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

0,3

Links

Table of n, a(n) for n=0..29.

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) = A110679(n) - A001076(n).

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

Cf. A001076, A005252, A110679.

Sequence in context: A055917 A056120 A255278 * A048711 A249184 A118101

Adjacent sequences: A294259 A294260 A294261 * A294263 A294264 A294265

Keyword

easy,nonn

Author

Jean-François Alcover and Paul Curtz, Oct 26 2017

Status

approved