A054491 -id:A054491 - cp's OEIS frontend

A077237 Combined Diophantine Chebyshev sequences A054491 and A077234.

1, 2, 6, 9, 23, 34, 86, 127, 321, 474, 1198, 1769, 4471, 6602, 16686, 24639, 62273, 91954, 232406, 343177, 867351, 1280754, 3236998, 4779839, 12080641, 17838602, 45085566, 66574569, 168261623

Offset: 0

Wolfdieter Lang, Nov 08 2002

-3*a(n)^2 + b(n)^2 = 13, with the companion sequence b(n)= A077238(n).

			3*a(2)^2 + 13 = 3*36+13 = 121 = 11^2 = A077238(2)^2.

Matthew House, Table of n, a(n) for n = 0..3478
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

CoefficientList[Series[(1 + x) (1 + x + x^2)/(1 - 4 x^2 + x^4), {x, 0, 28}], x] (* Michael De Vlieger, Feb 11 2017 *)

a(2*k)= A054491(k) and a(2*k+1)= A077234(k), k>=0.
G.f.: (1+x)*(1+x+x^2)/(1-4*x^2+x^4).
a(n) = 4*a(n-2) - a(n-4). - Matthew House, Feb 11 2017

A077236 a(n) = 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 11.

Original entry on oeis.org

4, 11, 40, 149, 556, 2075, 7744, 28901, 107860, 402539, 1502296, 5606645, 20924284, 78090491, 291437680, 1087660229, 4059203236, 15149152715, 56537407624, 211000477781, 787464503500, 2938857536219, 10967965641376

Offset: 0

Wolfdieter Lang, Nov 08 2002

a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n)= A054491(n).
Bisection (even part) of Chebyshev sequence with Diophantine property.
The odd part is A077235(n) with Diophantine companion A077234(n).

			11 = a(1) = sqrt(3*A054491(1)^2 + 13) = sqrt(3*6^2 + 13)= sqrt(121) = 11.

Luigi Cerlienco, Maurice Mignotte, and F. Piras, Suites récurrentes linéaires: Propriétés algébriques et arithmétiques, L'Enseignement Math., 33 (1987), 67-108. See Example 2, page 93.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A077238 (even and odd parts), A077235, A053120.

a:=[4,11];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Apr 28 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (4-5*x)/(1-4*x+x^2) )); // G. C. Greubel, Apr 28 2019

CoefficientList[Series[(4-5*x)/(1-4*x+x^2), {x,0,20}], x] (* or *) LinearRecurrence[{4,-1}, {4,11}, 30] (* G. C. Greubel, Apr 28 2019 *)

my(x='x+O('x^30)); Vec((4-5*x)/(1-4*x+x^2)) \\ G. C. Greubel, Apr 28 2019

a(n) = polchebyshev(n+1, 1, 2) + 2*polchebyshev(n, 1, 2); \\ Michel Marcus, Oct 13 2021

((4-5*x)/(1-4*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

a(n) = T(n+1,2) + 2*T(n,2), with T(n,x) Chebyshev's polynomials of the first kind, A053120. T(n,2) = A001075(n).
G.f.: (4-5*x)/(1-4*x+x^2).
From Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009: (Start)
a(n) = ((4+sqrt(3))*(2+sqrt(3))^n + (4-sqrt(3))*(2-sqrt(3))^n)/2. Offset 0.
a(n) = second binomial transform of 4,3,12,9,36. (End)
a(n) = (A054491(n+1) - A054491(n-1))/2 = sqrt(3*A054491(n-1)*A054491(n+1) + 52), n >= 1. - Klaus Purath, Oct 12 2021

Edited by N. J. A. Sloane, Sep 07 2018, replacing old definition with simple formula from Philippe Deléham, Nov 16 2008

A077235 Bisection (odd part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

5, 16, 59, 220, 821, 3064, 11435, 42676, 159269, 594400, 2218331, 8278924, 30897365, 115310536, 430344779, 1606068580, 5993929541, 22369649584, 83484668795, 311569025596, 1162791433589, 4339596708760, 16195595401451, 60442784897044, 225575544186725

Offset: 0

Wolfdieter Lang, Nov 08 2002

a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n) = A077234(n).

The even part is A077236(n) with Diophantine companion A054491(n).

			16 = a(1) = sqrt(3*A077234(1)^2 + 13) = sqrt(3*9^2 + 13)= sqrt(256) = 16.

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,-1)

Cf. A077238 (even and odd parts).

Vec((5-4*x)/(1-4*x+x^2) + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n) = 2*T(n+1, 2)+T(n, 2), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 2)= A001075(n).
G.f.: (5-4*x)/(1-4*x+x^2).
a(n) = 4*a(n-1)-a(n-2) with a(0)=5 and a(1)=16. - Philippe Deléham, Nov 16 2008

A153594 a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).

