A122652 -id:A122652 - cp's OEIS frontend

A001079 a(n) = 10*a(n-1) - a(n-2); a(0) = 1, a(1) = 5.

1, 5, 49, 485, 4801, 47525, 470449, 4656965, 46099201, 456335045, 4517251249, 44716177445, 442644523201, 4381729054565, 43374646022449, 429364731169925, 4250272665676801, 42073361925598085, 416483346590304049

Offset: 0

N. J. A. Sloane

Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(6). - Benoit Cloitre, Feb 14 2004
Appears to give all solutions >1 to the equation x^2=ceiling(x*r*floor(x/r)) where r=sqrt(6). - Benoit Cloitre, Feb 24 2004
a(n) and b(n) (A004189) are the nonnegative proper solutions to the Pell equation a(n)^2 - 6*(2*b(n))^2 = +1, n >= 0. The formula given below by Gregory V. Richardson follows. - Wolfdieter Lang, Jun 26 2013
a(n) are the integer square roots of (A032528 + 1). They are also the values of m where (A032528(m) - 1) has integer square roots. See A122653 for the integer square roots of (A032528 - 1), and see A122652 for the values of m where (A032528(m) + 1) has integer square roots. - Richard R. Forberg, Aug 05 2013
a(n) are also the values of m where floor(2m^2/3) has integer square roots, excluding m = 0. The corresponding integer square roots are given by A122652(n). - Richard R. Forberg, Nov 21 2013
Except for the first term, positive values of x (or y) satisfying x^2 - 10xy + y^2 + 24 = 0. - Colin Barker, Feb 09 2014
Dickson on page 384 gives the Diophantine equation "24x^2 + 1 = y^2" and later states "y_{n+1} = 10y_n - y_{n-1}" where y_n is this sequence. - Michael Somos, Jun 19 2023

			Pell equation: n = 0: 1^2 - 24*0^2 = +1, n = 1: 5^2 - 6*(1*2)^2 = 1, n = 2: 49^2 - 6*(2*10)^2 = +1. - _Wolfdieter Lang_, Jun 26 2013
G.f. = 1 + 5*x + 49*x^2 + 485*x^3 + 4801*x^4 + 47525*x^5 + 470449*x^6 + ...

Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163-166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From N. J. A. Sloane, May 30 2012
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 384.
L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 374.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.

T. D. Noe, Table of n, a(n) for n=0..200
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Leonhard Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.
Leonhard Euler, De solutione problematum diophanteorum per numeros integros, par. 18.
Tanya Khovanova, Recursive Sequences
Robert Phillips, Polynomials of the form 1+4ke+4ke^2, 2008.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (10,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A004189, A001078, A046173, A046172, A036353, A138281, A004189.

I:=[1,5]; [n le 2 select I[n] else 10*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 10 2016

A001079 := proc(n)
    option remember;
    if n <= 1 then
        op(n+1,[1,5]) ;
    else
        10*procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A001079(n),n=0..20) ; # R. J. Mathar, Apr 30 2017

Table[(-1)^n Round[N[Cos[2 n ArcSin[Sqrt[3]]], 50]], {n, 0, 20}] (* Artur Jasinski, Oct 29 2008 *)
a[ n_] := ChebyshevT[n, 5]; (* Michael Somos, Aug 24 2014 *)
CoefficientList[Series[(1-5*x)/(1-10*x+x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 20 2017 *)
a[n_] := 3^n*Sum[(2/3)^k*Binomial[2*n, 2*k], {k,0,n}]; Flatten[Table[a[n], {n,0,18}]] (* Detlef Meya, May 21 2024 *)

{a(n) = subst(poltchebi(n), 'x, 5)}; /* Michael Somos, Sep 05 2006 */

{a(n) = real((5 + 2*quadgen(24))^n)}; /* Michael Somos, Sep 05 2006 */

{a(n) = n = abs(n); polsym(1 - 10*x + x^2, n)[n+1] / 2}; /* Michael Somos, Sep 05 2006 */

