This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A308217 #78 Dec 06 2024 16:30:03
%S A308217 1,8,2,23,3,46,4,77,5,116,6,163,7,218,8,281,9,352,10,431,11,518,12,
%T A308217 613,13,716,14,827,15,946,16,1073,17,1208,18,1351,19,1502,20,1661,21,
%U A308217 1828,22,2003,23,2186,24,2377,25,2576,26,2783,27,2998,28,3221,29,3452
%N A308217 a(n) is the multiplicative inverse of A001844(n) modulo A001844(n+1); where A001844 is the sequence of centered square numbers.
%C A308217 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.
%H A308217 Robert Israel, <a href="/A308217/b308217.txt">Table of n, a(n) for n = 0..10000</a>
%H A308217 Daniel Hoyt, <a href="/A308217/a308217_3.png">Graph of A308215 and A308217 in relation to A001844</a>
%H A308217 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A308217 a(n) satisfies a(n)*(2*n*(n-1)+1) == 1 (mod 2*n*(n+1)+1).
%F A308217 Conjectures from _Colin Barker_, May 16 2019: (Start)
%F A308217 G.f.: (1 + 8*x - x^2 - x^3 + x^5) / ((1 - x)^3*(1 + x)^3).
%F A308217 a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
%F A308217 a(n) = (9 - 5*(-1)^n + (8-6*(-1)^n)*n - 2*(-1+(-1)^n)*n^2) / 4. (End)
%F A308217 From _Robert Israel_, Aug 11 2019: (Start)
%F A308217 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).
%F A308217 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).
%F A308217 Colin Barker's conjectures easily follow. (End)
%F A308217 E.g.f.: ((2 + 9*x)*cosh(x) + (7 + x + 2*x^2)*sinh(x))/2. - _Stefano Spezia_, Dec 06 2024
%p A308217 A001844:= n -> 2*n*(n+1)+1:
%p A308217 seq(1/A001844(n) mod A001844(n+1),n=0..100); # _Robert Israel_, Aug 11 2019
%t A308217 LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 8, 2, 23, 3, 46}, 30] (* _Georg Fischer_, Dec 06 2024 *)
%o A308217 (PARI) f(n) = 2*n*(n+1)+1; \\ A001844
%o A308217 a(n) = lift(1/Mod(f(n), f(n+1))); \\ _Michel Marcus_, May 16 2019
%Y A308217 Cf. A001844, A054552, A033951, A308215.
%K A308217 nonn,easy
%O A308217 0,2
%A A308217 _Daniel Hoyt_, May 15 2019