A321324 a(n) = (n^2 - c(n)) / 7 + 1 where c(n) = c(-n) = c(n+7) for all n in Z and a(n) = 1 if 0 <= n <=3 except a(1) = 0.

1, 0, 1, 1, 2, 4, 5, 8, 9, 12, 14, 17, 21, 24, 29, 32, 37, 41, 46, 52, 57, 64, 69, 76, 82, 89, 97, 104, 113, 120, 129, 137, 146, 156, 165, 176, 185, 196, 206, 217, 229, 240, 253, 264, 277, 289, 302, 316, 329, 344, 357, 372, 386, 401, 417, 432, 449, 464, 481

Offset

0,5

Comments

Let t(n) be a strong elliptic divisibility sequence as given in [Kimberling, p. 16] where x = y = z = w := (1 + sqrt(5))/2. Then, t(n) = (-1)^floor(n/7) * w^a(n) except t(7*k) = 0. Since t(n) is a generalized Somos-4 sequence, it satisfies t(n+2)*t(n-2) = w*w*t(n+1)*t(n-1) - w*t(n)*t(n) and t(n+3)*t(n-2) = w*t(n+2)*t(n-1) - w*t(n+1)*t(n) for all n in Z.

Links

Table of n, a(n) for n=0..58.

C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Formula

G.f.: 1/(1 - x) - x*(1 + x^3)*(1 - 2*x + x^2 - 2*x^3 + x^4)/((1 - x)^2*(1 - x^7)).

a(n) = a(-n) for all n in Z.

a(n) - a(n+1) - a(n+2) + a(n+3) = 0 if n = 7*k or 7*k+4, 2 if n = 7*k+2, else 0.

Example

G.f. = 1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 8*x^7 + 9*x^8 + 12*x^9 + ...
t(0) = 0, t(1) = 1, t(2) = t(3) = w, t(4) = 1 + w = w^2, t(5) = 2 + 3*w = w^4, t(6) = 3 + 5*w = w^5, t(7) = 0.

Mathematica

a[ n_] := (n^2 - {8, 4, 9, 9, 4, 8, 0}[[Mod[n, 7, 1]]]) / 7 + 1;
LinearRecurrence[{2, -1, 0, 0, 0, 0, 1, -2, 1}, {1, 0, 1, 1, 2, 4, 5, 8, 9}, 60] (* Harvey P. Dale, Jan 18 2023 *)

Prog

(PARI) {a(n) = (n^2 - [0, 8, 4, 9, 9, 4, 8][n%7+1]) / 7 + 1};

Crossrefs

Sequence in context: A339816 A002541 A239953 * A343013 A259558 A352778

Adjacent sequences: A321321 A321322 A321323 * A321325 A321326 A321327

Keyword

nonn,easy,changed

Author

Michael Somos, Nov 04 2018

Status

approved