x='x+O('x^30); Vec((1-5*x)/(1-10*x+x^2)) \\ G. C. Greubel, Dec 20 2017

For all members x of the sequence, 6*x^2 -6 is a square. Limit_{n->infinity} a(n)/a(n-1) = 5 + 2*sqrt(6). - Gregory V. Richardson, Oct 13 2002
a(n) = T(n, 5) = (S(n, 10)-S(n-2, 10))/2 with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 10) = A004189(n+1).
a(n) = sqrt(1+24*A004189(n)^2) (cf. Richardson comment).
a(n)*a(n+3) - a(n+1)*a(n+2) = 240. - Ralf Stephan, Jun 06 2005
Chebyshev's polynomials T(n,x) evaluated at x=5.
G.f.: (1-5*x)/(1-10*x+x^2). - Simon Plouffe in his 1992 dissertation
a(n)= ((5+2*sqrt(6))^n + (5-2*sqrt(6))^n)/2.
a(-n) = a(n).
a(n+1) = 5*a(n) + 2*(6*a(n)^2-6)^(1/2) - Richard Choulet, Sep 19 2007
(sqrt(2)+sqrt(3))^(2*n)=a(n)+A001078(n)*sqrt(6). - Reinhard Zumkeller, Mar 12 2008
a(n+1) = 2*A054320(n) + 3*A138288(n). - Reinhard Zumkeller, Mar 12 2008
a(n) = cosh(2*n* arcsinh(sqrt(2))). - Herbert Kociemba, Apr 24 2008
a(n) = (-1)^n * cos(2*n* arcsin(sqrt(3))). - Artur Jasinski, Oct 29 2008
a(n) = cos(2*n* arccos(sqrt(3))). - Artur Jasinski, Sep 10 2016
a(n) = A142238(2n-1) = A041006(2n-1) = A041038(2n-1), for all n > 0. - M. F. Hasler, Feb 14 2009
2*a(n)^2 = 3*A122652(n)^2 + 2. - Charlie Marion, Feb 01 2013
E.g.f.: cosh(2*sqrt(6)*x)*exp(5*x). - Ilya Gutkovskiy, Sep 10 2016
From Peter Bala, Aug 17 2022: (Start)
a(n) = (1/2)^n * [x^n] ( 10*x + sqrt(1 + 96*x^2) )^n.
The g.f. A(x) satisfies A(2*x) = 1 + x*B'(x)/B(x), where B(x) = 1/sqrt(1 - 20*x + 4*x^2) is the g.f. of A098270.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p >= 3 and positive integers n and k.
Sum_{n >= 1} 1/(a(n) - 3/a(n)) = 1/4.
Sum_{n >= 1} (-1)^(n+1)/(a(n) + 2/a(n)) = 1/6.
Sum_{n >= 1} 1/(a(n)^2 - 3) = 1/4 - 1/sqrt(24). (End)
a(n) = 3^n*Sum_{k=0..n} (2/3)^k*binomial(2*n, 2*k). - Detlef Meya, May 21 2024

Chebyshev comments from Wolfdieter Lang, Nov 08 2002

A122653 a(n) = 10*a(n-1) - a(n-2) with a(0)=0, a(1)=6.

Original entry on oeis.org

0, 6, 60, 594, 5880, 58206, 576180, 5703594, 56459760, 558894006, 5532480300, 54765908994, 542126609640, 5366500187406, 53122875264420, 525862252456794, 5205499649303520, 51529134240578406, 510085842756480540, 5049329293324226994, 49983207090485789400

Offset: 0

N. J. A. Sloane, Sep 21 2006

Kekulé numbers for the benzenoids P''(n).
a(n) are the integer square roots of A032528(m) - 1. A001079 gives the value of m where these roots occur. Also see A122652. - Richard R. Forberg, Aug 05 2013
Numbers n such that 6*n^2 + 9 is a square. - Colin Barker, Mar 17 2014

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 301).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (10,-1).

CoefficientList[Series[(6 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10,-1},{0,6},30] (* Harvey P. Dale, Dec 16 2014 *)

a(n)=if(n<2,(n%2)*6,10*a(n-1)-a(n-2)) \\ Benoit Cloitre, Sep 23 2006

G.f.: 6x/(1 - 10x + x^2). - Philippe Deléham, Nov 17 2008
a(n) = 6*A004189(n). - R. J. Mathar, Jun 22 2020
6*a(n)^2+9 = (3*A001079(n))^2  - detail of the Barker comment. - R. J. Mathar, Jun 22 2020

More terms and better definition from Benoit Cloitre, Sep 23 2006

A278438 Numbers m such that T(m) + 2*T(m+1) is a square, where T = A000217.

