A054477 -id:A054477 - cp's OEIS frontend

A002320 a(n) = 5*a(n-1) - a(n-2).

1, 3, 14, 67, 321, 1538, 7369, 35307, 169166, 810523, 3883449, 18606722, 89150161, 427144083, 2046570254, 9805707187, 46981965681, 225104121218, 1078538640409, 5167589080827, 24759406763726, 118629444737803

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Together with A002310 these are the two sequences satisfying the requirement that (a(n)^2 + a(n-1)^2)/(1 - a(n)*a(n-1)) be an integer; in both cases this integer is -5. - Floor van Lamoen, Oct 26 2001

From a posting to Netnews group sci.math by ksbrown(AT)seanet.com (K. S. Brown) on Aug 15 1996.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See p. 44.
Tanya Khovanova, Recursive Sequences
MathPages, N = (x^2 + y^2)/(1+xy) is a Square
Index entries for linear recurrences with constant coefficients, signature (5,-1).

Cf. A054477.

a002320 n = a002320_list !! n
a002320_list = 1 : 3 :
   (zipWith (-) (map (* 5) (tail a002320_list)) a002320_list)
-- Reinhard Zumkeller, Oct 16 2011

LinearRecurrence[{5,-1},{1,3},30] (* Harvey P. Dale, Nov 13 2014 *)

Sequences A002310, A002320 and A049685 have this in common: each one satisfies a(n+1) = (a(n)^2+5)/a(n-1) - Graeme McRae, Jan 30 2005
G.f.: (1-2x)/(1-5x+x^2). - Philippe Deléham, Nov 16 2008
a(n) = Sum_{k = 0..n} A238731(n,k)*2^k. - _Philippe Deléham, Mar 05 2014
E.g.f.: exp(5*x/2)*(sqrt(21)*cosh(sqrt(21)*x/2) + sinh(sqrt(21)*x/2))/sqrt(21). - Stefano Spezia, Jul 07 2025
From Peter Bala, Jul 07 2025: (Start)
a(n) = ( (4 + sqrt(21))*(5 - sqrt(21))^(n+1) - (4 - sqrt(21))*(5 + sqrt(21))^(n+1) )/(2^(n+1)*sqrt(21)).
Sum_{n >= 1} (-1)^(n+1)/(a(2*n) + 5/a(2*n)) = 1/15, since 5/(a(2*n) + 5/a(2*n)) = 1/a(2*n-1) + 1/a(2*n+1).
Sum_{n >= 1} (-1)^(n+1)/(a(2*n-1) + 5/a(2*n-1)) = 1/5, since 5/(a(2*n-1) + 5/a(2*n-1)) = 1/a(2*n-2) + 1/a(2*n). (End)

A002310 a(n) = 5*a(n-1) - a(n-2), with a(0) = 1 and a(1) = 2.

Original entry on oeis.org

1, 2, 9, 43, 206, 987, 4729, 22658, 108561, 520147, 2492174, 11940723, 57211441, 274116482, 1313370969, 6292738363, 30150320846, 144458865867, 692144008489, 3316261176578, 15889161874401, 76129548195427, 364758579102734, 1747663347318243, 8373558157488481, 40120127440124162

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Together with A002320 these are the two sequences satisfying ( a(n)^2+a(n-1)^2 )/(1 - a(n)a(n-1)) is an integer, in both cases this integer is -5. - Floor van Lamoen, Oct 26 2001

Limit_{n->oo} a(n+1)/a(n) = (5 + sqrt(21))/2 = A107905. - Wolfdieter Lang, Nov 17 2023

From a posting to Netnews group sci.math by ksbrown(AT)seanet.com (K. S. Brown) on Aug 15 1996.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 44, 56.
Margherita Maria Ferrari and Norma Zagaglia Salvi, Aperiodic Compositions and Classical Integer Sequences, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.8.
Tanya Khovanova, Recursive Sequences
MathPages, N = (x^2 + y^2)/(1+xy) is a Square
Index entries for linear recurrences with constant coefficients, signature (5,-1).

Cf. A002310, A002320, A004254, A049310, A049685, A054477, A107905.

a002310 n = a002310_list !! n
a002310_list = 1 : 2 :
   (zipWith (-) (map (* 5) (tail a002310_list)) a002310_list)
-- Reinhard Zumkeller, Oct 16 2011

LinearRecurrence[{5, -1}, {1, 2}, 25] (* T. D. Noe, Feb 22 2014 *)

Sequences A002310, A002320 and A049685 have this in common: each one satisfies a(n+1) = (a(n)^2+5)/a(n-1). - Graeme McRae, Jan 30 2005
G.f.: (1-3x)/(1-5x+x^2). - Philippe Deléham, Nov 16 2008
a(n) = S(n, 5) - 3*S(n-1, 5), for n >= 0, with the S-Chebyshev polynomial (see A049310) S(n, 5) = A004254(n+1). - Wolfdieter Lang, Nov 17 2023
E.g.f.: exp(5*x/2)*(21*cosh(sqrt(21)*x/2) - sqrt(21)*sinh(sqrt(21)*x/2))/21. - Stefano Spezia, Jul 07 2025

A003769 Number of perfect matchings (or domino tilings) in K_4 X P_n.

Original entry on oeis.org

3, 16, 75, 361, 1728, 8281, 39675, 190096, 910803, 4363921, 20908800, 100180081, 479991603, 2299777936, 11018898075, 52794712441, 252954664128, 1211978608201, 5806938376875, 27822713276176, 133306628004003, 638710426743841, 3060245505715200

Offset: 1

Frans J. Faase

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Colin Barker, Table of n, a(n) for n = 1..1000
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (4,4,-1).

Essentially the same as A005386. First differences of A099025.

Vec(x*(3 + 4*x - x^2) / ((1 + x)*(1 - 5*x + x^2)) + O(x^40)) \\ Colin Barker, Dec 16 2017

a(n) = 4a(n-1) + 4a(n-2) - a(n-3), n>3.
a(n) = (1/7)*(6*A030221(n) - A054477(n) + 2(-1)^n).
G.f.: x*(3+4*x-x^2)/((1+x)*(1-5*x+x^2)). - R. J. Mathar, Dec 16 2008
a(n) = 2^(-1-n)*((-1)^n*2^(2+n) + (5-sqrt(21))^(1+n) + (5+sqrt(21))^(1+n)) / 7. - Colin Barker, Dec 16 2017

cp's OEIS Frontend

A002320 a(n) = 5*a(n-1) - a(n-2).

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A002310 a(n) = 5*a(n-1) - a(n-2), with a(0) = 1 and a(1) = 2.

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A003769 Number of perfect matchings (or domino tilings) in K_4 X P_n.

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