A274977 a(n) = a(n-1) + 3*a(n-2) with n>1, a(0)=1, a(1)=6.
1, 6, 9, 27, 54, 135, 297, 702, 1593, 3699, 8478, 19575, 45009, 103734, 238761, 549963, 1266246, 2916135, 6714873, 15463278, 35607897, 81997731, 188821422, 434814615, 1001278881, 2305722726, 5309559369, 12226727547, 28155405654, 64835588295, 149301805257, 343808570142
Offset: 0
Examples
Table of similar sequences (not extendable on the left side) where this recurrence can be applied to the first two terms: ---------------------------------------------------------------------- (*) - - 1, -1, 2, -1, 5, 2, 17, 23, 74, 143, 365, ... A052533: - - 1, 0, 3, 3, 12, 21, 57, 120, 291, 651, 1524, ... (^) - 0, 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, ... A006138: - - 1, 2, 5, 11, 26, 59, 137, 314, 725, 1667, 3842, ... A105476: - - 1, 3, 6, 15, 33, 78, 177, 411, 942, 2175, 5001, ... (^) 0, 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, 6160, ... A105963: - - 1, 5, 8, 23, 47, 116, 257, 605, 1376, 3191, 7319, ... A274977: - - 1, 6, 9, 27, 54, 135, 297, 702, 1593, 3699, 8478, ... A075118: - 2, 1, 7, 10, 31, 61, 154, 337, 799, 1810, 4207, 9637, ... ---------------------------------------------------------------------- (*) see version A140165. (^) see A006130 and the signed versions A140167, A182228.
Links
- Bruno Berselli, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,3).
Programs
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GAP
a:=[1,6];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, Jan 15 2020
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Magma
[n le 2 select 5*n-4 else Self(n-1)+3*Self(n-2): n in [1..40]];
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Magma
R
:=PowerSeriesRing(Integers(), 32); Coefficients(R!((1 + 5*x)/(1- x-3*x^2))); // Marius A. Burtea, Jan 15 2020 -
Maple
seq(coeff(series((1+5*x)/(1-x-3*x^2), x, n+1), x, n), n = 0..40); # G. C. Greubel, Jan 15 2020
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Mathematica
RecurrenceTable[{a[n]==a[n-1] +3a[n-2], a[0]==1, a[1]==6}, a, {n,0,40}] Table[Round[Sqrt[3]^(n-1)*(Sqrt[3]*Fibonacci[n+1, 1/Sqrt[3]] + 5*Fibonacci[n, 1/Sqrt[3]])], {n,0,40}] (* G. C. Greubel, Jan 15 2020 *) LinearRecurrence[{1,3},{1,6},40] (* Harvey P. Dale, Jul 11 2023 *) -
PARI
v=vector(40); v[1]=1; v[2]=6; for(n=3, #v, v[n]=v[n-1]+3*v[n-2]); v
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Sage
from sage.combinat.sloane_functions import recur_gen2 a = recur_gen2(1, 6, 1, 3) [next(a) for n in range(40)]
Formula
G.f.: (1 + 5*x)/(1 - x - 3*x^2).
a(n) = ((13 + 11*sqrt(13))*(1 + sqrt(13))^n + (13 - 11*sqrt(13))*(1 - sqrt(13))^n)/(26*2^n).
3*a(n) + a(n+1) = 9*A105476(n+1).
a(n+1) - a(n) = 27*A105476(n-3) with n>2.
a(n) = 3^((n-1)/2)*( sqrt(3)*Fibonacci(n+1, 1/sqrt(3)) + 5*Fibonacci(n, 1/sqrt(3)) ). - G. C. Greubel, Jan 15 2020
E.g.f.: (1/13)*exp(x/2)*(13*cosh((sqrt(13)*x)/2) + 11*sqrt(13)*sinh((sqrt(13)*x)/2)). - Stefano Spezia, Jan 15 2020
Comments