A077985 -id:A077985 - cp's OEIS frontend

A251732 a(n) = 3^n*A123335(n). Rational parts of the integers in Q(sqrt(2)) giving the length of a Lévy C-curve variant at iteration step n.

1, -3, 27, -189, 1377, -9963, 72171, -522693, 3785697, -27418419, 198581787, -1438256493, 10416775041, -75444958683, 546420727467, -3957528992949, 28662960504897, -207595523965923, 1503539788339611, -10889598445730973, 78869448769442337, -571223078628232779

Offset: 0

Kival Ngaokrajang, Dec 07 2014

The irrational part is given in A251733.
Inspired by the Lévy C-curve, and generated using different construction rules as shown in the links.
The length of this variant Lévy C-curve is an integer in the real quadratic number field Q(sqrt(2)), namely L(n) = A(n) + B(n)*sqrt(2) with A(n) = a(n) = 3^n*A123335(n) and B(n) = A251733(n) = 3^n*A077985(n-1), with A077985(-1) =  0. See the construction rule and the illustration in the links.
The total length of the Lévy C-curve after n iterations is sqrt(2)^n, also an integer in Q(sqrt(2)) (see a comment on A077957). The fractal dimension of the Lévy C-curve is 2, but for this modified case it is log(3)/log(1+sqrt(2)) = 1.2464774357... .

			The first lengths a(n) + A251733(n)*sqrt(2) are:
1, -3 + 3*sqrt(2), 27 - 18*sqrt(2), -189 + 135*sqrt(2), 1377 - 972*sqrt(2), -9963 + 7047*sqrt(2), 72171 - 51030*sqrt(2), -522693 + 369603*sqrt(2), 3785697 - 2676888*sqrt(2), -27418419 + 19387755*sqrt(2), 198581787 - 140418522*sqrt(2), ... - _Wolfdieter Lang_, Dec 08 2014

Colin Barker, Table of n, a(n) for n = 0..1000
MathImages, Lévy's C-curve
Kival Ngaokrajang, Construction rule, Illustration of modified Lévy C curve
Wikipedia, Lévy C curve
Index entries for linear recurrences with constant coefficients, signature (-6,9).

Cf. A017910, A077985, A123335, A251733.

[Round(((3*(-1+Sqrt(2)))^n + (-3*(1+Sqrt(2)))^n)/2): n in [0..30]]; // G. C. Greubel, Nov 18 2017

LinearRecurrence[{-6,9}, {1,-3}, 30] (* G. C. Greubel, Nov 18 2017 *)

Vec(-(3*x+1) / (9*x^2-6*x-1) + O(x^100)) \\ Colin Barker, Dec 07 2014

a(n) = 3^n*A123335(n).
a(n) = -6*a(n-1) + 9*a(n-2). - Colin Barker, Dec 07 2014
G.f.: -(3*x+1)/(9*x^2-6*x-1). - Colin Barker, Dec 07 2014
a(n) = ((3*(-1+sqrt(2)))^n + (-3*(1+sqrt(2)))^n) / 2. - Colin Barker, Jan 21 2017
E.g.f.: exp(-3*x)*cosh(3*sqrt(2)*x). - Stefano Spezia, Feb 01 2023

More terms from Colin Barker, Dec 07 2014

Edited: Name specified, Q(sqrt(2))remarks given earlier in a comment to a first version, MathImages link added. - Wolfdieter Lang, Dec 07 2014

A215936 a(n) = -2*a(n-1) + a(n-2) for n > 2, with a(0) = a(1) = 1, a(2) = 0.

Original entry on oeis.org

1, 1, 0, 1, -2, 5, -12, 29, -70, 169, -408, 985, -2378, 5741, -13860, 33461, -80782, 195025, -470832, 1136689, -2744210, 6625109, -15994428, 38613965, -93222358, 225058681, -543339720, 1311738121, -3166815962, 7645370045, -18457556052, 44560482149

Offset: 0

Michael Somos, Aug 28 2012

BINOMIAL transform is A052955.

Essentially the same as A000129, A069306, A048624, A215928, A077985, and A176981. - R. J. Mathar, Sep 08 2013

			G.f. = 1 + x + x^3 - 2*x^4 + 5*x^5 - 12*x^6 + 29*x^7 - 70*x^8 + 169*x^9 - 408*x^10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. C. Firengiz and A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
Index entries for linear recurrences with constant coefficients, signature (-2,1).

Cf. A001542, A001653, A052955, A077985.

