A001254 -id:A001254 - cp's OEIS frontend

A171089 a(n) = 2*(Lucas(n)^2 - (-1)^n).

6, 4, 16, 34, 96, 244, 646, 1684, 4416, 11554, 30256, 79204, 207366, 542884, 1421296, 3720994, 9741696, 25504084, 66770566, 174807604, 457652256, 1198149154, 3136795216, 8212236484, 21499914246, 56287506244, 147362604496, 385800307234, 1010038317216

Offset: 0

R. J. Mathar, Sep 08 2010

In Thomas Koshy's book on Fibonacci and Lucas numbers, the formula for even-indexed Lucas numbers in terms of squares of Lucas numbers (A001254) is erroneously given as L(2n) = 2L(n)^2 + 2(-1)^(n - 1) on page 404 as Identity 34.7. - Alonso del Arte, Sep 07 2010

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A001254.

I:=[6, 4, 16]; [n le 3 select I[n] else 2*Self(n-1) + 2*Self(n-2) - Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

f[n_] := 2 (LucasL@n^2 - (-1)^n); Array[f, 27, 0] (* Robert G. Wilson v, Sep 10 2010 *)
CoefficientList[Series[2*(3 - 4*x - 2*x^2)/((1 + x)*(x^2 - 3*x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)

a(n) = round(2^(1-n)*((-2)^n+(3-sqrt(5))^n+(3+sqrt(5))^n)) \\ Colin Barker, Oct 01 2016

Vec(2*(3-4*x-2*x^2)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Oct 01 2016

a(n) = 2*(A000032(n))^2 -2*(-1)^n.
a(n) = 2*A047946(n).
a(n) = 2*a(n-1) + 2*a(n-2) -a(n-3).
G.f.: 2*(3-4*x-2*x^2)/( (1+x)*(x^2-3*x+1) ).
a(n) = 2^(1-n)*((-2)^n+(3-sqrt(5))^n+(3+sqrt(5))^n). - Colin Barker, Oct 01 2016

a(21) onwards from Robert G. Wilson v, Sep 10 2010

A203570 Bisection of A201207 (half-convolution of the Lucas sequence A000032 with itself); even part.

Original entry on oeis.org

4, 7, 27, 84, 270, 826, 2488, 7353, 21461, 61960, 177344, 503892, 1422892, 3996619, 11173935, 31114236, 86328978, 238764238, 658478176, 1811322045, 4970928809, 13613135152, 37208048132, 101518052904, 276527670100, 752102592271

Offset: 0

Wolfdieter Lang, Jan 03 2012

nonn

The odd part of the bisection of A201207 is given in A203574.
See a comment on A201204 for the definition of the half-convolution of a sequence with itself, and the rule for the o.g.f.s of the bisection. Here the o.g.f. is (Lconve(x) + L2(x))/2, with the o.g.f. Lconve(x) = (4-11*x+11*x^2+x^3)/
(1-3*x+x^2)^2 of A203573 and the o.g.f. L2(x)= (4-7*x-x^2)/   ((1+x)*(1-3*x+x^2)) of A001254. This leads to the o.g.f. given in the formula section.

Cf. A201207, A203574.

a(n) = A201207(2*n), n>=0.
a(n) = (2*(4*n+6)*F(2*n+1)-4*(n+1)*F(2*n))/4 + (-1)^n, with the Fibonacci numbers F(n)=A000045(n).
O.g.f.: (4-13*x+4*x^3+12*x^2)/((1-3*x+x^2)^2*(1+x)). See a comment above.

A216243 Partial sums of the squares of Lucas numbers (A000032).

Original entry on oeis.org

4, 5, 14, 30, 79, 200, 524, 1365, 3574, 9350, 24479, 64080, 167764, 439205, 1149854, 3010350, 7881199, 20633240, 54018524, 141422325, 370248454, 969323030, 2537720639, 6643838880, 17393796004, 45537549125, 119218851374, 312119004990, 817138163599, 2139295485800

Offset: 0

R. J. Mathar, Mar 14 2013

Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Cf. A001654.

A001254 := proc(n)
        A000032(n)^2 ;
end proc;
A := proc(n)
        add( A001254(i),i=0..n) ;
end proc:

Accumulate[LucasL[Range[0,30]]^2] (* or *) LinearRecurrence[{3,0,-3,1},{4,5,14,30},30] (* Harvey P. Dale, Oct 13 2019 *)

a(n) = Sum_{i=0..n} A001254(i) = A002878(n) +A176040(n) = A215602(n)+2.
G.f.: ( -4+7*x+x^2 ) / ( (x-1)*(1+x)*(x^2-3*x+1) ).
a(n) = -7*A064831(n) -A064831(n-1) +4*A064831(n+1).
a(n) = L(2*n+1) + 2 + (-1)^n, for L(n) the Lucas sequence A000032(n). - Greg Dresden, Jan 26 2021

A383039 Sum of the legs of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000032(n) and its long leg and hypotenuse are consecutive natural numbers.

