A088320 -id:A088320 - cp's OEIS frontend

A086927 a(n) = 10*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 10.

2, 10, 102, 1030, 10402, 105050, 1060902, 10714070, 108201602, 1092730090, 11035502502, 111447755110, 1125513053602, 11366578291130, 114791295964902, 1159279537940150, 11707586675366402, 118235146291604170

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 21 2003

a(n+1)/a(n) converges to (5+sqrt(26)) = 10.099019...

Lim a(n)/a(n+1) as n approaches infinity = 0.099019... = 1/(5+sqrt(26)) = (sqrt(26)-5).

			a(4) = 10402 = 10*a(3) + a(2) = 10*1030 + 102 = (5+sqrt(26))^4 + (5-sqrt(26))^4 =  10401.999903 + 0.000097 = 10402.

Stefano Arnone, C Falcolini, F Moauro, M Siccardi, On Numbers in Different Bases: Symmetries and a Conjecture, Experimental Mathematics, Vol 26 2016, pp 197-209; http://dx.doi.org/10.1080/10586458.2016.1149125

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (10,1).

Cf. A036336.

I:=[2,10]; [n le 2 select I[n] else 10*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016

RecurrenceTable[{a[0] == 2, a[1] == 10, a[n] == 10 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)

a(n) = (5+sqrt(26))^n + (5-sqrt(26))^n.
G.f.: (2-10*x)/(1-10*x-x^2). - Philippe Deléham, Nov 20 2008
a(n) = 2*A088320(n). - R. J. Mathar, Feb 06 2020

More terms from Jon E. Schoenfield, May 15 2010

A089926 a(n) = 12*a(n-1) + a(n-2), a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 73, 882, 10657, 128766, 1555849, 18798954, 227143297, 2744518518, 33161365513, 400680904674, 4841332221601, 58496667563886, 706801342988233, 8540112783422682, 103188154744060417, 1246797969712147686

Offset: 0

Paul Barry, Nov 15 2003

The family of recurrences a(n) = 2*k*a(n-1) + a(n-2), a(0)=1, a(1)=k has solution a(n) = ((k+sqrt(k^2+1))^n + (k-sqrt(k^2+1))^n)/2; a(n) = Sum_{j=0..floor(n/2)} C(n,2k)*(k^2+1)^jk^(n-2j); a(n) = T(n,ki)*(-i)^n; e.g.f. exp(kx)*cosh(sqrt(k^2+1)*x).

Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (12,1).

Cf. A088320, A088317, A005667, A001077.

Essentially the same as A041060.

E.g.f.: exp(6x)*cosh(sqrt(37)x);
a(n) = ((6+sqrt(37))^n + (6-sqrt(37))^n)/2;
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*37^k*6^(n-2k).
a(n) = T(n, 6i)*(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2 = -1.
G.f.: (1-6x)/(1-12*x-x^2). - Philippe Deléham, Nov 21 2008

cp's OEIS Frontend

A086927 a(n) = 10*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 10.

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