A077237 -id:A077237 - 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

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

cp's OEIS Frontend

A077238 Combined Diophantine Chebyshev sequences A077236 and A077235.

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