A077236 -id:A077236 - cp's OEIS frontend

A077238 Combined Diophantine Chebyshev sequences A077236 and A077235.

4, 5, 11, 16, 40, 59, 149, 220, 556, 821, 2075, 3064, 7744, 11435, 28901, 42676, 107860, 159269, 402539, 594400, 1502296, 2218331, 5606645, 8278924, 20924284, 30897365, 78090491, 115310536, 291437680, 430344779, 1087660229, 1606068580, 4059203236, 5993929541

Offset: 0

Wolfdieter Lang, Nov 08 2002

a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n)= A077237(n).
Positive values of x (or y) satisfying x^2 - 4xy + y^2 + 39 = 0. - Colin Barker, Feb 06 2014
Positive values of x (or y) satisfying x^2 - 14xy + y^2 + 624 = 0. - Colin Barker, Feb 16 2014

			11 = a(2) = sqrt(3*A077237(2)^2 + 13) = sqrt(3*6^2 + 13)= sqrt(121) = 11.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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) (4 + 9 x + 4 x^2)/(1 - 4 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 07 2014 *)
LinearRecurrence[{0,4,0,-1},{4,5,11,16},40] (* Harvey P. Dale, Oct 23 2015 *)

a(2*k)= A077236(k) and a(2*k+1)= A077235(k), k>=0.
G.f.: (1-x)*(4+9*x+4*x^2)/(1-4*x^2+x^4).
a(n) = 4*a(n-2)-a(n-4). - Colin Barker, Feb 06 2014

More terms from Colin Barker, Feb 06 2014

A054491 a(n) = 4*a(n-1) - a(n-2), a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 23, 86, 321, 1198, 4471, 16686, 62273, 232406, 867351, 3236998, 12080641, 45085566, 168261623, 627960926, 2343582081, 8746367398, 32641887511, 121821182646, 454642843073, 1696750189646, 6332357915511, 23632681472398

Offset: 0

Barry E. Williams, May 04 2000

Bisection (even part) of Chebyshev sequence with Diophantine property.

The odd part is A077234 with Diophantine companion A077235.

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.
Andrej Dujella and László Szalay, Four squares from three numbers, arXiv:2506.14013 [math.NT], 2025. See p. 2.
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 sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A001353, A001834, A077234, A077235.

a:=[1,6];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 15 2020

I:=[1,6]; [n le 2 select I[n] else 4*Self(n-1) -Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 15 2020

seq( simplify(ChebyshevU(n,2) +2*ChebyshevU(n-1,2)), n=0..30); # G. C. Greubel, Jan 15 2020

Table[ChebyshevU[n, 2] +2*ChebyshevU[n-1, 2], {n,0,30}] (* G. C. Greubel, Jan 15 2020 *)
LinearRecurrence[{4,-1},{1,6},30] (* Harvey P. Dale, Sep 04 2021 *)

a(n) = if (n==0, 1, if (n==1, 6, 4*a(n-1)-a(n-2))) \\ Michel Marcus, Jun 23 2013

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

[chebyshev_U(n,2) +2*chebyshev_U(n-1,2) for n in (0..30)]; # G. C. Greubel, Jan 15 2020

-3*a(n)^2 + A077236(n)^2 = 13.
a(n) = ( 6*((2+sqrt(3))^n-(2-sqrt(3))^n) - ((2+sqrt(3))^(n-1)-(2-sqrt(3))^(n-1)) )/(2*sqrt(3)).
a(n) = 6*S(n-1, 4) - S(n-2, 4) = S(n, 4) + 2*S(n-1, 4), with S(n, x) := U(n, x/2) Chebyshev's polynomials of 2nd kind, A049310. S(-1, x) := 0, S(-2, x) := -1, S(n, 4)= A001353(n+1).
G.f.: (1+2*x)/(1-4*x+x^2).
a(n+1) = A001353(n+2) + 2*A001353(n+1). - Creighton Dement, Nov 28 2004. Comment from Vim Wenders, Mar 26 2008: This is easily verified using a(n) = (6*( (2+sqrt(3))^n - (2-sqrt(3))^n ) - ( (2+sqrt(3))^(n-1) - (2-sqrt(3))^(n-1) ))/(2*sqrt(3)) and A001353(n) = ( (2+sqrt(3))^n - (2-sqrt(3))^n )/(2*sqrt(3)).
a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-7)^k. - Philippe Deléham, Mar 05 2014
E.g.f.: (1/3)*exp(2*x)*(3*cosh(sqrt(3)*x) + 4*sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Jan 27 2020

Chebyshev comments from Wolfdieter Lang, Nov 08 2002

A162766 a(n) = 3*a(n-2) for n > 2; a(1) = 4, a(2) = 3.

Original entry on oeis.org

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, Jul 13 2009

nonn

Binomial transform is A162559. Second binomial transform is A077236.

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

Cf. A074324, A166552, A162559, A077236, A162436.

