A294262 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.
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
Offset: 0
Links
- Hermann Stamm-Wilbrandt, 6 interlaced bisections
- Index entries for linear recurrences with constant coefficients, signature (3,5,1).
Programs
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Magma
[(Fibonacci(3*n+1) +(-1)^n)/2 : n in [0..30]]; // G. C. Greubel, Apr 19 2019
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Mathematica
LinearRecurrence[{3,5,1},{1,1,7},30] -
PARI
{a(n) = (fibonacci(3*n+1) +(-1)^n)/2}; \\ 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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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
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