A189748
a(n) = 5*a(n-1) + 4*a(n-2) with a(1)=5, a(2)=4.
Original entry on oeis.org
5, 4, 40, 216, 1240, 7064, 40280, 229656, 1309400, 7465624, 42565720, 242691096, 1383718360, 7889356184, 44981654360, 256465696536, 1462255100120, 8337138286744, 47534711834200, 271022112317976, 1545249408926680, 8810335493905304, 50232675105233240
Offset: 1
Cf.
A000045,
A000079,
A105476,
A159612,
A080040,
A135522,
A103435,
A189732,
A189734,
A189735,
A189736,
A189737,
A189738,
A189739,
A189741,
A189742,
A189743,
A189744,
A189745,
A189746,
A189747,
A189749.
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LinearRecurrence[{5,4},{5,4},40]
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a[1]:5$ a[2]:4$ a[n]:=5*a[n-1]+4*a[n-2]$ makelist(a[n], n, 1, 23); /* Bruno Berselli, May 24 2011 */
A206776
a(n) = 3*a(n-1) + 2*a(n-2) for n>1, a(0)=2, a(1)=3.
Original entry on oeis.org
2, 3, 13, 45, 161, 573, 2041, 7269, 25889, 92205, 328393, 1169589, 4165553, 14835837, 52838617, 188187525, 670239809, 2387094477, 8501763049, 30279478101, 107841960401, 384084837405, 1367938433017, 4871984973861, 17351831787617, 61799465310573
Offset: 0
G.f. = 2 + 3*x + 13*x^2 + 45*x^3 + 161*x^4 + 573*x^5 + 2041*x^6 + 7269*x^7 + ...
- Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994. Exercise 7.49(c), pages 379, 573.
Cf.
A189736 (same recurrence but with initial values reversed).
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[n le 1 select n+2 else 3*Self(n)+2*Self(n-1): n in [0..25]];
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A206776 := proc(n)
option remember ;
if n <= 1 then
n+2 ;
else
3*procname(n-1)+2*procname(n-2) ;
end if;
end proc:
seq(A206776(n),n=0..30) ; # R. J. Mathar, Feb 18 2024
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RecurrenceTable[{a[n] == 3 a[n - 1] + 2 a[n - 2], a[0] == 2, a[1] == 3}, a[n], {n, 25}]
LinearRecurrence[{3,2},{2,3},30] (* Harvey P. Dale, Apr 29 2014 *)
a[ n_] := If[ n < 0, (-2)^n a[ -n], ((3 + Sqrt[17])/2)^n + ((3 - Sqrt[17])/2)^n // Expand]; (* Michael Somos, Oct 13 2016 *)
a[ n_] := If[ n < 0, (-2)^n a[ -n], Boole[n == 0] + SeriesCoefficient[ ((1 + 3*x + Sqrt[1 + 6*x + 17*x^2])/2)^n, {x, 0, n}]]; (* Michael Somos, Oct 13 2016 *)
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a[0]:2$ a[1]:3$ a[n]:=3*a[n-1]+2*a[n-2]$ makelist(a[n], n, 0, 25);
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Vec((2-3*x)/(1-3*x-2*x^2) + O(x^30)) \\ Michel Marcus, Jun 26 2015
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{a(n) = 2 * real(( (3 + quadgen(68)) / 2 )^n)}; /* Michael Somos, Oct 13 2016 */
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{a(n) = my(w = quadgen(-8)); simplify(w^n * subst(2 * polchebyshev(n), x, -3/4*w))}; /* Michael Somos, Oct 13 2016 */
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for(n=0,25,print1(round(((3+sqrt(17))/2)^n+((3-sqrt(17))/2)^n),", ")) \\ Hugo Pfoertner, Nov 19 2018
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