A097778 -id:A097778 - cp's OEIS frontend

A004254 a(n) = 5*a(n-1) - a(n-2) for n > 1, a(0) = 0, a(1) = 1.

0, 1, 5, 24, 115, 551, 2640, 12649, 60605, 290376, 1391275, 6665999, 31938720, 153027601, 733199285, 3512968824, 16831644835, 80645255351, 386394631920, 1851327904249, 8870244889325, 42499896542376, 203629237822555, 975646292570399, 4674602225029440, 22397364832576801

Offset: 0

N. J. A. Sloane

Nonnegative values of y satisfying x^2 - 21*y^2 = 4; values of x are in A003501. - Wolfdieter Lang, Nov 29 2002
a(n) is equal to the permanent of the (n-1) X (n-1) Hessenberg matrix with 5's along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else. - John M. Campbell, Jun 09 2011
For n >= 1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,2,3,4}. - Milan Janjic, Jan 25 2015
From Klaus Purath, Jul 26 2024: (Start)
For any three consecutive terms (x, y, z), y^2 - xz = 1 always applies.
a(n) = (t(i+2n) - t(i))/(t(i+n+1) - t(i+n-1)) where (t) is any recurrence t(k) = 4t(k-1) + 4t(k-2) - t(k-3) or t(k) = 5t(k-1) - t(k-2) without regard to initial values.
In particular, if the recurrence (t) of the form (4,4,-1) has the same three initial values as the current sequence, a(n) = t(n) applies.
a(n) = (t(k+1)*t(k+n) - t(k)*t(k+n+1))/(y^2 - xz) where (t) is any recurrence of the current family with signature (5,-1) and (x, y, z) are any three consecutive terms of (t), for integer k >= 0. (End)

			G.f. = x + 5*x^2 + 24*x^3 + 115*x^4 + 551*x^5 + 2640*x^6 + 12649*x^7 + ...

F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 0..1467 (terms 0..200 from T. D. Noe)
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
K. Andersen, L. Carbone, and D. Penta, 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.
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 , Example 12
Noureddine Chair, Exact two-point resistance, and the simple random walk on the complete graph minus N edges, Ann. Phys. 327, No. 12, 3116-3129 (2012), B(7).
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Dale Gerdemann, Fractal images from (5,-1) recursion, YouTube Video, Nov 05 2014.
Dale Gerdemann, Fractal images from (5,-1) recursion: Selections in detail, YouTube Video, Nov 05 2014.
Frank A. Haight, Letter to N. J. A. Sloane, Sep 06 1976
Frank A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971. [Annotated scanned copy]
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=5, q=-1.
A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=7.
Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.
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
F. M. van Lamoen, Square wreaths around hexagons, Forum Geometricorum, 6 (2006) 311-325.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (5,-1).

Partial sums of A004253.
Cf. A000027, A001906, A001353, A003501, A030221. a(n) = sqrt((A003501(n)^2 - 4)/21).
First differences of a(n) are in A004253, partial sums in A089817.
Cf. A004253.
INVERT transformation yields A001109. - R. J. Mathar, Sep 11 2008
Cf. A054493, A107905.

[ n eq 1 select 0 else n eq 2 select 1 else 5*Self(n-1)-Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 19 2011

A004254:=1/(1-5*z+z**2); # Simon Plouffe in his 1992 dissertation

a[n_]:=(MatrixPower[{{1,3},{1,4}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
a[ n_] := ChebyshevU[2 n - 1, Sqrt[7]/2] / Sqrt[7]; (* Michael Somos, Jan 22 2017 *)

{a(n) = subst(4*poltchebi(n+1) - 10*poltchebi(n), x, 5/2) / 21}; /* Michael Somos, Dec 04 2002 */

{a(n) = imag((5 + quadgen(84))^n) / 2^(n-1)}; /* Michael Somos, Dec 04 2002 */

{a(n) = polchebyshev(n - 1, 2, 5/2)}; /* Michael Somos, Jan 22 2017 */

{a(n) = simplify( polchebyshev( 2*n - 1, 2, quadgen(28)/2) / quadgen(28))}; /* Michael Somos, Jan 22 2017 */