[1,1] cat [n le 2 select (n-1) else -2*Self(n-1)+Self(n-2): n in [1..35] ]; // Vincenzo Librandi, Sep 09 2013

CoefficientList[Series[(1 + 3 x + x^2)/(1 + 2 x - x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 09 2013 *)
a[ n_] := With[ {m = If[ n < 1, 1 - n, n], s = If[ n < 1, (-1)^n, 1]}, s SeriesCoefficient[ x (1 + 2 x) / (1 + 2 x - x^2), {x, 0, m}]]; (* Michael Somos, Mar 19 2019 *)

{a(n) = my(m=n, s=1); if(n<1, m=1-n; s=(-1)^n); s * polcoeff( x * (1 + 2*x) / (1 + 2*x - x^2) + x * O(x^m), m)}; /* Michael Somos, Mar 19 2019 */

G.f.: 1 / (1 - x / (1 + x / (1 + x / (1 + x)))) = (1 + 3*x + x^2) / (1 + 2*x - x^2).
a(n + 3) = A077985(n). a(n) * a(n+2) - a(n+1)^2 = -(-1)^n.
a(2*n + 1) = A001653(n). a(2*n + 2) = -A001542(n).
a(n) = Sum_{k=0..n} A147746(n,k)*(-1)^(n-k). - Philippe Deléham, Aug 30 2012
G.f.: 1 + x + x^2/(1-x)  - G(0)*x^2 /(2-2*x), where G(k)= 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 10 2013
a(n) = (-1)^n a(1-n) = A000129(-1-n) if n < 0. a(n-2) = 2*a(n-1) + a(n) if n<1 or n>2. - Michael Somos, Mar 19 2019
E.g.f.: exp(-x)*(4*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))/2 - 1. - Stefano Spezia, Oct 31 2024

A176981 Expansion of 2+(1-2x)/(-1+2x+x^2).

Original entry on oeis.org

1, 0, -1, -2, -5, -12, -29, -70, -169, -408, -985, -2378, -5741, -13860, -33461, -80782, -195025, -470832, -1136689, -2744210, -6625109, -15994428, -38613965, -93222358, -225058681, -543339720, -1311738121, -3166815962, -7645370045, -18457556052, -44560482149

Offset: 0

Roger L. Bagula, Apr 30 2010

It is essentially A000129, A077985 and A069306 except for signs and offsets.

			1 - x^2 - 2*x^3 - 5*x^4 - 12*x^5 - 29*x^6 - 70*x^7 - 169*x^8 - 408*x^9 - 985*x^10 + ...

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

Cf. A000129.

a[0] = 1; a[n_] := a[n] = a[n - 1] - Sqrt[2*a[n - 1]^2 + (-1)^n]; Table[a[n], {n, 0, 30}]
Join[{1}, LinearRecurrence[{2, 1}, {0, -1}, 30]] (* or *) Join[{1}, Rest[ CoefficientList[Series[1 + (1 - 2 x)/(-1 + 2 x + x^2), {x, 0, 30}], x]]] (* Harvey P. Dale, Dec 24 2011 *)
FullSimplify[Join[{1}, Table[((1 - Sqrt[2])^(n-1) - (1 + Sqrt[2])^(n-1)) / 2^(3/2), {n, 1, 30}]]] (* Vaclav Kotesovec, Sep 01 2025 *)

a(n) = a(n-1) - sqrt(2*a(n-1)^2 + (-1)^n) = a(n-1)*(1-sqrt(2+(-1)^n/a(n-1)^2)) for n>0.
So lim_{n->infinity} a(n+1)/a(n) = 1+sqrt(2).
Matches the A000045 formula: Fibonacci(n) = Fibonacci(n-1)*(1 + sqrt(5+4*(-1)^(n-1)/Fibonacci(n-1)^2))/2.
a(0)=1, a(1)=0, a(2)=-1, a(n) = 2*a(n-1)+a(n-2). - Harvey P. Dale, Dec 24 2011
G.f.: 1 / (1 + x^2 / (1 - 2*x / (1 - x / (1 + x)))). - Michael Somos, Jan 03 2013
G.f.: 1 - Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 + x)/( x*(4*k+4 + x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013
For n>0, a(n) = ((1 - sqrt(2))^(n-1) - (1 + sqrt(2))^(n-1)) / 2^(3/2). - Vaclav Kotesovec, Sep 01 2025

Name corrected by Jason Yuen, Sep 01 2025

A161734 a(n) = ((2+sqrt(2))(5+sqrt(2))^n+(2-sqrt(2))(5-sqrt(2))^n)/4.

Original entry on oeis.org

1, 6, 37, 232, 1469, 9354, 59753, 382388, 2449561, 15700686, 100666957, 645553792, 4140197909, 26554241874, 170317866833, 1092431105228, 7007000115121, 44944085730966, 288279854661877, 1849084574806552, 11860409090842349, 76075145687872794

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 17 2009

Fifth binomial transform of A016116. Fourth binomial transform of the sequence of the absolute values of A077985. Third binomial transform of A007052. Second binomial transform of A086351. - R. J. Mathar, Jun 18 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A016116, A077985, A000129, A007052, A086351.