Original entry on oeis.org

7, 799, 78407, 7683199, 752875207, 73774087199, 7229107670407, 708378777612799, 69413891098384007, 6801852948864019999, 666512175097575576007, 65311391306613542428799, 6399849835873029582446407, 627119972524250285537319199, 61451357457540654953074835207

Offset: 1

Bruno Berselli, Nov 23 2016

It is well known that T(m) + k*T(m+1) is always a square for k=1. For k=3, the nonnegative values of m are the terms of A278310.
Square roots of T(m) + 2*T(m+1) are listed by A168520 (after 0).
Negative values of m for which T(m) + 2*T(m+1) is a square: -1, -2, -82, -7922, -776162, ...

Colin Barker, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Subsequence of A056220.
Cf. A000217, A001078, A001079, A122652, A168520, A278620.
Cf. A278310: numbers m such that T(m) + 3*T(m+1) is a square.

Iv:=[7, 799]; [n le 2 select Iv[n] else 98*Self(n-1)-Self(n-2)+112: n in [1..20]];

P:=proc(q) local n; for n from 1 to q do if type(sqrt((3*n^2+7*n+4)/2),integer) then print(n); fi; od; end: P(10^9); #  Paolo P. Lava, Nov 25 2016

Table[((5 + 2 Sqrt[6])^(2 n) + (5 - 2 Sqrt[6])^(2 n))/12 - 7/6, {n, 1, 20}]
RecurrenceTable[{a[1] == 7, a[2] == 799, a[n] == 98 a[n - 1] - a[n - 2] + 112}, a, {n, 1, 20}]
LinearRecurrence[{99,-99,1},{7,799,78407},20] (* Harvey P. Dale, Oct 18 2024 *)

Vec(x*(7 + 106*x - x^2)/((1 - x)*(1 - 98*x + x^2)) + O(x^20)) \\ Colin Barker, Nov 27 2016

def A278438():
    a, b = 7, 799
    yield a
    while True:
        yield b
        a, b = b, 98*b - a + 112
a = A278438(); print([next(a) for  in range(15)]) # _Peter Luschny, Nov 24 2016

O.g.f.: x*(7 + 106*x - x^2)/((1 - x)*(1 - 98*x + x^2)).
E.g.f.: (exp((5-2*sqrt(6))^2*x) + exp((5+2*sqrt(6))^2*x) - 14*exp(x))/12 + 1.
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n>3.
a(n) = 98*a(n-1) - a(n-2) + 112 for n>2.
a(n) = a(-n) = ((5 + 2*sqrt(6))^(2*n) + (5 - 2*sqrt(6))^(2*n))/12 - 7/6.
a(n) = A001079(2*n)/6 - 7/6.
a(n) = 2*A001078(n)^2 - 1 = A122652(n)^2/2 - 1.
a(n) = -A278620(n+1) + 106*A278620(n) + 7*A278620(n-1).
Lim_{n -> infinity} a(n)/a(n-1) = (5 + 2*sqrt(6))^2.

A378908 Square array, read by descending antidiagonals, where each row n comprises the integers w >= 1 such that A000037(n)*w^2+4 is a square.

Original entry on oeis.org

4, 24, 2, 140, 8, 1, 816, 30, 3, 4, 4756, 112, 8, 40, 6, 27720, 418, 21, 396, 96, 2, 161564, 1560, 55, 3920, 1530, 12, 12, 941664, 5822, 144, 38804, 24384, 70, 456, 6, 5488420, 21728, 377, 384120, 388614, 408, 17316, 120, 1, 31988856, 81090, 987, 3802396

Offset: 1

Charles L. Hohn, Dec 10 2024

Also, integers w >= 1 for each row n >= 1 such that z+(1/z) is an integer, where x = A000037(n), y = w*sqrt(x), and z = (y+ceiling(y))/2.
All terms of row n are positive integer multiples of T(n, 1).
Limit_{k->oo} T(n, k+1)/T(n, k) = (sqrt(b^2-4)+b)/2 where b=T(n, 2)/T(n, 1).

