A077242 -id:A077242 - cp's OEIS frontend

A077241 Combined Diophantine Chebyshev sequences A054488 and A077413.

1, 2, 8, 13, 47, 76, 274, 443, 1597, 2582, 9308, 15049, 54251, 87712, 316198, 511223, 1842937, 2979626, 10741424, 17366533, 62605607, 101219572, 364892218, 589950899, 2126747701, 3438485822, 12395593988, 20040964033, 72246816227, 116807298376

Offset: 0

Wolfdieter Lang, Nov 08 2002

-8*a(n)^2 + b(n)^2 = 17, with the companion sequence b(n)= A077242(n).
The number a > 0 belongs to the sequence A077241, if a^2 belongs to the sequence A034856. - Alzhekeyev Ascar M, Apr 27 2012
Numbers k such that k^2 + 2 is a triangular number (see A214838). - Alex Ratushnyak, Mar 07 2013

			8*a(2)^2 + 17 = 8*8^2+17 = 529 = 23^2 = A077242(2)^2.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

I:=[1,2,8,13]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 18 2014

LinearRecurrence[{0, 6, 0, -1}, {1, 2, 8, 13}, 30] (* Bruno Berselli, Mar 10 2013 *)
CoefficientList[Series[(1 + x) (1 + x + x^2)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 18 2014 *)

makelist(expand((-1)^n*((4-5*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor((n+1)/2))+(4+5*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor((n+1)/2)))/8), n, 0, 30); /* Bruno Berselli, Mar 10 2013 */

a(2k) = A054488(k) and a(2k+1)= A077413(k) for k>=0.
G.f.: (1+x)*(1+x+x^2)/(1-6*x^2+x^4).
a(n) = (-1)^n*((4-5*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor((n+1)/2))+(4+5*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor((n+1)/2)))/8. [Bruno Berselli, Mar 10 2013]

A077239 Bisection (odd part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

7, 37, 215, 1253, 7303, 42565, 248087, 1445957, 8427655, 49119973, 286292183, 1668633125, 9725506567, 56684406277, 330380931095, 1925601180293, 11223226150663, 65413755723685, 381259308191447, 2222142093424997, 12951593252358535, 75487417420726213

Offset: 0

Wolfdieter Lang, Nov 08 2002

a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A077413(n).

The even part is A077240(n) with Diophantine companion A054488(n).

			37 = a(1) = sqrt(8*A077413(1)^2 +17) = sqrt(8*13^2 + 17)= sqrt(1369) = 37.

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 (6,-1).

Cf. A077242 (even and odd parts).

Table[2*ChebyshevT[n+1, 3] + ChebyshevT[n, 3], {n, 0, 19}]  (* Jean-François Alcover, Dec 19 2013 *)

Vec((7-5*x)/(1-6*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015

a(n) = 6*a(n-1) - a(n-2), a(-1) := 5, a(0)=7.
a(n) = 2*T(n+1, 3)+T(n, 3), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 3)= A001541(n).
G.f.: (7-5*x)/(1-6*x+x^2).
a(n) = (((3-2*sqrt(2))^n*(-8+7*sqrt(2))+(3+2*sqrt(2))^n*(8+7*sqrt(2))))/(2*sqrt(2)). - Colin Barker, Oct 12 2015

A077240 Bisection (even part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

5, 23, 133, 775, 4517, 26327, 153445, 894343, 5212613, 30381335, 177075397, 1032071047, 6015350885, 35060034263, 204344854693, 1191009093895, 6941709708677, 40459249158167, 235813785240325, 1374423462283783, 8010726988462373, 46689938468490455

Offset: 0

Wolfdieter Lang, Nov 08 2002

a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A054488(n).

The odd part is A077239(n) with Diophantine companion A077413(n).

			23 = a(1) = sqrt(8*A054488(1)^2 + 17) = sqrt(8*8^2 + 17)= sqrt(529) = 23.

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 (6,-1).

Cf. A077242 (even and odd parts).

