A295619 a(n) = a(n-1) + 3*a(n-2) - 2*a(n-3) - 2*a(n-4), where a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4.

1, 2, 3, 4, 7, 9, 16, 21, 37, 50, 87, 121, 208, 297, 505, 738, 1243, 1853, 3096, 4693, 7789, 11970, 19759, 30705, 50464, 79121, 129585, 204610, 334195, 530613, 864808, 1379037, 2243845, 3590114, 5833959, 9358537, 15192496, 24419961, 39612457, 63770274

Offset

0,2

Comments

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = a(n-1) + 3*a(n-2) - 2*a(n-3) - 2*a(n-4), where a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4.

G.f.: (1 + x - 2 x^2 - 3 x^3)/(1 - x - 3 x^2 + 2 x^3 + 2 x^4).

Mathematica

LinearRecurrence[{1, 3, -2, -2}, {1, 2, 3, 4}, 50]
nxt[{a_, b_, c_, d_}]:={b, c, d, d+3c-2b-2a}; NestList[nxt, {1, 2, 3, 4}, 40][[;; , 1]] (* Harvey P. Dale, Jan 16 2026 *)

Crossrefs

Cf. A001622, A000045, A295620, A295621.

Sequence in context: A320357 A250255 A070763 * A319645 A071114 A078696

Adjacent sequences: A295616 A295617 A295618 * A295620 A295621 A295622

Keyword

nonn,easy

Author

Clark Kimberling, Nov 25 2017

Status

approved