			n=row index; x=nonsquare integer of index n (A000037(n)):
 n  x    T(n, k)
------+---------------------------------------------------------------------
 1  2 |  4,   24,   140,     816,      4756,       27720,        161564, ...
 2  3 |  2,    8,    30,     112,       418,        1560,          5822, ...
 3  5 |  1,    3,     8,      21,        55,         144,           377, ...
 4  6 |  4,   40,   396,    3920,     38804,      384120,       3802396, ...
 5  7 |  6,   96,  1530,   24384,    388614,     6193440,      98706426, ...
 6  8 |  2,   12,    70,     408,      2378,       13860,         80782, ...
 7 10 | 12,  456, 17316,  657552,  24969660,   948189528,   36006232404, ...
 8 11 |  6,  120,  2394,   47760,    952806,    19008360,     379214394, ...
 9 12 |  1,    4,    15,      56,       209,         780,          2911, ...
10 13 |  3,   33,   360,    3927,     42837,      467280,       5097243, ...
11 14 |  8,  240,  7192,  215520,   6458408,   193536720,    5799643192, ...
12 15 |  2,   16,   126,     992,      7810,       61488,        484094, ...
13 17 | 16, 1056, 69680, 4597824, 303386704, 20018924640, 1320945639536, ...
14 18 |  8,  272,  9240,  313888,  10662952,   362226480,   12305037368, ...
...

Rows from n = 1 (nonzero terms): A005319, A052530, A001906, A122652, A001080*2, A001542, A239365, A075844, A001353, A075835, A068204*2, A001090*2, A121740*2, A202299*2.

Cf. A000037, A002420, A033999, A048942, A180495, A298675.

row(n)={my(v=List()); for(t=3, oo, if((t^2-4)%x>0 || !issquare((t^2-4)/x), next); listput(v, sqrtint((t^2-4)/x)); break); listput(v, v[1]*sqrtint(v[1]^2*x+4)); while(#v<10, listput(v, v[#v]*(v[2]/v[1])-v[#v-1])); Vec(v)}
for(n=1, 20, x=n+floor(1/2+sqrt(n)); print (n, " ", x, " ", row(n)))

For x = A000037(n) (nonsquare integer of index n):
If x is not the sum of 2 squares, then T(n, 1) = A048942(n); otherwise, T(n, 1) is a positive integer multiple of A048942(n).
For j in {-2, 1, 2, 4}, if x-j is a square (except 2-2=0^2 or 5-1=2^2), then T(n, 1) = (4/abs(j))*sqrt(x-j) and T(n, 2) = T(n, 1)^3/(4/abs(j)) + sign(j)*2*T(n, 1).
For j in {1, 4}, if x+j is a square, then T(n, 1) = 2/sqrt(4/j) and T(n, 2) = (4/j)*sqrt(x+j).
For k >= 2, T(n, k) = T(n, k-1)*sqrt(T(n, 1)^2*x+4) - [k>=3]*T(n, k-2).
T(n, 2) = Sum_{i=0..oo}(T(n, 1)^(2-2*i) * x^((1-2*i)/2) * A002420(i) * A033999(i)).
If T(n, 1) is even, then T(n, 2) = T(n, 1)*A180495(n); if T(n, 1) is odd and x is even, then T(n, 2) = T(n, 1)*sqrt(A180495(n)+2); if T(n, 1) and x are both odd, then T(n, 2) is a factor of T(n, 1)*A180495(n).
For k >= 3, T(n, k) = T(n, k-1)*(T(n, 2)/T(n, 1)) - T(n, k-2) = T(n, 1)*A298675(T(n, 2)/T(n, 1), k-1) + T(n, k-2) = sqrt((A298675(T(n, 2)/T(n, 1), k)^2-4)/x).

cp's OEIS Frontend

A001079 a(n) = 10*a(n-1) - a(n-2); a(0) = 1, a(1) = 5.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

PARI

Formula

Extensions

A122653 a(n) = 10*a(n-1) - a(n-2) with a(0)=0, a(1)=6.

Views

Author

Keywords

Comments

References

Links

Programs

Mathematica

PARI

Formula

Extensions

A278438 Numbers m such that T(m) + 2*T(m+1) is a square, where T = A000217.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A378908 Square array, read by descending antidiagonals, where each row n comprises the integers w >= 1 such that A000037(n)*w^2+4 is a square.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula