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 A032199 #41 Jan 01 2025 23:31:47
%S A032199 3,11,33,114,397,1524,5873,23492,95033,390262,1617513,6764338,
%T A032199 28478813,120631428,513569941,2196269672,9429031893,40621573968,
%U A032199 175544747673,760719000950,3304818106413,14389871196680
%N A032199 "CIK" (necklace, indistinct, unlabeled) transform of 3,5,7...
%C A032199 The auxiliary sequence (c(n): n>=1) that appears in the formula section below can be determined via its g.f. C(x) = Sum_{n>=1} c(n)*x^n. We have C(x) = x*(dB(x)/dx)/(1-B(x)), where B(x) = 3*x + 5*x^2 + 7*x^3 + 9*x^4 + ... = (3*x - x^2)/(1-x)^2. We get C(x) = (3*x + x^2)/((1-x)*(2*x^2 - 5*x + 1)) = -2*x/(1-x) + (5*x - 4*x^2)/(2*x^2 - 5*x +1). The second term of the last equation is the g.f. of the sequence (A159289(n-1): n>=1). - _Petros Hadjicostas_, Jan 06 2018
%H A032199 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H A032199 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%F A032199 From _Petros Hadjicostas_, Jan 06 2018: (Start)
%F A032199 a(n) = (1/n)*Sum_{d|n} phi(d)*c(n/d), where c(n) = -2 + A159289(n-1) = -2 + r1^n + r2^n with r1 = (5 - sqrt(17))/2 and r2 = (5 + sqrt(17))/2 (for n >= 1). (Notice the different offsets between this sequence and sequence A159289.)
%F A032199 a(n) = -2 + (1/n)*Sum_{d|n} phi(n/d)*A159289(d-1).
%F A032199 G.f.: -Sum_{n >= 1} (phi(n)/n)*log(1-B(x^n)), where B(x) = 3*x + 5*x^2 + 7*x^3 + 9*x^4 + ... = (3*x-x^2)/(1-x)^2.
%F A032199 G.f.: -2*x/(1-x) - Sum_{n>=1} (phi(n)/n)*log(1-5*x^n+2*x^(2*n)).
%F A032199 (End)
%e A032199 From _Petros Hadjicostas_, Jan 06 2018: (Start)
%e A032199 We give examples to illustrate the meaning of the CIK transform, whose theory is outlined by C. G. Bower in the weblink above. We are forced, however, to switch the roles of sequences (a(n): n>=1) and (b(n): n>=1) in the weblink above about transforms.
%e A032199 We assume we have boxes of different sizes and colors. Here, b(n) = number of colors a box holding n balls can be, while a(n) = number of ways we can have a collection of boxes so that the total number of balls is n. We have (a(n): n>=1) = CIK((b(n): n>=1)).
%e A032199 Since b(1) = 3, b(2) = 5, b(3) = 7, etc., a box holding 1 ball can be one of 3 colors, a box holding 2 balls can be one of 5 colors, a box holding 3 balls can be one of 7 colors, and so on.
%e A032199 The meaning of the CIK transform entails that boxes of the same size and color are indistict and unlabeled, while they form a necklace (on a circle). (The boxes differ only in their size—how many balls they can hold—and in their color.)
%e A032199 We have a(1) = 3 because we have 1 ball in one box (that can hold 1 ball) and the box can be of one of 3 colors.
%e A032199 To prove that a(2) = 11, we consider two cases. In the first case, we have a single box that can hold 2 balls. This box can be one of 5 colors. In the second case, we have 2 identical boxes each holding 1 ball. Each such box can be one of 3 colors, say a, b, and c. We have 6 possibilities on a circle: aa, ab, ac, bb, bc, cc. Hence, a(2) = 5 + 6 = 11.
%e A032199 To prove that a(3) = 33, we consider three cases giving a total of 7 + 15 + 11 = 33 possibilities. In the first case, we have 1 box holding 3 balls. This case gives rise to 7 possibilities. In the second case, we have 2 boxes, one that can hold 2 balls and one that can hold 1 ball. In this case, we have 5 x 3 = 15 possibilities (even on a circle). In the third case, we have three identical boxes on a circle, each holding 1 ball. If the colors are a, b, and c, we have 11 possibilities on a circle: aaa, aab, aac, abb, abc, acb, acc, bbb, bbc, bcc, and ccc.
%e A032199 (End)
%Y A032199 Cf. A159289.
%K A032199 nonn
%O A032199 1,1
%A A032199 _Christian G. Bower_
%E A032199 Name edited by _Petros Hadjicostas_, Jan 06 2018