OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).
FORMULA
a(n) = 10*a(n-2) - a(n-4).
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - a(n-4) + a(n-5).
G.f.: x*(1+7*x+7*x^2+x^3)/(1-10*x^2+x^4).
a(2*n+1) = ((1+r)*(5+2*r)^n + (1-r)*(5-2*r)^n)/2 where r=sqrt(6) and 0<=n.
a(2*n+2) = ((7+3*r)*(5+2*r)^n + (7-3*r)*(5-2*r)^n)/2 where r=sqrt(6) and 0<=n.
a(n) = -((5-2*r)^(1/4)*((2*r+5)^((-1)^n/4+n/2)*(-1)^n - r*(2*r+5)^((-1)^n/4+n/2)) + (2*r+5)^(1/4)*((5-2*r)^((-1)^n/4+n/2)*(-1)^n + (5-2*r)^((-1)^n/4+n/2)*r))/(2*(5-2*r)^(1/4)*(2*r+5)^(1/4)) with r=sqrt(6) and 1<=n. - Alexander R. Povolotsky, Sep 01 2012
a(n) = b(n) +7*b(n-1) +7*b(n-2) +b(n-3), where b(n) = (1/2)*(1 +(-1)^n)* ChebyshevU(n/2, 5). - G. C. Greubel, Apr 28 2022
MAPLE
a(1)=1: a(2)=7: a(3)=17: a(4)=71:
for n from 5 to 20 do
a(n)=10*a(n-2)-a(n-4):
printf("%9d%20d\n", n, a(n)):
end do:
MATHEMATICA
LinearRecurrence[{0, 10, 0, -1}, {1, 7, 17, 71}, 50] (* G. C. Greubel, Feb 22 2017 *)
PROG
(PARI)
a(n) = if(n<1, 0, if(n<5, [1, 7, 17, 71][n], 10*a(n-2)-a(n-4) ) );
/* Joerg Arndt, Sep 03 2012 */
(SageMath)
def b(n): return (1/2)*(1+(-1)^n)*chebyshev_U(n//2, 5)
def A216073(n): return b(n) +7*b(n-1) +7*b(n-2) +b(n-3)
[A216073(n) for n in (0..50)] # G. C. Greubel, Apr 28 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, Sep 01 2012
STATUS
approved