Original entry on oeis.org

1, 8, 51, 304, 1769, 10200, 58603, 336224, 1927953, 11052712, 63358307, 363181200, 2081791609, 11932977272, 68400527259, 392075513536, 2247397253921, 12882196355400, 73841406542227, 423262699717616, 2426163312691977, 13906891405206808

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

Second binomial transform of A054491. Fourth binomial transform of 1 followed by A162766 and of A074324 without initial term 1.
First differences are in A161728.
Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(3) = 5.73205080756887729....

G. C. Greubel, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (8,-13).

Cf. A002194 (decimal expansion of sqrt(3)), A054491, A074324, A161728, A162766.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ];  // Klaus Brockhaus, Dec 31 2008

I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1)-13*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016

Join[{a=1,b=8},Table[c=8*b-13*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
LinearRecurrence[{8,-13},{1,8},40] (* Harvey P. Dale, Aug 16 2012 *)

a(n)=([0,1; -13,8]^(n-1)*[1;8])[1,1] \\ Charles R Greathouse IV, Sep 04 2016

[lucas_number1(n,8,13) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009

G.f.: x/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009
a(n) = 8*a(n-1) - 13*a(n-2) for n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
E.g.f.: sinh(sqrt(3)*x)*exp(4*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016
a(n) = Sum_{k=0..n-1} A027907(n,2k+1)*3^k. - J. Conrad, Aug 30 2016
a(n) = Sum_{k=0..n-1} A083882(n-1-k)*4^k. - J. Conrad, Sep 03 2016

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009

A077234 Bisection (odd part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

2, 9, 34, 127, 474, 1769, 6602, 24639, 91954, 343177, 1280754, 4779839, 17838602, 66574569, 248459674, 927264127, 3460596834, 12915123209, 48199896002, 179884460799, 671337947194, 2505467327977, 9350531364714, 34896658130879, 130236101158802, 486047746504329

Offset: 0

Wolfdieter Lang, Nov 08 2002

-3*a(n)^2 + b(n)^2 = 13, with the companion sequence b(n) = A077235(n).

The even part is A054491(n) with Diophantine companion A077236(n).

			3*a(1)^2 + 13 = 3*81+13 = 256 = 16^2 = A077235(1)^2.

Colin Barker, Table of n, a(n) for n = 0..1000
Andrej Dujella and László Szalay, Four squares from three numbers, arXiv:2506.14013 [math.NT], 2025. See p. 2.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A001353, A049310, A054491, A077235, A077236, A077237 (even and odd parts).

Vec((2+x)/(1-4*x+x^2) + O(x^50)) \\ Colin Barker, Jun 16 2015

a(n) = 2*S(n, 4)+S(n-1, 4), with S(n, x) = U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(-1, x) = 0 and S(n, 4) = A001353(n+1).
G.f.: (2+x)/(1-4*x+x^2).
a(n) = 4*a(n-1)-a(n-2) with a(0)=2 and a(1)=9. - Philippe Deléham, Nov 16 2008
E.g.f.: exp(2*x)*(6*cosh(sqrt(3)*x) + 5*sqrt(3)*sinh(sqrt(3)*x))/3. - Stefano Spezia, Oct 19 2023

A054485 Expansion of (1+3x)/(1-4x+x^2).

Original entry on oeis.org

1, 7, 27, 101, 377, 1407, 5251, 19597, 73137, 272951, 1018667, 3801717, 14188201, 52951087, 197616147, 737513501, 2752437857, 10272237927, 38336513851, 143073817477, 533958756057, 1992761206751, 7437086070947, 27755583077037

Offset: 0

Barry E. Williams, May 06 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A054491.

a:=[1,7];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 19 2020

I:=[1, 7]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in[1..30]]; // Vincenzo Librandi, Jun 23 2012

seq( simplify(ChebyshevU(n,2) +3*ChebyshevU(n-1,2)), n=0..30); # G. C. Greubel, Jan 19 2020

LinearRecurrence[{4,-1},{1,7},40] (* Vincenzo Librandi, Jun 23 2012 *)
Table[ChebyshevU[n, 2] +3*ChebyshevU[n-1, 2], {n,0,30}] (* G. C. Greubel, Jan 19 2020 *)

Vec((1+3*x)/(1-4*x+x^2) + O(x^30)) \\ Michel Marcus, Mar 20 2015

vector(31, n, polchebyshev(n-1,1,2) +5*polchebyshev(n-2,2,2) ) \\ G. C. Greubel, Jan 19 2020

[chebyshev_U(n,2) + 3*chebyshev_U(n-1,2) for n in (0..30)] # G. C. Greubel, Jan 19 2020

a(n) = (7*((2+sqrt(3))^n - (2-sqrt(3))^n) - ((2+sqrt(3))^(n-1) - (2-sqrt(3))^(n-1)))/2*sqrt(3).
a(n) = 4*a(n-1) - a(n-2), a(0)=1, a(0)=7.
a(n) = ChebyshevU(n,2) + 3*Chebyshev(n-1,2) = ChebyshevT(n,2) + 5*ChebyshevU(n-1,2). - G. C. Greubel, Jan 19 2020

A166552 a(n) = 3*a(n-2) for n > 2; a(1) = 1; a(2) = 4.