[lucas_number1(n,5,1) for n in range(27)] # Zerinvary Lajos, Jun 25 2008

G.f.: x/(1-5*x+x^2).
a(n) = (((5+sqrt(21))/2)^n-((5-sqrt(21))/2)^n)/sqrt(21). - Barry E. Williams, Aug 29 2000
a(n) = S(2*n-1, sqrt(7))/sqrt(7) = S(n-1, 5); S(n, x)=U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.
A003501(n) = sqrt(21*a(n)^2 + 4).
a(n) = Sum_{k=0..n-1} binomial(n+k, 2*k+1)*2^k. - Paul Barry, Nov 30 2004
[A004253(n), a(n)] = [1,3; 1,4]^n * [1,0]. - Gary W. Adamson, Mar 19 2008
a(n+1) = Sum_{k=0..n} Gegenbauer_C(n-k,k+1,2). - Paul Barry, Apr 21 2009
a(n+1) = Sum_{k=0..n} A101950(n,k)*4^k. - Philippe Deléham, Feb 10 2012
From Peter Bala, Dec 23 2012: (Start)
Product {n >= 1} (1 + 1/a(n)) = (1/3)*(3 + sqrt(21)).
Product {n >= 2} (1 - 1/a(n)) = (1/10)*(3 + sqrt(21)). (End)
From Michael Somos, Jan 22 2017: (Start)
A054493(2*n - 1) = 7 * a(n)^2 for all n in Z.
a(n) = -a(-n) for all n in Z.
0 = -1 + a(n)*(+a(n) - 5*a(n+1)) + a(n+1)*(+a(n+1)) for all n in Z. (End)
Limit_{n->oo} a(n+1)/a(n) = (5 + sqrt(21))/2 = A107905. - Wolfdieter Lang, Nov 15 2023
From Klaus Purath, Jul 26 2024: (Start)
a(n) = 4(a(n-1) + a(n-2)) - a(n-3).
a(n) = 6(a(n-1) - a(n-2)) + a(n-3).
In general, for all sequences of the form U(n) = P*U(n-1) - U(n-2) the following applies:
U(n) = (P-1)*U(n-1) + (P-1)*U(n-2) - U(n-3).
U(n) = (P+1)*U(n-1) - (P+1)*U(n-2) + U(n-3). (End)
a(n) = (5*a(n-1)+sqrt(21*a(n-1)^2+4))/2 for n>0. - Alexandru Petrescu, Apr 15 2025
From Peter Bala, May 22 2025: (Start)
Product_{n >= 1} ((a(2*n) + 1)/(a(2*n) - 1))^2 = 7/3.
Product_{n >= 1} ((a(2*n+1) + 1)/(a(2*n+1) - 1))^2 = 25/21.
The o.g.f. A(x) satisfies A(x) + A(-x) + 10*A(x)*A(-x) = 0. The o.g.f. for A097778 equals -1/x * A(sqrt(x))*A(-sqrt(x)). (End)
E.g.f.: 2*exp(5*x/2)*sinh(sqrt(21)*x/2)/sqrt(21). - Stefano Spezia, Jul 02 2025

A350917 a(0) = 1, a(1) = 2, and a(n) = 23*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

1, 2, 41, 937, 21506, 493697, 11333521, 260177282, 5972743961, 137112933817, 3147624733826, 72258255944177, 1658792261982241, 38079963769647362, 874180374439907081, 20068068648348215497, 460691398537569049346, 10575834097715739919457, 242783492848924449098161, 5573444501427546589338242, 127946440039984647105681401, 2937194676418219336841333977

Offset: 0

Max Alekseyev, Jan 21 2022

One of 10 linear second-order recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916.
Other properties for all n:
(a(n)+1)*(a(n+2)+1) = (a(n+1)+1)*(a(n+1)+26);
((105*a(n) - 20)^2 - 50^2) / 21 is an integer square.

Sebastian et al., Are there infinitely many positive integer solutions to (3+3k+l)^2=m(kl-k^3-1)?, MathOverflow, 2022.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A350916.

Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350919, A350920, A350921, A350922, A350923, A350924, A350925, A350926.

a(n) = 17/42*A090731(n) - 15/2*A097778(n-1) + 4/21.

G.f.: ( -1+22*x-17*x^2 ) / ( (x-1)*(x^2-23*x+1) ). - R. J. Mathar, Feb 07 2022

A350926 a(0) = 1, a(1) = 17, and a(n) = 23*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

1, 17, 386, 8857, 203321, 4667522, 107149681, 2459775137, 56467678466, 1296296829577, 29758359401801, 683145969411842, 15682598937070561, 360016629583211057, 8264699881476783746, 189728080644382815097, 4355481154939327963481, 99986338482960160344962, 2295330303953144359970641

Offset: 0

Max Alekseyev, Jan 22 2022

One of 10 linear second-order recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916.

Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A350916.

Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350917, A350919, A350920, A350921, A350922, A350923, A350925, A350925.

LinearRecurrence[{24,-24,1},{1,17,386},20] (* Harvey P. Dale, Jun 12 2022 *)

G.f.: (1 - 7*x + 2*x^2)/((1 - x)*(1 - 23*x + x^2)). - Stefano Spezia, Jan 22 2022

21*a(n) = 4+17*A097778(n)-38*A097778(n-1). - R. J. Mathar, Feb 07 2022

A090731 a(n) = 23a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 23.

