A123102 -id:A123102 - cp's OEIS frontend

A247594 a(n) = a(n-1) + a(n-2) + 3*a(n-3) with a(0) = 1, a(1) = 2, a(2) = 5.

1, 2, 5, 10, 21, 46, 97, 206, 441, 938, 1997, 4258, 9069, 19318, 41161, 87686, 186801, 397970, 847829, 1806202, 3847941, 8197630, 17464177, 37205630, 79262697, 168860858, 359740445, 766389394, 1632712413, 3478323142, 7410203737, 15786664118, 33631837281

Offset: 0

Michael Somos, Sep 20 2014

a(n) is the number of words of length n in {A,B,C} such that no two consecutive letters are B and every letter C is adjacent to exactly one letter B.

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 21*x^4 + 46*x^5 + 97*x^6 + 206*x^7 + ...
a(3) = 10 with words [AAA, AAB, ABA, ABC, ACB, BAA, BAB, BCA, CBA, CBC].

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
David Beckwith, Problem 11776, The American Mathematical Monthly, 121 (2014), 455. See solution, 123 (May, 2016), 508-510.
Roberto Tauraso, Solution of Problem 11776.
Index entries for linear recurrences with constant coefficients, signature (1,1,3).

Cf. A103143, A123102.

a247594 n = a247594_list !! n
a247594_list = 1 : 2 : 5 : zipWith (+)
   (tail $ zipWith (+) a247594_list $ tail a247594_list)
   (map (* 3) a247594_list)
-- Reinhard Zumkeller, Sep 21 2014

I:=[1,2,5]; [n le 3 select I[n] else Self(n-1)+Self(n-2)+3*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 13 2015

LinearRecurrence[{1, 1, 3}, {1, 2, 5}, 40] (* Vincenzo Librandi, Aug 13 2015 *)

{a(n) = if( n<0, polcoeff( (2*x + x^2 + x^3) / (3 + x + x^2 - x^3) + x * O(x^-n), -n), polcoeff( (1 + x + 2*x^2) / (1 - x - x^2 - 3*x^3) + x * O(x^n), n))};

first(m)={my(v=vector(m));v[1]=1;v[2]=2;v[3]=5;for(i=4,m,v[i]=v[i-1]+v[i-2]+3*v[i-3]);v;} /* Anders Hellström, Aug 12 2015 */

G.f.: (1 + x + 2*x^2) / (1 - x - x^2 - 3*x^3).
0 = a(n) - a(n-1) - a(n-2) - 3*a(n-3) for all n in Z.
From Greg Dresden, Aug 05 2022: (Start)
a(n) = b(n+3) - b(n) for b(n) = A103143(n).
a(n) = c(n+2) - 2*c(n-1) for c(n) = A123102(n). (End)

A356411 Sum of powers of roots of x^3 - x^2 - x - 3.

Original entry on oeis.org

3, 1, 3, 13, 19, 41, 99, 197, 419, 913, 1923, 4093, 8755, 18617, 39651, 84533, 180035, 383521, 817155, 1740781, 3708499, 7900745, 16831587, 35857829, 76391651, 162744241, 346709379, 738628573, 1573570675, 3352327385, 7141783779

Offset: 0

Greg Dresden, Aug 05 2022

a(n) is the sum of the n-th powers of the three roots of x^3 - x^2 - x - 3. These roots are c1 = 2.130395..., c2 = -0.5651977... - i*1.0434274..., and c3 = -0.5651977... + i*1.0434274..., and so a(n) = c1^n + c2^n + c3^n. The real parts of c2 and c3 are A273065.

a(n) can also be determined by Vieta's formulas and Newton's identities. For example, a(3) by definition is c1^3 + c2^3 + c3^3, and from Newton's identities this equals e1^3 - 3*e1*e2 + 3*e3 for e1, e2, e3 the elementary symmetric polynomials of x^3 - x^2 - x - 3. From Vieta's formulas we have e1 = 1, e2 = -1, and e3 = 3, giving us e1^3 - 3*e1*e2 + 3*e3 = 1 + 3 + 9 = 13, as expected.

			For n=3, a(3) = (2.130395...)^3 + (-0.5651977... - i*1.0434274...)^3 + (-0.5651977... + i*1.0434274...)^3 = 13.

Index entries for linear recurrences with constant coefficients, signature (1,1,3).

Cf. A103143, A123102, A247594, A356463, A273065 (Re c2,c3).

LinearRecurrence[{1, 1, 3}, {3, 1, 3}, 40]

polsym(x^3 - x^2 - x - 3, 35) \\ Joerg Arndt, Aug 11 2022

a(n) = a(n-1) + a(n-2) + 3*a(n-3) with a(0)=3, a(1)=1, a(2) = 3.

G.f.: (3 - 2*x - x^2)/(1 - x - x^2 - 3*x^3).

cp's OEIS Frontend

A247594 a(n) = a(n-1) + a(n-2) + 3*a(n-3) with a(0) = 1, a(1) = 2, a(2) = 5.

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A356411 Sum of powers of roots of x^3 - x^2 - x - 3.

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