Original entry on oeis.org

1, 4, 3, 12, 9, 36, 27, 108, 81, 324, 243, 972, 729, 2916, 2187, 8748, 6561, 26244, 19683, 78732, 59049, 236196, 177147, 708588, 531441, 2125764, 1594323, 6377292, 4782969, 19131876, 14348907, 57395628, 43046721, 172186884, 129140163

Offset: 1

Klaus Brockhaus, Oct 16 2009

nonn

Interleaving of A000244 (powers of 3) and 4*A000244.
a(n) = A074324(n); A074324 has the additional term a(0)=1.
First differences are in A162852.
Second binomial transform is A054491. Fourth binomial transform is A153594.

G. C. Greubel, Table of n, a(n) for n = 1..1000[Terms 1 through 300 were computed by Vincenzo Librandi; Terms 301 through 1000 by G. C. Greubel, May 17 2016]
Index entries for linear recurrences with constant coefficients, signature (0,3)

Equals A162766 preceded by 1.

Cf. A000244 (powers of 3), A074324, A162852, A054491, A153594.

[ n le 2 select 3*n-2 else 3*Self(n-2): n in [1..35] ];

LinearRecurrence[{0, 3}, {1, 4}, 50] (* G. C. Greubel, May 17 2016 *)

a(n)=3^(n\2)*(4/3)^!bittest(n,0) \\ M. F. Hasler, Dec 03 2014

a(n) = (7+(-1)^n)*3^(1/4*(2*n-5+(-1)^n))/2.
G.f.: x*(1+4*x)/(1-3*x^2).
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = 3^floor((n-1)/2)*4^(1-n%2). - M. F. Hasler, Dec 03 2014
E.g.f.: (sqrt(3)*sinh(sqrt(3)*x) + 4*cosh(sqrt(3)*x) - 4)/3. - Ilya Gutkovskiy, May 17 2016

A105968 a(n) = 4a(n-1) - a(n-2) - 2(-1)^n, a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 13, 50, 185, 692, 2581, 9634, 35953, 134180, 500765, 1868882, 6974761, 26030164, 97145893, 362553410, 1353067745, 5049717572, 18845802541, 70333492594, 262488167833, 979619178740, 3655988547125, 13644335009762, 50921351491921, 190041070957924

Offset: 0

Creighton Dement, Apr 28 2005

This sequence is the (type 1A) "jbasejfor" transformation of the sequence (-1, -1, -1, -1, ..) with respect to the floretion given in the program code. Under the same conditions, the jbasejfor transformation of the sequence (1, 1, 1, 1, ...) is A006253 [Number of perfect matchings (or domino tilings) in C_4 X P_n]; the jbasejfor transformation of the sequence (1, -1, 1, -1, ...) is A001075 [Chebyshev's T(n,x) polynomials evaluated at x=2]; the jbasejfor transformation of the sequence (-1, 1, -1, 1, ...) is A001353 [3*a(n)^2 + 1 is a perfect square]. In this sense, the sequences (a(n)), A006253, A001075 and A001353 form a "quartett".

Floretion Algebra Multiplication Program, FAMP Code: 4jbasejforseq[ + .25'i + .25'j + .25'k + .25i' + .25j' + .25k' + .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' + .25'ki' + .25'kj' + .25e]. ForType: 1A. 1vesforseq = (-1, -1, -1, -1, ..).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Cf. A001075, A001353, A006253, A054491.

a:=[1,4,13];; for n in [4..30] do a[n]:=3*a[n-1]+3*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 15 2020

I:=[1,4,13]; [n le 3 select I[n] else 3*Self(n-1) +3*Self(n-2) -Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2020

seq( simplify((4*ChebyshevU(n,2) -5*ChebyshevU(n-1,2) -(-1)^n)/3), n = 0..30); # G. C. Greubel, Jan 15 2020

Table[(4*ChebyshevU[n, 2] -5*ChebyshevU[n-1, 2] -(-1)^n)/3, {n,0,30}] (* G. C. Greubel, Jan 15 2020 *)
nxt[{n_,a_,b_}]:={n+1,b,4b-a-2(-1)^(n+1)}; NestList[nxt,{1,1,4},30][[;;,2]] (* or *) LinearRecurrence[ {3,3,-1},{1,4,13},30] (* Harvey P. Dale, Apr 03 2024 *)

