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A294262
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a(n) = 3*a(n-1) + 5*a(n-2) + a(n-3), with a(0) = a(1) = 1 and a(2) = 7, a linear recurrence which is a trisection of A005252.
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3
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1, 1, 7, 27, 117, 493, 2091, 8855, 37513, 158905, 673135, 2851443, 12078909, 51167077, 216747219, 918155951, 3889371025, 16475640049, 69791931223, 295643364939, 1252365390981, 5305104928861, 22472785106427, 95196245354567, 403257766524697, 1708227311453353, 7236167012338111, 30652895360805795, 129847748455561293, 550043889183050965
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OFFSET
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0,3
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LINKS
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FORMULA
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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) = 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.
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MATHEMATICA
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LinearRecurrence[{3, 5, 1}, {1, 1, 7}, 30]
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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