A001079 -id:A001079 - cp's OEIS frontend

A322790 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..k} binomial(2k,2j)(n+1)^(k-j)n^j.

1, 1, 1, 1, 3, 1, 1, 17, 5, 1, 1, 99, 49, 7, 1, 1, 577, 485, 97, 9, 1, 1, 3363, 4801, 1351, 161, 11, 1, 1, 19601, 47525, 18817, 2889, 241, 13, 1, 1, 114243, 470449, 262087, 51841, 5291, 337, 15, 1, 1, 665857, 4656965, 3650401, 930249, 116161, 8749, 449, 17, 1

Offset: 0

Seiichi Manyama, Dec 26 2018

			Square array begins:
   1,  1,   1,    1,      1,       1,         1, ...
   1,  3,  17,   99,    577,    3363,     19601, ...
   1,  5,  49,  485,   4801,   47525,    470449, ...
   1,  7,  97, 1351,  18817,  262087,   3650401, ...
   1,  9, 161, 2889,  51841,  930249,  16692641, ...
   1, 11, 241, 5291, 116161, 2550251,  55989361, ...
   1, 13, 337, 8749, 227137, 5896813, 153090001, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Columns 0-3 give A000012, A005408, A069129(n+1), A322830.
Rows 0-9 give A000012, A001541, A001079, A011943(n+1), A023039, A077422, A097308, A068203, A056771, A078986.
Main diagonal gives A173174.
A(n-1,n) gives A173148(n).
Cf. A322699, A322747.

A[0, k_] := 1; A[n_, k_] := Sum[Binomial[2 k, 2 j]*(n + 1)^(k - j)*n^j, {j, 0, k}]; Table[A[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 26 2018 *)

a(n) = 2 * A322699(n) + 1.
A(n,k) + sqrt(A(n,k)^2 - 1) = (sqrt(n+1) + sqrt(n))^(2*k).
A(n,k) - sqrt(A(n,k)^2 - 1) = (sqrt(n+1) - sqrt(n))^(2*k).
A(n,0) = 1, A(n,1) = 2*n+1 and A(n,k) = (4*n+2) * A(n,k-1) - A(n,k-2) for k > 1.
A(n,k) = T_{k}(2*n+1) where T_{k}(x) is a Chebyshev polynomial of the first kind.
T_1(x) = x. So A(n,1) = 2*n+1.

A041038 Numerators of continued fraction convergents to sqrt(24).

Original entry on oeis.org

4, 5, 44, 49, 436, 485, 4316, 4801, 42724, 47525, 422924, 470449, 4186516, 4656965, 41442236, 46099201, 410235844, 456335045, 4060916204, 4517251249, 40198926196, 44716177445, 397928345756

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Cf. A010480, A041039.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[24],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011*)
Numerator[Convergents[Sqrt[24], 30]] (* Vincenzo Librandi, Oct 28 2013 *)

a(2n) = 2*A041006(2n) ; a(2n-1) = A041006(2n-1) = A001079(n). [From M. F. Hasler, Feb 13 2009]

G.f.: (4+5*x+4*x^2-x^3)/(1-10*x^2+x^4)

A077250 Bisection (odd part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

11, 103, 1019, 10087, 99851, 988423, 9784379, 96855367, 958769291, 9490837543, 93949606139, 930005223847, 9206102632331, 91131021099463, 902104108362299, 8929910062523527, 88396996516872971, 875040055106206183, 8662003554545188859, 85744995490345682407

Offset: 0

Wolfdieter Lang, Nov 08 2002

a(n)^2 - 24*b(n)^2 = 25, with the companion sequence b(n) = A077249(n).

The even part is A077409(n) with Diophantine companion A077251(n).

			103 = a(1) = sqrt(24*A077249(1)^2 + 25) = sqrt(24*21^2 + 25) = sqrt(10609) = 103.

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 (10,-1).

CoefficientList[Series[(11 - 7 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)

a(n)= 2*polchebyshev(n+1,1,5)+polchebyshev(n,1,5) \\ Charles R Greathouse IV, Jun 11 2011

Vec((11-7*x)/(1-10*x+x^2) + O(x^30)) \\ Colin Barker, Jun 15 2015

a(n) = 10*a(n-1)- a(n-2), a(-1)=7, a(0)=11.
a(n) = 2*T(n+1, 5)+T(n, 5), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 5)= A001079(n).
a(n) = sqrt(25 + 24*A077249(n)^2).
G.f.: (11-7*x)/(1-10*x+x^2).

A077409 Bisection (even part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

7, 59, 583, 5771, 57127, 565499, 5597863, 55413131, 548533447, 5429921339, 53750679943, 532076878091, 5267018100967, 52138104131579, 516114023214823, 5109002128016651, 50573907256951687, 500630070441500219, 4955726797158050503, 49056637901139004811

Offset: 0

Wolfdieter Lang, Nov 08 2002

a(n)^2 - 24*b(n)^2 = 25, with the companion sequence b(n) = A077251(n).

