A288511 a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - 4*a(n-8) + 4*a(n-9) - 3*a(n-10) + 2*a(n-11) - 3*a(n-12) + 2*a(n-13) for n >= 16, with initial values as shown.

2, 4, 5, 7, 9, 12, 17, 23, 33, 48, 70, 103, 152, 228, 343, 515, 779, 1180, 1787, 2715, 4124, 6264, 9526, 14483, 22025, 33504, 50957, 77519, 117929, 179396, 272930, 415215, 631680, 961032, 1462067, 2224347, 3384083, 5148432, 7832727, 11916547, 18129540

Offset

0,1

Comments

Conjecture: a(n) is the number of letters (0's and 1's) in the n-th iteration of the mapping 00->0101, 10->001, starting with 00; see A288508.

Links

Clark Kimberling, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (2, -1, 2, -4, 4, -4, 4, -4, 4, -3, 2, -3, 2).

Formula

G.f.: x*(2 - x^2 - 3*x^3 - x^5 - x^7 - x^9 - x^10 - 3*x^11 + x^12 + 2*x^13 + x^14- 2*x^15) / ((1 - x)^2*(1 - x^2 - 2*x^3)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Jun 12 2017

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - 4*a(n-8) + 4*a(n-9) - 3*a(n-10) + 2*a(n-11) - 3*a(n-12) + 2*a(n-13) for n >= 16, where a(0)=2, a(1)=4, a(2)=5, a(3)=7, a(4)=9, a(5)=12, a(6)=17, a(7)=23, a(8)=33, a(9)=48, a(10)=70, a(11)=103, a(12)=152, a(13)=228, a(14)=343, a(15)=515.

Mathematica

Join[{2, 4, 5}, LinearRecurrence[{2, -1, 2, -4, 4, -4, 4, -4, 4, -3, 2, -3, 2}, {7, 9, 12, 17, 23, 33, 48, 70, 103, 152, 228, 343, 515}, 40]]

Prog

(PARI) Vec(x*(2 - x^2 - 3*x^3 - x^5 - x^7 - x^9 - x^10 - 3*x^11 + x^12 + 2*x^13 + x^14- 2*x^15) / ((1 - x)^2*(1 - x^2 - 2*x^3)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Jun 12 2017

Crossrefs

Cf. A288508.

Sequence in context: A275534 A069355 A212661 * A249058 A105771 A193494

Adjacent sequences: A288508 A288509 A288510 * A288512 A288513 A288514

Keyword

nonn,easy

Author

Clark Kimberling, Jun 12 2017

Status

approved