090725, 6220049810, 23110593298, 85867345570, 319039636721, 1185390110881, 4404311472106, 16364198176874

Offset: 0

Author

Jonathan Vos Post, May 16 2005

Keywords

• easy
• nonn

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.

Examples

			a(0) = A073817(0)^2 + A073817(1)^2 = 4^2 + 1^2 = 16 + 1 = 17.
a(1) = A073817(1)^2 + A073817(2)^2 = 1^2 + 3^2 = 1 + 9 = 10.
a(2) = A073817(2)^2 + A073817(3)^2 = 3^2 + 7^2 = 9 + 49 = 58.
a(3) = A073817(3)^2 + A073817(4)^2 = 7^2 + 15^2 = 49 + 225 = 274.
a(4) = A073817(4)^2 + A073817(5)^2 = 15^2 + 26^2 = 225 + 676 = 901 = 30^2 + 1.
a(5) = A073817(5)^2 + A073817(6)^2 = 26^2 + 51^2 = 676 + 2601 = 3277.
		

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).

Crossrefs

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

Programs

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

Magma

R:=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

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 *)

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

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

Formula

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

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

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