The odd part is A077250(n) with Diophantine companion A077249(n).

			59 = a(1) = sqrt(24*A077251(1)^2 + 25) = sqrt(24*12^2 + 25) = sqrt(3481) = 59.

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 (10,-1).

I:=[7,59]; [n le 2 select I[n] else 10*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018

CoefficientList[Series[(7 - 11 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10,-1}, {7,59}, 30] (* G. C. Greubel, Jan 18 2018 *)

a(n)=if(n<0,0,subst(poltchebi(n+1)+2*poltchebi(n),x,5))

Vec((7-11*x)/(1-10*x+x^2) + O(x^30)) \\ Colin Barker, Jun 15 2015

a(n)=polchebyshev(n+1,,5)+2*polchebyshev(n,,5) \\ Charles R Greathouse IV, Jun 15 2015

a(n)=([0,1;-1,10]^n*[7;59])[1,1] \\ Charles R Greathouse IV, Jun 15 2015

a(n) = 10*a(n-1)- a(n-2), a(-1)=11, a(0)=7.
a(n) = T(n+1, 5)+2*T(n, 5), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 5) = A001079(n).
a(n) = sqrt(24*A077251(n)^2 + 25).
G.f.: (7-11*x)/(1-10*x+x^2).

A080872 a(n)a(n+3) - a(n+1)a(n+2) = 4, given a(0)=a(1)=1, a(2)=5.

Original entry on oeis.org

1, 1, 5, 9, 49, 89, 485, 881, 4801, 8721, 47525, 86329, 470449, 854569, 4656965, 8459361, 46099201, 83739041, 456335045, 828931049, 4517251249, 8205571449, 44716177445, 81226783441, 442644523201, 804062262961, 4381729054565, 7959395846169, 43374646022449, 78789896198729, 429364731169925

Offset: 0

Paul D. Hanna, Feb 22 2003

Seiichi Manyama, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0, 10, 0, -1).

Cf. A080871, A080873, A080874, A080875.

Bisections are A001079 and A072256.

CoefficientList[Series[(-x^3-5 x^2+x+1)/(x^4-10 x^2+1),{x,0,30}],x] (* or *) LinearRecurrence[{0,10,0,-1},{1,1,5,9},30] (* Harvey P. Dale, May 06 2012 *)

Vec( (-x^3 - 5*x^2 + x + 1)/(x^4 - 10*x^2 + 1) + O(x^66) ) \\ Joerg Arndt, Jan 29 2016

G.f.: (-x^3 - 5*x^2 + x + 1)/(x^4 - 10*x^2 + 1).
a(n) = (3+sqrt(3))/12*(sqrt(3)-sqrt(2))^n+(3-sqrt(3))/12*(-sqrt(3)+sqrt(2))^n+(3+sqrt(3))/12*(sqrt(3)+sqrt(2))^n+(3-sqrt(3))/12*(-sqrt(3)-sqrt(2))^n. [Richard Choulet, Dec 03 2008]
a(n+4) = 10*a(n+2)-a(n). [Richard Choulet, Dec 04 2008]

A084765 a(n) = 2*a(n-1)^2 - 1, a(0)=1, a(1)=5.

Original entry on oeis.org

1, 5, 49, 4801, 46099201, 4250272665676801, 36129635465198759610694779187201, 2610701117696295981568349760414651575095962187244375364404428801

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jun 04 2003

Product_{k>=1} (1 + 1/a(k)) = sqrt(3/2) (see A010527).
A subsequence of A001079 (cf. formula), which must contain any prime occurring in A001079. The initial term a(0)=1 seems rather unnatural; using the recurrence relation it would yield the constant sequence 1,1,1,... Note that this sequence corresponds to sequence b(n) in Shallit's paper, which starts only at offset n=1. - M. F. Hasler, Sep 27 2009
Since if x is even (x^2-2)/2 = 2*y^2-1 and 10 is even from a(1) onward this is a reduced version of the LL sequence starting with 10 (A135927) as it is reduced by dividing by 2 it is also the difference between two possible LL sequences. - Roderick MacPhee, May 31 2015
For n >= 3, a(n) == 201 (mod 1000) if n is even, a(n) == 801 (mod 1000) if n is odd. - Robert Israel, Jun 01 2015
The next term -- a(8) -- has 128 digits. - Harvey P. Dale, Mar 28 2020

G. C. Greubel, Table of n, a(n) for n = 0..10
Jeffrey Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
H. S. Wilf, Limit of a sequence, Elementary Problem E 1093, Amer. Math. Monthly 61 (1954), 424-425.

