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 A308217 a(n) is the multiplicative inverse of A001844(n) modulo A001844(n+1); where A001844 is the sequence of centered square numbers. 4
 1, 8, 2, 23, 3, 46, 4, 77, 5, 116, 6, 163, 7, 218, 8, 281, 9, 352, 10, 431, 11, 518, 12, 613, 13, 716, 14, 827, 15, 946, 16, 1073, 17, 1208, 18, 1351, 19, 1502, 20, 1661, 21, 1828, 22, 2003, 23, 2186, 24, 2377, 25, 2576, 26, 2783, 27, 2998, 28, 3221, 29, 3452 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The sequence explores the relationship between the terms of A001844, the sums of consecutive squares. The sequence is an interleaving of A033951 (a number spiral arm) and the natural numbers. The gap between the lower values of A308215 and the upper values of A308217 increase by 3n; each successive gap increasing by 6. LINKS Robert Israel, Table of n, a(n) for n = 0..10000 Daniel Hoyt, Graph of A308215 and A308217 in relation to A001844 FORMULA a(n) satisfies a(n)*(2*n*(n-1)+1) == 1 (mod 2*n*(n+1)+1). Conjectures from Colin Barker, May 16 2019: (Start) G.f.: (1 + 8*x - x^2 - x^3 + x^5) / ((1 - x)^3*(1 + x)^3). a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5. a(n) = (9 - 5*(-1)^n + (8-6*(-1)^n)*n - 2*(-1+(-1)^n)*n^2) / 4. (End) From Robert Israel, Aug 11 2019: (Start) a(n) = 1 + n/2 if n is even, since 0 < 1+n/2 < A001844(n+1) and (1+n/2)*A001844(n)-1 = (n/2)*A001844(n+1). a(n) = n^2 + 7/2*(n+1) if n is odd, since 0 < n^2+7/2*(n+1) < A001844(n+1) and (n^2+7/2*(n+1))*A001844(n)-1 = (n^2+3*k/2+1/2)*A001844(n+1). Colin Barker's conjectures easily follow. (End) MAPLE A001844:= n -> 2*n*(n+1)+1: seq(1/A001844(n) mod A001844(n+1), n=0..100); # Robert Israel, Aug 11 2019 PROG (PARI) f(n) = 2*n*(n+1)+1; \\ A001844 a(n) = lift(1/Mod(f(n), f(n+1))); \\ Michel Marcus, May 16 2019 CROSSREFS Cf. A001844, A054552, A033951, A308215. Sequence in context: A161593 A008866 A180728 * A248297 A340008 A006708 Adjacent sequences:  A308214 A308215 A308216 * A308218 A308219 A308220 KEYWORD nonn AUTHOR Daniel Hoyt, May 15 2019 STATUS approved

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Last modified October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)