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URL: https://oeis.org/A106854

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A106854
Expansion of 1/(1-x*(1-5*x)).
11
1, 1, -4, -9, 11, 56, 1, -279, -284, 1111, 2531, -3024, -15679, -559, 77836, 80631, -308549, -711704, 831041, 4389561, 234356, -21713449, -22885229, 85682016, 200108161, -228301919, -1228842724, -87333129, 6056880491, 6493546136, -23790856319, -56258586999, 62695694596, 343988629591
OFFSET
0,3
COMMENTS
Row sums of Riordan array (1,x*(1-5*x)). In general, a(n) = Sum_{k=0..n}(-1)^(n-k)*binomial(k,n-k)*r^(n-k), yields the row sums of the Riordan array (1,x*(1-k*x)).
LINKS
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae, Vol. 38, No. 2 (2024), pp. 284-313. See p. 16.
FORMULA
a(n) = ((1+sqrt(-19))^(n+1)-(1-sqrt(-19))^(n+1))/(2^(n+1)*sqrt(-19)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k, n-k)*5^(n-k).
a(n) = 5^(n/2)(cos(-n*arccot(sqrt(19)/19))-sqrt(19)sin(-n*arccot(sqrt(19)/19))/19).
a(n) = a(n-1)-5*a(n-2), a(0)=1, a(1)=1. - Philippe Deléham, Oct 21 2008
a(n) = Sum_{k=0..n} A109466(n,k)*5^(n-k). - Philippe Deléham, Oct 25 2008
G.f.: Q(0)/2, where Q(k) = 1 + 1/( 1 - x*(2*k+1 -5*x)/( x*(2*k+2 -5*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2013
E.g.f.: exp(x/2) * (cos(sqrt(19)*x/2) + sin(sqrt(19)*x/2)/sqrt(19)). - Amiram Eldar, Feb 10 2026
MATHEMATICA
Join[{a=1, b=1}, Table[c=b-5*a; a=b; b=c, {n, 80}]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2011 *)
CoefficientList[Series[1/(1-x(1-5x)), {x, 0, 40}], x] (* or *) LinearRecurrence[ {1, -5}, {1, 1}, 40] (* Harvey P. Dale, Jan 21 2012 *)
PROG
(SageMath) [lucas_number1(n, 1, 5) for n in range(1, 35)] # Zerinvary Lajos, Jul 16 2008
(PARI) Vec(1/(1-x+5*x^2) + O(x^99)) \\ Altug Alkan, Sep 06 2016
(Magma) I:=[1, 1]; [n le 2 select I[n] else Self(n-1) - 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, May 08 2005
STATUS
approved