OFFSET
0,3
LINKS
Hermann Stamm-Wilbrandt, 6 interlaced bisections
Index entries for linear recurrences with constant coefficients, signature (3,5,1).
FORMULA
G.f.: (1 - 2*x - x^2)/(1 - 3*x - 5*x^2 - x^3).
a(n) = (1/20)*(10*(-1)^n + (2-sqrt(5))^n*(5-sqrt(5)) + (2+sqrt(5))^n*(5+sqrt(5))).
a(n) = A005252(3*n).
a(n) = 4*a(n-1) + a(n-2) + 2*(-1)^n for n >= 2.
a(n) = Sum_{k=0..floor(3*n/4)} binomial(3*n-2*k, 2*k).
a(n) = (Fibonacci(3*n + 1) + (-1)^n)/2.
a(2*n) = A232970(2*n); a(2*n+1) = A049651(2*n+1). See "6 interlaced bisections" link. - Hermann Stamm-Wilbrandt, Apr 18 2019
MATHEMATICA
LinearRecurrence[{3, 5, 1}, {1, 1, 7}, 30]
PROG
(bc)
a=1
b=1
c=7
print 0, " ", a, "\n"
print 1, " ", b, "\n"
print 2, " ", c, "\n"
for(x=3; x<=1000; ++x){
d=3*c+5*b+1*a
print x, " ", d, "\n"
a=b
b=c
c=d
} # Hermann Stamm-Wilbrandt, Apr 18 2019
(PARI) {a(n) = (fibonacci(3*n+1) +(-1)^n)/2}; \\ G. C. Greubel, Apr 19 2019
(Magma) [(Fibonacci(3*n+1) +(-1)^n)/2 : n in [0..30]]; // G. C. Greubel, Apr 19 2019
(Sage) [(fibonacci(3*n+1) +(-1)^n)/2 for n in (0..30)] # G. C. Greubel, Apr 19 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jean-François Alcover and Paul Curtz, Oct 26 2017
STATUS
approved