Original entry on oeis.org

7, 1, 17, 31, 97, 241, 647, 1681, 4417, 11551, 30257, 79201, 207367, 542881, 1421297, 3720991, 9741697, 25504081, 66770567, 174807601, 457652257, 1198149151, 3136795217, 8212236481, 21499914247, 56287506241, 147362604497, 385800307231, 1010038317217, 2644314644401, 6922905616007

Offset: 0

Miguel-Ángel Pérez García-Ortega, Apr 13 2025

			For n=3, the short leg is A382379(3,1) = 5 and the long leg is A382379(3,2) = 12 so the sum of the legs is then a(2) = 5 + 12 = 17.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A000032, A382379, A382409, A382410, A001254.

a=Table[LucasL[n],{n,0,30}];Apply[Join,Map[{2#^2-1}&,a]]

a(n) = A382379(n,1) + A382379(n,2).
a(n) = 2*Lucas(n)^2 - 1.
a(n) = 2*A001254(n) - 1.

A105949 Powers of 3-Step Lucas numbers (A001644).

Original entry on oeis.org

1, 3, 9, 11, 21, 27, 39, 49, 71, 81, 121, 131, 241, 243, 343, 441, 443, 729, 815, 1331, 1499, 1521, 2187, 2401, 2757, 5041, 5071, 6561, 9261, 9327, 14641, 16807, 17155, 17161, 19683, 31553, 58035, 58081, 59049, 59319, 106743, 117649, 161051, 177147

Offset: 1

Jonathan Vos Post, Apr 27 2005

A001644 3-Step Lucas numbers. A000032 Lucas numbers. A001254 Squares of Lucas numbers. A075155 Cubes of Lucas numbers. A099923 Fourth powers of Lucas numbers. A103325 Fifth powers of Lucas numbers. A103324 Square array T(n,k) read by antidiagonals: powers of Lucas numbers. A105317 Powers of Fibonacci numbers.

Eric Weisstein's World of Mathematics, Lucas n-Step Number.

Cf. A001644, A105317.

{A001644(n)} U {A001644(n)^2} U {A001644(n)^3}...

A140328 Sum of two squares of Lucas numbers (A000032).

Original entry on oeis.org

2, 5, 8, 10, 13, 17, 18, 20, 25, 32, 50, 53, 58, 65, 98, 122, 125, 130, 137, 170, 242, 325, 328, 333, 340, 373, 445, 648, 842, 845, 850, 857, 890, 965, 1165, 1682, 2210, 2213, 2218, 2225, 2258, 2330, 2533, 3050, 4418, 5777, 5780, 5785, 5792, 5825, 5897, 6100

Offset: 1

Jonathan Vos Post, May 26 2008

This is to A045702 as Lucas numbers (A000032) are to Fibonacci numbers A000045. Hypotenuse squared of right triangle whose legs are both Lucas numbers.

			a(1) = 2 because A001254(1) = 1^2 = 1 and 1 + 1 = 2.
a(17) = 125 because A001254(5) = 11^2 = 121, A001254(0) = 2^2 = 4 and 125 + 4 = 125.

Cf. A000032, A001254, A045702.

{k = i + j such that i is in A000032^2 and j is in A000032^2} = {k = i + j such that i is in A001254 and j is in A001254}.

A235944 Digital roots of squares of Lucas numbers.

Original entry on oeis.org

4, 1, 9, 7, 4, 4, 9, 4, 4, 7, 9, 1, 4, 1, 9, 7, 4, 4, 9, 4, 4, 7, 9, 1, 4, 1, 9, 7, 4, 4, 9, 4, 4, 7, 9, 1, 4, 1, 9, 7, 4, 4, 9, 4, 4, 7, 9, 1, 4, 1, 9, 7, 4, 4, 9, 4, 4, 7, 9, 1, 4, 1, 9, 7, 4, 4, 9, 4, 4, 7, 9, 1, 4, 1, 9, 7, 4, 4, 9, 4, 4, 7, 9, 1, 4, 1, 9, 7, 4, 4, 9, 4, 4, 7, 9, 1, 4, 1, 9, 7, 4, 4, 9, 4, 4, 7, 9, 1

Offset: 0

Colin Barker, Jan 17 2014

The sequence is periodic with period 12.

			a(5)=4 because A000032[5]=11 and the digital root of 11*11 = 121 is 4.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000032, A001254, A216676.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{4, 1, 9, 7, 4, 4, 9, 4, 4, 7, 9, 1},108] (* Ray Chandler, Aug 27 2015 *)
PadRight[{},120,{4,1,9,7,4,4,9,4,4,7,9,1}] (* Harvey P. Dale, Feb 18 2018 *)

Vec(-(x^11+9*x^10+7*x^9+4*x^8+4*x^7+9*x^6+4*x^5+4*x^4+7*x^3+9*x^2+x+4)/(x^12-1) + O(x^100))

a(n) = A010888(A001254(n)).
a(n) = a(n-12).
G.f.: -(x^11 +9*x^10 +7*x^9 +4*x^8 +4*x^7 +9*x^6 +4*x^5 +4*x^4 +7*x^3 +9*x^2 +x +4) / (x^12 -1).

Extended by Ray Chandler, Aug 27 2015

cp's OEIS Frontend

A171089 a(n) = 2*(Lucas(n)^2 - (-1)^n).

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A203570 Bisection of A201207 (half-convolution of the Lucas sequence A000032 with itself); even part.

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A216243 Partial sums of the squares of Lucas numbers (A000032).

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A383039 Sum of the legs of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000032(n) and its long leg and hypotenuse are consecutive natural numbers.

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A105949 Powers of 3-Step Lucas numbers (A001644).

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A140328 Sum of two squares of Lucas numbers (A000032).

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A235944 Digital roots of squares of Lucas numbers.

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