Table[ChebyshevT[n+1, 3] + 2*ChebyshevT[n, 3], {n, 0, 19}]  (* Jean-François Alcover, Dec 19 2013 *)
LinearRecurrence[{6,-1},{5,23},30] (* Harvey P. Dale, Mar 29 2017 *)

Vec((5-7*x)/(1-6*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015

a(n) = 6*a(n-1) - a(n-2), a(-1) = 7, a(0) = 5.
a(n) = T(n+1, 3)+2*T(n, 3), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 3)= A001541(n).
G.f.: (5-7*x)/(1-6*x+x^2).
a(n) = (((3-2*sqrt(2))^n*(-4+5*sqrt(2))+(3+2*sqrt(2))^n*(4+5*sqrt(2))))/(2*sqrt(2)). - Colin Barker, Oct 12 2015

A144796 Expansion of x(5+215x-1253x^2-23x^3)/((1+34x+x^2)(1-34x+x^2)).

Original entry on oeis.org

5, 215, 4517, 248087, 5212613, 286292183, 6015350885, 330380931095, 6941709708677, 381259308191447, 8010726988462373, 439972911271998743, 9244372002975869765, 507728358348578357975, 10667997280707165246437

Offset: 1

Richard Choulet, Sep 21 2008

The original definition "Numbers n such that (n^2-17)/2 is a square" describes A077242. - R. J. Mathar, Nov 27 2011

Index entries for linear recurrences with constant coefficients, signature (0,1154,0,-1).

Rest[CoefficientList[Series[x (5+215x-1253x^2-23x^3)/((1+34x+x^2)(1-34x+x^2)),{x,0,30}],x]] (* or *) LinearRecurrence[{0,1154,0,-1},{5,215,4517,248087},30] (* Harvey P. Dale, Apr 04 2014 *)

Vec(x(5+215*x-1253*x^2-23*x^3)/((1+34*x+x^2)*(1-34*x+x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 1154*a(n-2) - a(n-4).

A359438 For n >= 0, let S be the sequence of numbers m such that (m^2 - 2n^2 + 1)/2 is a square. Then a(n) is the number k such that S(j) = 6S(j-k) - S(j-2k) for all j for which S(j-2k) is defined.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 2, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 3, 4, 4, 4, 4, 2, 4, 2, 8, 2, 2, 4, 2, 2, 2, 6, 2, 2, 4, 4, 2, 2, 4, 2, 4, 8, 4, 2, 4, 6, 2, 4, 4, 2, 2, 2, 8, 4, 4, 4, 2, 4, 4, 4, 2, 4, 4, 2, 4, 4, 4, 2, 2, 8, 4, 4, 2, 4

Offset: 0

Jon E. Schoenfield, Dec 31 2022

nonn

			For n = 0, {S(j)} = A002315 (the NSW numbers), which satisfies S(j) = 6*S(j-1) - S(j-2), so a(0) = 1.
For n = 1, {S(j)} = A001541, which also satisfies S(j) = 6*S(j-1) - S(j-2), so a(1) = 1.
For n = 2, {S(j)} = A077443, which satisfies S(j) = 6*S(j-2) - S(j-4), so a(2) = 2.
For n = 5, {S(j)} = A106525, which satisfies S(j) = 6*S(j-3) - S(j-6), so a(5) = 3.

Cf. A000005, A002315, A001541, A077443, A077242, A106525.

a(0) = 1; for n >= 1, a(n) = A000005(2*n^2 - 1).

cp's OEIS Frontend

A077241 Combined Diophantine Chebyshev sequences A054488 and A077413.

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A077239 Bisection (odd part) of Chebyshev sequence with Diophantine property.

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A077240 Bisection (even part) of Chebyshev sequence with Diophantine property.

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A144796 Expansion of x(5+215x-1253x^2-23x^3)/((1+34x+x^2)(1-34x+x^2)).

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A359438 For n >= 0, let S be the sequence of numbers m such that (m^2 - 2*n^2 + 1)/2 is a square. Then a(n) is the number k such that S(j) = 6*S(j-k) - S(j-2k) for all j for which S(j-2k) is defined.

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A359438 For n >= 0, let S be the sequence of numbers m such that (m^2 - 2n^2 + 1)/2 is a square. Then a(n) is the number k such that S(j) = 6S(j-k) - S(j-2k) for all j for which S(j-2k) is defined.