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 A116732 #32 Jul 26 2025 14:12:50
%S A116732 0,0,0,1,1,2,4,6,11,19,32,56,96,165,285,490,844,1454,2503,4311,7424,
%T A116732 12784,22016,37913,65289,112434,193620,333430,574195,988811,1702816,
%U A116732 2932392,5049824,8696221,14975621,25789274,44411292,76479966,131704911,226806895
%N A116732 a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) with a(0) = a(1) = a(2) = 0, a(3) = 1.
%C A116732 This sequence is an example of a "symmetric" quartic recurrence and has some expected divisibility properties.
%C A116732 a(n-3) counts partially ordered partitions of (n-3) into parts 1,2,3 where only the order of the adjacent 1's and 3's are unimportant (see example). - _David Neil McGrath_, Jul 25 2015
%H A116732 Michael De Vlieger, <a href="/A116732/b116732.txt">Table of n, a(n) for n = 0..4239</a>
%H A116732 Jarib R. Acosta, Yadira Caicedo, Juan P. Poveda, José L. Ramírez, and Mark Shattuck, <a href="https://www.emis.de/journals/JIS/VOL22/Shattuck/shattuck13.html">Some New Restricted n-Color Composition Functions</a>, J. Int. Seq., Vol. 22 (2019), Article 19.6.4.
%H A116732 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1).
%F A116732 G.f.: x^3/(x^4 - x^3 - x^2 - x + 1).
%F A116732 a(n) = -a(2-n) for all n in Z. - _Michael Somos_, Jul 25 2025
%e A116732 Partially ordered partitions of (n-3) into parts 1,2,3 where only the order of adjacent 1's and 3's are unimportant. E.g., a(n-3)=a(6)=19. These are (33),(321),(312),(231),(123),(132),(3111),(2211),(1122),(1221),(2112),(2121),(1212),(21111),(12111),(11211),(11121),(11112),(111111). - _David Neil McGrath_, Jul 25 2015
%e A116732 G.f. = x^3 + x^4 + 2*x^5 + 4*x^6 + 6*x^7 + 11*x^8 + 19*x^9 + 32*x^10 + ... - _Michael Somos_, Jul 25 2025
%t A116732 LinearRecurrence[{1, 1, 1, -1}, {0, 0, 0, 1}, 40] (* _Vladimir Joseph Stephan Orlovsky_, Feb 02 2012 *)
%t A116732 CoefficientList[Series[x^3/(1-x-x^2-x^3+x^4),{x,0,40}],x] (* _Harvey P. Dale_, Mar 25 2018 *)
%o A116732 (PARI) v=[0,0,0,1];for(i=1,40,v=concat(v,v[#v]+v[#v-1]+v[#v-2]-v[#v-3]));v \\ _Derek Orr_, Aug 27 2015
%o A116732 (PARI) {a(n) = if(n<0, -a(2-n), polcoeff(x^3/(1 - x - x^2 - x^3 + x^4 + x*O(x^n)), n))} /* _Michael Somos_, Jul 25 2025 */
%Y A116732 Close to A000786 (& A048239), A115992, A115993. Cf. A116201.
%K A116732 nonn,easy
%O A116732 0,6
%A A116732 _R. K. Guy_, Mar 23 2008
%E A116732 More terms from _Max Alekseyev_, Mar 23 2008
%E A116732 Name clarified by _Michael Somos_, Jul 25 2025