Cf. A001079, A002812, A010527, A084764, A135927.

[n le 2 select 5^(n-1) else 2*Self(n-1)^2-1: n in [1..10]]; // Vincenzo Librandi, Jun 02 2015

1,seq(expand((5+2*sqrt(6))^(2^n)+(5-2*sqrt(6))^(2^n))/2, n=0..10); # Robert Israel, Jun 01 2015

a[n_]:= a[n]= If[n<2, 5^n, 2 a[n-1]^2 -1]; Table[a[n], {n,0,10}]
Join[{1}, NestList[2 #^2 - 1 &, 5, 10]] (* Harvey P. Dale, Mar 28 2020 *)

first(m)={my(v=[1,5]);for(i=3,m,v=concat(v, 2*v[i-1]^2 - 1));v;} \\ Anders Hellström, Aug 22 2015

def A084765(n): return 1 if n==0 else chebyshev_T(2^(n-1), 5)
[A084765(n) for n in range(11)] # G. C. Greubel, May 17 2023

a(n+1) = (x^(2^n) + y^(2^n))/2, with x = 5 + 2*sqrt(6), y = 5 - 2*sqrt(6).
a(n) = A001079(2^(n-1)) with a(0) = 1. - M. F. Hasler, Sep 27 2009
4*sqrt(6)/11 = Product_{n >= 1} (1 - 1/(2*a(n))). See A002812 for some general properties of the recurrence a(n+1) = 2*a(n)^2 - 1. - Peter Bala, Nov 11 2012
a(n) = cos(2^(n-1)*arccos(5)) for n >= 1. - Peter Luschny, Oct 12 2022

A097726 Pell equation solutions (5a(n))^2 - 26b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

Original entry on oeis.org

1, 103, 10505, 1071407, 109273009, 11144775511, 1136657829113, 115927953794015, 11823514629160417, 1205882564220568519, 122988198035868828521, 12543590317094399940623, 1279323224145592925115025, 130478425272533383961791927, 13307520054574259571177661529

Offset: 0

Wolfdieter Lang, Aug 31 2004

a(-1) = -1. - Artur Jasinski, Feb 10 2010

5*a(n) gives the x-values in the solution to the Pell equation x^2 - 26*y^2 = -1. - Colin Barker, Aug 24 2013

			(x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 = -1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (102,-1).

Cf. A097725 for S(n, 102).
Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121. - Artur Jasinski, Feb 10 2010
Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

Table[(1/5) Round[N[Sinh[(2 n - 1) ArcSinh[5]], 100]], {n, 1, 50}] (* Artur Jasinski, Feb 10 2010 *)
CoefficientList[Series[(1 + x)/(1 - 102 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 13 2014 *)
LinearRecurrence[{102,-1},{1,103},20] (* Harvey P. Dale, Aug 20 2017 *)

x='x+O('x^99); Vec((1+x)/(1-102*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 102*x + x^2).
a(n) = S(n, 2*51) + S(n-1, 2*51) = S(2*n, 2*sqrt(26)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 5*i)/(5*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 102*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=103. - Philippe Deléham, Nov 18 2008
a(n) = (1/5)*sinh((2*n-1)*arcsinh(5)), n >= 1. - Artur Jasinski, Feb 10 2010

More terms from Harvey P. Dale, Aug 20 2017

A122652 a(0) = 0, a(1) = 4; for n > 1, a(n) = 10*a(n-1) - a(n-2).

Original entry on oeis.org

0, 4, 40, 396, 3920, 38804, 384120, 3802396, 37639840, 372596004, 3688320200, 36510605996, 361417739760, 3577666791604, 35415250176280, 350574834971196, 3470333099535680, 34352756160385604, 340057228504320360, 3366219528882817996, 33322138060323859600

Offset: 0

N. J. A. Sloane, Sep 21 2006

Kekulé numbers for the benzenoids P_2(n).
a(n) are the values of m where A032528(m) - 1 has integer square roots. The roots are given by A001079. - Richard R. Forberg, Aug 05 2013
Numbers n such that 6*n^2 + 4 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. 283, K{P_2(n)}).

Michael De Vlieger, Table of n, a(n) for n = 0..1004
Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
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.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (10,-1).

Cf. A001078, A001079, A032528.

