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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover and Paul Curtz, Oct 26 2017

STATUS

approved

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Last modified June 17 22:17 EDT 2019. Contains 324200 sequences. (Running on oeis4.)