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 A257792 #23 Sep 08 2022 08:46:12
%S A257792 1,1,2,4,7,14,26,49,92,174,328,618,1166,2197,4143,7811,14726,27764,
%T A257792 52344,98687,186058,350784,661347,1246865,2350768,4432000,8355837,
%U A257792 15753609,29700940,55996428,105572414,199040101,375258649,707490872,1333862213,2514786376
%N A257792 Expansion of 1/(1-x-x^2-x^3-x^5+x^8-x^9).
%C A257792 This sequence counts partially ordered partitions of (n) in two distinct ways. It partitions (n) into parts containing (1,2,3,5,9) where the adjacent order of 3's and 5's are unimportant, example (1), and it partitions (n) into parts containing (1,2,3,4,5,6) where the adjacent order of the odd numbers is unimportant, example (2). The sign "=" is used within a bracket to indicate that the arrangements are counted as one.
%H A257792 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H A257792 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,0,1,0,0,-1,1).
%F A257792 G.f.: 1/(1-x-x^2-x^3-x^5+x^8-x^9).
%F A257792 a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-5) - a(n-8) + a(n-9).
%e A257792 Example (1):Partial order of (n) into parts (1,2,3,5,9) where the adjacent order of 3's and 5's is unimportant. a(8)=92 These are (53=35)=1,(521)=6,(5111)=4,(332)=3,(3311)=6,(3221)=12,(32111)=20,(311111)=6,(2222)=1,(22211)=10,(221111)=15,(2111111)=7,(11111111)=1.
%e A257792 Example (2):Partial order of (n) into parts (1,2,3,4,5,6) where the adjacent order of all odd numbers (i.e. 1,3,5) is unimportant. a(6)=26 These are (6),(51=15),(42),(24),(411),(141),(114),(33),(321),(123),(231=213),(312=132),(3111=1311=1131=1113),(222),(2211),(1122),(1221),(2112),(2121),(1212),(21111),(12111),(11211),(11121),(11112),(111111).
%t A257792 CoefficientList[Series[1/(1 - x - x^2 - x^3 - x^5 + x^8 - x^9), {x, 0, 80}], x] (* _Vincenzo Librandi_, May 09 2015 *)
%t A257792 LinearRecurrence[{1,1,1,0,1,0,0,-1,1},{1,1,2,4,7,14,26,49,92},36] (* _Ray Chandler_, Jul 14 2015 *)
%o A257792 (Magma) I:=[1,1,2,4,7,14,26,49,92]; [n le 9 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)+Self(n-5)-Self(n-8)+Self(n-9): n in [1..40]]; // _Vincenzo Librandi_, May 09 2015
%o A257792 (Sage) m = 40; L.<x> = PowerSeriesRing(ZZ, m); f = 1/(1-x-x^2-x^3-x^5+x^8-x^9); print(f.coefficients()) # _Bruno Berselli_, May 12 2015
%K A257792 nonn,easy
%O A257792 0,3
%A A257792 _David Neil McGrath_, May 08 2015
%E A257792 More terms from _Vincenzo Librandi_, May 09 2015