[Floor(((2+Sqrt(2))*(5+Sqrt(2))^n+(2-Sqrt(2))*(5-Sqrt(2))^n)/4): n in [0..30]]; // Vincenzo Librandi, Aug 18 2011

CoefficientList[Series[(1-4*z)/(23*z^2-10*z+1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 12 2011 *)
LinearRecurrence[{10,-23}, {1,6}, 50] (* G. C. Greubel, Apr 03 2018 *)

F=nfinit(x^2-2); for(n=0, 20, print1(nfeltdiv(F, ((2+x)*(5+x)^n+(2-x)*(5-x)^n), 4)[1], ",")) \\ Klaus Brockhaus, Jun 19 2009

a(n) = 10*a(n-1) - 23*a(n-2). - R. J. Mathar, Jun 18 2009
G.f.: (1-4*x)/(1-10*x+23*x^2). - R. J. Mathar, Jun 18 2009
E.g.f.: exp(5*x)*(2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x))/2. - G. C. Greubel, Apr 03 2018

Extended by R. J. Mathar and Klaus Brockhaus, Jun 18 2009

Edited by Klaus Brockhaus, Jul 05 2009

A249577 List of triples (r,s,t): the matrix M = [[1,4,4][1,3,2][1,2,1]] is raised to successive negative powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.

Original entry on oeis.org

2, -1, 1, -4, 3, -2, 10, -7, 5, -24, 17, -12, 58, -41, 29, -140, 99, -70, 338, -239, 169, -816, 577, -408, 1970, -1393, 985, -4756, 3363, -2378, 11482, -8119, 5741, -27720, 19601, -13860, 66922, -47321, 33461, -161564, 114243, -80782, 390050, -275807, 195025, -941664, 665857, -470832

Offset: 0

Russell Walsmith, Nov 01 2014

The sequence comprises, in reverse order, numbers to the right of a(0) in A249576.

			M^-1 = [[1,-4,4][-1,3,-2][1,-2,1]]. sqrt(M[1,3]) = 2; M[3,3] = M[1,1] = -1; M[3,1] = 1. Hence a(0) = 2; a(1) = -1; a(2) = 1.

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

Cf. A249576, A000129, A001333, A163271, A077985.

LinearRecurrence[{0,0,-2,0,0,1},{2,-1,1,-4,3,-2},50] (* Harvey P. Dale, Aug 02 2024 *)

Vec(-(x^4+x^2-x+2)/(x^6-2*x^3-1) + O(x^100)) \\ Colin Barker, Nov 02 2014

a(n) = -2*a(n-3)+a(n-6); G.f.: -(x^4+x^2-x+2) / (x^6-2*x^3-1). - Colin Barker, Nov 02 2014

A111645 Expansion of (x+1)(1-3x)/((x^2+4x+1)(x^2-2*x-1)).

Original entry on oeis.org

-1, 8, -37, 156, -625, 2436, -9341, 35464, -133809, 502896, -1885317, 7056580, -26384961, 98589388, -368228797, 1374944336, -5133041825, 19160828056, -71518973861, 266936079404, -996276071249, 3718290672596, -13877182280637, 51791152239960, -193289149920721

Offset: 0

Creighton Dement, Aug 10 2005

In reference to the program code, the sequence of Pell numbers A000126 is given by 1kbaseseq[C*J]. A001353 is 1ibaseiseq[C*J].

Floretion Algebra Multiplication Program, FAMP Code: 1jbasejseq[C*J] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and J = + j' + k' + 1.5'ii' + .5'jj' + .5'kk' + .5e

Index entries for linear recurrences with constant coefficients, signature (-6,-8,2,1).

Cf. A111639, A111640, A111641, A111642, A111643, A111644, A000126.

CoefficientList[Series[(x+1)(1-3x)/((x^2+4x+1)(x^2-2x-1)),{x,0,30}],x] (* or *) LinearRecurrence[{-6,-8,2,1},{-1,8,-37,156},30] (* Harvey P. Dale, Nov 19 2015 *)

a(0)=-1, a(1)=8, a(2)=-37, a(3)=156, a(n)=-6*a(n-1)-8*a(n-2)+2*a(n-3)+a(n-4). - Harvey P. Dale, Nov 19 2015

2*a(n) = -7*A125905(n)-A125905(n-1) -A077985(n-1)+5*A077985(n). - R. J. Mathar, Sep 11 2019

cp's OEIS Frontend

A251732 a(n) = 3^n*A123335(n). Rational parts of the integers in Q(sqrt(2)) giving the length of a Lévy C-curve variant at iteration step n.

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A215936 a(n) = -2*a(n-1) + a(n-2) for n > 2, with a(0) = a(1) = 1, a(2) = 0.

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A176981 Expansion of 2+(1-2*x)/(-1+2*x+x^2).

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A161734 a(n) = ((2+sqrt(2))*(5+sqrt(2))^n+(2-sqrt(2))*(5-sqrt(2))^n)/4.

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A249577 List of triples (r,s,t): the matrix M = [[1,4,4][1,3,2][1,2,1]] is raised to successive negative powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.

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A111645 Expansion of (x+1)*(1-3*x)/((x^2+4*x+1)*(x^2-2*x-1)).

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A176981 Expansion of 2+(1-2x)/(-1+2x+x^2).

A161734 a(n) = ((2+sqrt(2))(5+sqrt(2))^n+(2-sqrt(2))(5-sqrt(2))^n)/4.

A111645 Expansion of (x+1)(1-3x)/((x^2+4x+1)(x^2-2*x-1)).