[ n le 2 select 5-n else 3*Self(n-2): n in [1..34] ];

a(n)=3^(n\2)*4^(n%2) \\ M. F. Hasler, Dec 03 2014

a(n) = (5-3*(-1)^n)*3^(1/4*(2*n-1+(-1)^n))/2.
G.f.: x*(4+3*x)/(1-3*x^2).
a(n) = A074324(n+1) = A166552(n+1) = 3^floor(n/2)*4^(n%2), where n%2 = 0 for n even, 1 for n odd. - M. F. Hasler, Dec 03 2014

G.f. corrected by Klaus Brockhaus, Sep 18 2009

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

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

A162559 a(n) = ((4+sqrt(3))(1+sqrt(3))^n + (4-sqrt(3))(1-sqrt(3))^n)/2.

Original entry on oeis.org

4, 7, 22, 58, 160, 436, 1192, 3256, 8896, 24304, 66400, 181408, 495616, 1354048, 3699328, 10106752, 27612160, 75437824, 206099968, 563075584, 1538351104, 4202853376, 11482408960, 31370524672, 85705867264, 234152783872, 639717302272, 1747740172288, 4774914949120

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

nonn

Binomial transform of A162766. Inverse binomial transform of A077236.

Index entries for linear recurrences with constant coefficients, signature (2,2).

Cf. A162766, A077236.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((4+r)*(1+r)^n+(4-r)*(1-r)^n)/2: n in [0..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 13 2009

seq((1/2)*simplify((4+sqrt(3))*(1+sqrt(3))^n+(4-sqrt(3))*(1-sqrt(3))^n), n = 0 .. 27); # Emeric Deutsch, Jul 16 2009

LinearRecurrence[{2,2},{4,7},30] (* Harvey P. Dale, Sep 21 2018 *)

a(n) = 2*a(n-1) + 2*a(n-2) for n > 1; a(0) = 4, a(1) = 7.

G.f.: (4-x)/(1-2*x-2*x^2).

Edited by Klaus Brockhaus, Paolo P. Lava and Emeric Deutsch, Jul 13 2009

Two different extensions were received. This version was rechecked by N. J. A. Sloane, Jul 19 2009

A162561 a(n) = ((4+sqrt(3))(5+sqrt(3))^nv+v(4-sqrt(3))(5-sqrt(3))^n)/2.

Original entry on oeis.org

4, 23, 142, 914, 6016, 40052, 268168, 1800536, 12105664, 81444848, 548123872, 3689452064, 24835795456, 167190009152, 1125512591488, 7576945713536, 51008180122624, 343388995528448, 2311709992586752, 15562542024241664

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

nonn

Third binomial transform of A077236. Fourth binomial transform of A162559. Fifth binomial transform of A162766.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -22).

Cf. A077236, A162559, A162766.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((4+r)*(5+r)^n+(4-r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 14 2009

LinearRecurrence[{10,-22},{4,23},30] (* Harvey P. Dale, Mar 27 2013 *)

a(n) = 10*a(n-1) - 22*a(n-2) for n > 2; a(0) = 4, a(1) = 23.

G.f.: (4-17*x)/(1-10*x+22*x^2).

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 14 2009

A164310 a(n) = 6a(n-1) - 6a(n-2) for n > 1; a(0) = 4, a(1) = 15.

Original entry on oeis.org

4, 15, 66, 306, 1440, 6804, 32184, 152280, 720576, 3409776, 16135200, 76352544, 361304064, 1709709120, 8090430336, 38284327296, 181163381760, 857274326784, 4056665670144, 19196348060160, 90838094340096, 429850477679616

Offset: 0

Klaus Brockhaus, Aug 12 2009

Binomial transform of A077236. Inverse binomial transform of A083882 without initial 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-6).

Cf. A077236, A083882.

[ n le 2 select 11*n-7 else 6*Self(n-1)-6*Self(n-2): n in [1..22] ];

LinearRecurrence[{6,-6}, {4,15}, 50] (* or *) CoefficientList[Series[(4 - 9*x)/(1 - 6*x + 6*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 13 2017 *)

x='x+O('x^50); Vec((4-9*x)/(1-6*x+6*x^2)) \\ G. C. Greubel, Sep 13 2017

a(n) = ((4+sqrt(3))*(3+sqrt(3))^n + (4-sqrt(3))*(3-sqrt(3))^n)/2.
G.f.: (4-9*x)/(1-6*x+6*x^2).
E.g.f.: (4*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x))*exp(3*x). - G. C. Greubel, Sep 13 2017

cp's OEIS Frontend

A077238 Combined Diophantine Chebyshev sequences A077236 and A077235.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

Formula

Extensions

A054491 a(n) = 4*a(n-1) - a(n-2), a(0)=1, a(1)=6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A162766 a(n) = 3*a(n-2) for n > 2; a(1) = 4, a(2) = 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A162559 a(n) = ((4+sqrt(3))*(1+sqrt(3))^n + (4-sqrt(3))*(1-sqrt(3))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A162561 a(n) = ((4+sqrt(3))*(5+sqrt(3))^nv+v(4-sqrt(3))*(5-sqrt(3))^n)/2.

Views

Author

A162559 a(n) = ((4+sqrt(3))(1+sqrt(3))^n + (4-sqrt(3))(1-sqrt(3))^n)/2.

A162561 a(n) = ((4+sqrt(3))(5+sqrt(3))^nv+v(4-sqrt(3))(5-sqrt(3))^n)/2.

A164310 a(n) = 6a(n-1) - 6a(n-2) for n > 1; a(0) = 4, a(1) = 15.