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A164301 a(n) = ((1+4*sqrt(2))*(5+sqrt(2))^n + (1-4*sqrt(2))*(5-sqrt(2))^n)/2. 8
1, 13, 107, 771, 5249, 34757, 226843, 1469019, 9472801, 60940573, 391531307, 2513679891, 16131578849, 103501150997, 663985196443, 4259325491499, 27321595396801, 175251467663533, 1124117982508907, 7210396068827811, 46249247090573249, 296653361322692837 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of A164300. Fifth binomial transform of A164587. Inverse binomial transform of A164598.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)

Index entries for linear recurrences with constant coefficients, signature (10,-23).

FORMULA

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 13.

G.f.: (1+3*x)/(1-10*x+23*x^2).

E.g.f.: ( cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x) )*exp(5*x). - G. C. Greubel, Sep 13 2017

From G. C. Greubel, Mar 12 2021: (Start)

a(n) = 2*A083880(n) + 8*A081182(n).

a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*4^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

MATHEMATICA

LinearRecurrence[{10, -23}, {1, 13}, 20] (* Harvey P. Dale, Oct 15 2015 *)

PROG

(MAGMA) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+4*r)*(5+r)^n+(1-4*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

(PARI) my(x='x+O('x^50)); Vec((1+3*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Sep 13 2017

(Sage) [( (1+3*x)/(1-10*x+23*x^2) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

CROSSREFS

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), this sequence (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

Cf. A081182, A083880, A164587.

Sequence in context: A218093 A132261 A142364 * A322499 A087305 A168117

Adjacent sequences:  A164298 A164299 A164300 * A164302 A164303 A164304

KEYWORD

nonn

AUTHOR

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

EXTENSIONS

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009

STATUS

approved

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Last modified August 7 23:50 EDT 2022. Contains 355995 sequences. (Running on oeis4.)