A086902 a(n) = 7*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 7.

2, 7, 51, 364, 2599, 18557, 132498, 946043, 6754799, 48229636, 344362251, 2458765393, 17555720002, 125348805407, 894997357851, 6390330310364, 45627309530399, 325781497023157, 2326097788692498, 16608466017870643, 118585359913786999, 846705985414379636, 6045527257814444451

Offset

0,1

Comments

From Johannes W. Meijer, Jun 12 2010: (Start)

In general sequences with recurrence a(n) = k*a(n-1)+a(n-2) with a(0)=2 and a(1)=k [and a(-1)=0] have generating function (2-k*x)/(1-k*x-x^2). If k is odd (k>=3) they satisfy a(3n+1) = b(5n), a(3n+2)=b(5*n+3), a(3n+3)=2*b(5n+4) where b(n) is the sequence of numerators of continued fraction convergents to sqrt(k^2+4). [If k is even then a(n)/2, for n>=1, is the sequence of numerators of continued fraction convergents to sqrt(k^2/4+1).]

For the sequence given above k=7 which implies that it is associated with A041090.

For a similar statement about sequences with recurrence a(n) = k*a(n-1)+a(n-2) but with a(0)=1 [and a(-1)=0] see A054413; a sequence that is associated with A041091.

For more information follow the Khovanova link and see A087130, A140455 and A178765.

(End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences.

Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Hadamard Product of the Generalized Fibonacci, Lucas, Catalan, and Harmonic Numbers, Journal of Integer Sequences, Vol. 28 (2025), Article 25.8.8. See p. 5.

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 (7,1).

Formula

a(n) = ((7+sqrt(53))/2)^n + ((7-sqrt(53))/2)^n.

a(n+1)/a(n) converges to (7+sqrt(53))/2 = 7.14005... = A176439.

Lim a(n)/a(n+1) as n approaches infinity = 0.1400549... = 2/(7+sqrt(53)) = (sqrt(53)-7)/2 = 1/A176439 = A176439 - 7.

E.g.f.: 2*exp(7*x/2)*cosh(sqrt(53)*x/2); a(n) = 2^(1-n)sum{k=0..floor(n/2), C(n, 2k)53^k7^(n-2k)}. a(n) = 2*T(n, 7i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry, Nov 15 2003

G.f.: (2-7x)/(1-7x-x^2). - Philippe Deléham, Nov 16 2008

From Johannes W. Meijer, Jun 12 2010: (Start)

a(2n+1) = 7*A097837(n), a(2n) = A099368(n).

a(3n+1) = A041090(5n), a(3n+2) = A041090(5*n+3), a(3n+3) = 2*A041090(5n+4).

Limit_{k->oo} a(n+k)/a(k) = (A086902(n) + A054413(n-1)*sqrt(53))/2.

Limit_{n->oo} A086902(n)/A054413(n-1) = sqrt(53). (End)

Example

a(4) = 7*a(3) + a(2) = 7*364 + 51 = 2599.

Mathematica

RecurrenceTable[{a[0] == 2, a[1] == 7, a[n] == 7 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
LinearRecurrence[{7, 1}, {2, 7}, 30] (* Harvey P. Dale, May 25 2023 *)

Prog

(PARI) a(n)=([0, 1; 1, 7]^n*[2; 7])[1, 1] \\ Charles R Greathouse IV, Apr 06 2016
(Magma) I:=[2, 7]; [n le 2 select I[n] else 7*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016

Crossrefs

Cf. A176439.

Cf. A000032 (k=1), A006497 (k=3), A087130 (k=5), A086902 (k=7), A087798 (k=9), A001946 (k=11), A088316 (k=13), A090301 (k=15), A090306 (k=17). - Johannes W. Meijer, Jun 12 2010

Sequence in context: A340027 A362340 A324513 * A381940 A265042 A385835

Adjacent sequences: A086899 A086900 A086901 * A086903 A086904 A086905

Keyword

nonn,easy

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 18 2003

Status

approved