Original entry on oeis.org

2, 23, 527, 12098, 277727, 6375623, 146361602, 3359941223, 77132286527, 1770682648898, 40648568638127, 933146396028023, 21421718540006402, 491766380024119223, 11289205022014735727, 259159949126314802498

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004

A Chebyshev T-sequence with Diophantine property.

a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 21*(5*b)^2 =+4 with companion sequence b(n)=A097778(n-1), n>=1; b(0):=0.

			(x;y) = (0;2), (23;1), (527;23), (12098;528), ... give the
nonnegative integer solutions to x^2 - 21*(5*y)^2 = 4.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Indranil Ghosh, Table of n, a(n) for n = 0..733
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (23,-1).

Cf. A037088, A051502.
a(n)=sqrt(4 + 21*(5*A097778(n-1))^2), n>=1.
Cf. A077428, A078355 (Pell +4 equations).

a[0] = 2; a[1] = 23; a[n_] := 23a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{23,-1},{2,23},30] (* Harvey P. Dale, Feb 20 2012 *)

[lucas_number2(n,23,1) for n in range(0,20)] # Zerinvary Lajos, Jun 26 2008

a(n) = S(n, 23) - S(n-2, 23) = 2*T(n, 23/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 23)=A097778(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
a(n) = ap^n + am^n, with ap := (23+5*sqrt(21))/2 and am := (23-5*sqrt(21))/2.
G.f.: (2-23*x)/(1-23*x+x^2).

Chebyshev and Pell comments from Wolfdieter Lang, Sep 10 2004

A334673 a(n) = 23*a(n-1) - a(n-2) + 1 for n > 1, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 24, 552, 12673, 290928, 6678672, 153318529, 3519647496, 80798573880, 1854847551745, 42580695116256, 977501140122144, 22439945527693057, 515141245996818168, 11825808712399124808, 271478459139183052417, 6232178751488811080784, 143068632825103471805616

Offset: 0

Francesca Arici, Sep 11 2020

Michael De Vlieger, Table of n, a(n) for n = 0..735
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A004253, A004254, A030221, A097778 (first differences).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

CoefficientList[Series[x/((1 - x) (x^2 - 23 x + 1)), {x, 0, 18}], x] (* Michael De Vlieger, Apr 07 2021 *)

a(n) = A004254(n)*A004254(n+1)/5 = A160695(n+1)/5.
G.f.: x/((1-x)*(x^2-23*x+1)). - Alois P. Heinz, Sep 11 2020
From Klaus Purath, Jun 18 2025: (Start)
a(n) = (A004253(n+1)^2 - 1) / 15.
a(n) = (A030221(n)^2 - 1) / 35.
a(n) + a(n+1) = A004253(n+1)^2. (End)

a(13)-a(14) corrected and more terms added by Alois P. Heinz, Sep 11 2020

A362357 Bisection of Chebyshev {S(n, 5)}_{n>=0}; the even part.

Original entry on oeis.org

1, 24, 551, 12649, 290376, 6665999, 153027601, 3512968824, 80645255351, 1851327904249, 42499896542376, 975646292570399, 22397364832576801, 514163744856696024, 11803368766871431751, 270963317893186234249

Offset: 0

Wolfdieter Lang, Apr 26 2023

The odd part of this bisection is given by 5*A097778(n), for n >= 0.

Index entries for sequences related to Chebyshev polynomials.`
Index entries for linear recurrences with constant coefficients, signature (23,-1).

Cf. A004254, A049310, A097778.

Table[ChebyshevU[2*n, 5/2], {n, 0, 20}] (* Vaclav Kotesovec, May 27 2023 *)

a(n) = polchebyshev(2*n, 2, 5/2); \\ Michel Marcus, May 27 2023

a(n) = S(2*n, 5) = S(n, 23) + S(n-1, 23), with the Chebyshev S polynomials (see A049310), S(-1, x) = 0, S(n, 5) = A004254(n+1) and S(n, 23) = A097778(n).
O.g.f.: (1 + x)/(1 - 23*x + x^2).
a(n) = 23*a(n-1) - a(n-2), for n >= 0, with a(-1) = -1 and a(-2) = -24.

cp's OEIS Frontend

A004254 a(n) = 5*a(n-1) - a(n-2) for n > 1, a(0) = 0, a(1) = 1.

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A350917 a(0) = 1, a(1) = 2, and a(n) = 23*a(n-1) - a(n-2) - 4 for n >= 2.

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A350926 a(0) = 1, a(1) = 17, and a(n) = 23*a(n-1) - a(n-2) - 4 for n >= 2.

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A090731 a(n) = 23a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 23.

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A334673 a(n) = 23*a(n-1) - a(n-2) + 1 for n > 1, a(0)=0, a(1)=1.

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A362357 Bisection of Chebyshev {S(n, 5)}_{n>=0}; the even part.

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