Vec((1-x)*(1+2*x)/((1+x)*(1-4*x+x^2)) + O(x^30)) \\ Colin Barker, May 25 2015

[(4*chebyshev_U(n,2) -5*chebyshev_U(n-1,2) -(-1)^n)/3 for n in (0..30)] # G. C. Greubel, Jan 15 2020

G.f.: (1-x)*(1+2*x)/((1+x)*(1-4*x+x^2)).
a(n) + a(n+1) = A054491(n+1) - A054491(n).
a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3). - Colin Barker, May 25 2015
a(n) = ( 4*ChebyshevU(n,2) - 5*ChebyshevU(n-1,2) - (-1)^n )/3. - G. C. Greubel, Jan 15 2020
E.g.f.: (exp(2*x)*(4*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - cosh(x) + sinh(x))/3. - Stefano Spezia, Sep 19 2023

A055269 a(n) = 4*a(n-1) - a(n-2) + 3 with a(0)=1, a(1)=7.

Original entry on oeis.org

1, 7, 30, 116, 437, 1635, 6106, 22792, 85065, 317471, 1184822, 4421820, 16502461, 61588027, 229849650, 857810576, 3201392657, 11947760055, 44589647566, 166410830212, 621053673285, 2317803862931, 8650161778442, 32282843250840, 120481211224921, 449642001648847

Offset: 0

Barry E. Williams, May 10 2000

Also partial sums of A054491.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A001834, A054491.

I:=[1,7,30]; [n le 3 select I[n] else 5*Self(n-1) - 5*Self(n-2) + Self(n-3): n in [1..40]]; // G. C. Greubel, Mar 16 2020

A055269:= n-> simplify((5*ChebyshevU(n, 2) - 3*ChebyshevU(n-1, 2) - 3)/2); seq( A055269(n), n=0..40); # G. C. Greubel, Mar 16 2020

LinearRecurrence[{5,-5,1},{1,7,30},40] (* or *) CoefficientList[ Series[ (1+2*x)/(1-5*x+5*x^2-x^3),{x,0,40}],x] (* Harvey P. Dale, Dec 01 2013 *)
Table[(5*ChebyshevU[n, 2] -3*ChebyshevU[n-1, 2] - 3)/2, {n,0,40}] (* G. C. Greubel, Mar 16 2020 *)

[(5*chebyshev_U(n, 2) - 3*chebyshev_U(n-1, 2) - 3)/2 for n in (0..40)] # G. C. Greubel, Mar 16 2020

G.f.: (1+2*x)/((1-x)*(1-4*x+x^2)).
a(n) = ( ( (17 - 5*(2-sqrt(3)))*(2+sqrt(3))^n + (5*(2+sqrt(3))-17)*(2-sqrt(3))^n )/(4*sqrt(3)) ) - 3/2.
a(n) = (5*ChebyshevU(n, 2) - 3*ChebyshevU(n-1, 2) - 3)/2. - G. C. Greubel, Mar 16 2020

Corrected by T. D. Noe, Nov 07 2006

A109731 a(n) = - 4*a(n-2) - a(n-4), a(0) = 1, a(1) = -4, a(2) = -6, a(3) = 15.

Original entry on oeis.org

1, -4, -6, 15, 23, -56, -86, 209, 321, -780, -1198, 2911, 4471, -10864, -16686, 40545, 62273, -151316, -232406, 564719, 867351, -2107560, -3236998, 7865521, 12080641, -29354524, -45085566, 109552575, 168261623, -408855776

Offset: 0

Creighton Dement, Aug 09 2005

Sequence A002530 and A002531 are also generated by the floretion given in the program code.

Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (0, -4, 0, -1).

Cf. A054491, A001353, A002530.

Floretion Algebra Multiplication Program, FAMP Code: 1lestesseq[A*B] with A = + .25'i + .25i' + 'ij' + .25'jk' + .25'kj' and B = + j' + k' + 'ii'.

LinearRecurrence[{0,-4,0,-1},{1,-4,-6,15},40] (* Harvey P. Dale, Mar 05 2013 *)

a(2n) = ((-1)^n)*A054491(n), a(2n+1) = ((-1)^n+1)*A001353(n+1). G.f. (1-4*x-2*x^2-x^3)/(x^4+4*x^2+1)

cp's OEIS Frontend

A077237 Combined Diophantine Chebyshev sequences A054491 and A077234.

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A077236 a(n) = 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 11.

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A077235 Bisection (odd part) of Chebyshev sequence with Diophantine property.

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A153594 a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).

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A077234 Bisection (odd part) of Chebyshev sequence with Diophantine property.

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A054485 Expansion of (1+3*x)/(1-4*x+x^2).

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A054485 Expansion of (1+3x)/(1-4x+x^2).

A105968 a(n) = 4a(n-1) - a(n-2) - 2(-1)^n, a(0) = 1, a(1) = 4.