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%I #21 Jan 05 2025 19:51:41
%S 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,
%T 97,104,113,120,129,137,146,156,165,176,185,196,206,217,229,240,253,
%U 264,277,289,302,316,329,344,357,372,386,401,417,432,449,464,481
%N 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.
%C 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.
%H C. Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,1,-2,1).
%F 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)).
%F a(n) = a(-n) for all n in Z.
%F 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.
%e 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 + ...
%e 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.
%t a[ n_] := (n^2 - {8, 4, 9, 9, 4, 8, 0}[[Mod[n, 7, 1]]]) / 7 + 1;
%t 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 *)
%o (PARI) {a(n) = (n^2 - [0, 8, 4, 9, 9, 4, 8][n%7+1]) / 7 + 1};
%K nonn,easy,changed
%O 0,5
%A _Michael Somos_, Nov 04 2018