CoefficientList[Series[(4 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10, -1}, {0, 4}, 21] (* Jean-François Alcover, Jan 07 2019 *)

a(n)=if(n<2,(n%2)*4,10*a(n-1)-a(n-2)) \\ Benoit Cloitre, Sep 23 2006

G.f.: 4*x/(1 - 10*x + x^2). - Philippe Deléham, Nov 17 2008
3*a(n)^2 + 2 = 2*A001079(n)^2. - Charlie Marion, Feb 01 2013
a(n) = (2*arcsinh(sqrt(2))*sinh(2*n*arcsinh(sqrt(2)))/log(sqrt(2) + sqrt(3)))/sqrt(6). - Artur Jasinski, Aug 09 2016
a(n) = 2*A001078(n). - Bruno Berselli, Nov 25 2016
E.g.f.: sqrt(6)*exp(5*x)*sinh(2*sqrt(6)*x)/3. - Franck Maminirina Ramaharo, Jan 07 2019

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

A138281 a(n) = floor((sqrt(2) + sqrt(3))^n).

Original entry on oeis.org

1, 3, 9, 31, 97, 308, 969, 3051, 9601, 30210, 95049, 299052, 940897, 2960313, 9313929, 29304086, 92198401, 290080547, 912670089, 2871501385, 9034502497, 28424933309, 89432354889, 281377831710, 885289046401, 2785353383794, 8763458109129, 27572156006234

Offset: 0

Reinhard Zumkeller, Mar 12 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A089078, A110117, A121503, A135611, A135798.

[Floor((Sqrt(2) + Sqrt(3))^n): n in [0..50]]; // G. C. Greubel, Jan 27 2018

Table[Floor[(Sqrt[2] + Sqrt[3])^n], {n, 0, 50}] (* G. C. Greubel, Jan 27 2018 *)

for(n=0,50, print1(floor((sqrt(2) + sqrt(3))^n), ", ")) \\ G. C. Greubel, Jan 27 2018

a(2*n) = floor(A001079(n) + A001078(n)*sqrt(6));
(sqrt(2) + sqrt(3))^(2*n) = A001079(n) + A001078(n)*sqrt(6);
a(2*n+1) = floor(A054320(n)*sqrt(2) + A138288(n)*sqrt(3));
(sqrt(2)+sqrt(3))^(2*n+1) = A054320(n)*sqrt(2) + A138288(n)*sqrt(3).

Terms a(16) and a(18) corrected, terms a(19) onward added by G. C. Greubel, Jan 27 2018

A098297 Member r=12 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 12, 121, 1200, 11881, 117612, 1164241, 11524800, 114083761, 1129312812, 11179044361, 110661130800, 1095432263641, 10843661505612, 107341182792481, 1062568166419200, 10518340481399521, 104120836647576012

Offset: 0

Wolfdieter Lang, Oct 18 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

Cf. A097784, A098296.

a:=[0,1,12];; for n in [4..30] do a[n]:=11*a[n-1]-11*a[n-2]+ a[n-3]; od; a; # G. C. Greubel, May 24 2019

I:=[0,1,12]; [n le 3 select I[n] else 11*Self(n-1)-11*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, May 24 2019

LinearRecurrence[{11,-11,1}, {0,1,12}, 30] (* G. C. Greubel, May 24 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1+x)/((1-x)*(1-10*x+x^2)))) \\ G. C. Greubel, May 24 2019

(x*(1+x)/((1-x)*(1-10*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

a(n) = (T(n, 5)-1)/4 with Chebyshev's polynomials of the first kind evaluated at x=5: T(n, 5) = A001079(n) = ((5 + 2*sqrt(6))^n + (5 - 2*sqrt(6))^n)/2.
a(n) = 10*a(n-1) - a(n-2) + 2, n >= 2, a(0)=0, a(1)=1.
a(n) = 11*a(n-1) - 11*a(n-2) + a(n-3), n >= 3, a(0)=0, a(1)=1, a(2)=12.
G.f.: x*(1+x)/((1-x)*(1-10*x+x^2)) = x*(1+x)/(1-11*x+11*x^2-x^3) (from the Stephan link, see A092184).
a(n) = A132596(n) / 2. - Peter Bala, Dec 31 2012

cp's OEIS Frontend

A322790 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..k} binomial(2*k,2*j)*(n+1)^(k-j)*n^j.

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A041038 Numerators of continued fraction convergents to sqrt(24).

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A077250 Bisection (odd part) of Chebyshev sequence with Diophantine property.

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A077409 Bisection (even part) of Chebyshev sequence with Diophantine property.

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A080872 a(n)*a(n+3) - a(n+1)*a(n+2) = 4, given a(0)=a(1)=1, a(2)=5.

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A084765 a(n) = 2*a(n-1)^2 - 1, a(0)=1, a(1)=5.

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A097726 Pell equation solutions (5*a(n))^2 - 26*b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

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A322790 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..k} binomial(2k,2j)(n+1)^(k-j)n^j.

A080872 a(n)a(n+3) - a(n+1)a(n+2) = 4, given a(0)=a(1)=1, a(2)=5.

A097726 Pell equation solutions (5a(n))^2 - 26b(n)^2 = -1 with b(n):=A097727(n), n >= 0.