A077235 -id:A077235 - 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

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

A298678 Start with the hexagonal tile of the Shield tiling and recursively apply the substitution rule. a(n) is the number of hexagonal tiles after n iterations.

Original entry on oeis.org

1, 0, 7, 12, 73, 216, 919, 3204, 12409, 45408, 171271, 635580, 2379241, 8865000, 33113527, 123523572, 461111833, 1720661616, 6422058919, 23966525484, 89446140169, 333813840888, 1245817611991, 4649439829860, 17351975261881, 64758394108800, 241681735391047

Offset: 0

Felix Fröhlich, Jan 24 2018

The following substitution rules apply to the tiles:
triangle with 6 markings -> 1 hexagon
triangle with 4 markings -> 1 square, 2 triangles with 4 markings
square                   -> 1 square, 4 triangles with 6 markings
hexagon                  -> 7 triangles with 6 markings, 3 triangles with 4 markings, 3 squares
For n > 0, a(n) is also the number of triangles with 6 markings after n iterations when starting with the hexagon.
a(n) is also the number of triangles with 6 markings after n iterations when starting with the triangle with 6 markings.
a(n) is also the number of hexagons after n iterations when starting with the triangle with 6 markings.

Colin Barker, Table of n, a(n) for n = 0..1000
F. Gähler, Matching rules for quasicrystals: the composition-decomposition method, Journal of Non-Crystalline Solids, 153-154 (1993), 160-164.
Tilings Encyclopedia, Shield
Index entries for linear recurrences with constant coefficients, signature (2,7,-2).

Cf. A298679, A298680, A298681, A298682, A298683.

/* The function substitute() takes as argument a 4-element vector, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons that are to be substituted. The function returns a vector w, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons resulting from the substitution. */
substitute(v) = my(w=vector(4)); for(k=1, #v, while(v[1] > 0, w[4]++; v[1]--); while(v[2] > 0, w[3]++; w[2]=w[2]+2; v[2]--); while(v[3] > 0, w[3]++; w[1]=w[1]+4; v[3]--); while(v[4] > 0, w[1]=w[1]+7; w[2]=w[2]+3; w[3]=w[3]+3; v[4]--)); w
terms(n) = my(v=[0, 0, 0, 1], i=0); while(1, print1(v[4], ", "); i++; if(i==n, break, v=substitute(v)))

Vec((1-2*x)/((1+2*x)*(1-4*x+x^2)) + O(x^40)) \\ Colin Barker, Jan 25 2018

G.f.: (1-2*x)/((1+2*x)*(1-4*x+x^2)). - Joerg Arndt, Jan 25 2018
13*a(n) = A077235(n) + 8*(-2)^n. - Bruno Berselli, Jan 25 2018
From Colin Barker, Jan 25 2018: (Start)
a(n) = (1/26)*((-1)^n*2^(4+n) + (5-2*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(5+2*sqrt(3))).
a(n) = 2*a(n-1) + 7*a(n-2) - 2*a(n-3) for n>2.
(End)

More terms from Colin Barker, Jan 25 2018

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

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A077238 Combined Diophantine Chebyshev sequences A077236 and A077235.

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

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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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A298678 Start with the hexagonal tile of the Shield tiling and recursively apply the substitution rule. a(n) is the number of hexagonal tiles after n iterations.

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

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