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 A165188 #31 Sep 08 2022 08:45:47
%S A165188 1,0,3,1,6,3,11,6,18,11,27,18,39,27,54,39,72,54,94,72,120,94,150,120,
%T A165188 185,150,225,185,270,225,321,270,378,321,441,378,511,441,588,511,672,
%U A165188 588,764,672,864,764,972,864,1089,972,1215,1089,1350,1215,1495,1350
%N A165188 Interleaving of A014125 and zero followed by A014125.
%C A165188 This sequence convolved with A000217 (without initial term 0) yields A164680.
%C A165188 See A164680 for similar convolutions.
%C A165188 A165188 convolved with A000217 yields sequence A164680. This is to be expected since A000217 can be associated with partition 1+1+1, A164680 with partition 1+1+1+2+2+2+3 and A165188 with partition 2+2+2+3 by observing their unreduced generating functions and verified by generating the sequences by converting the partitions into finite sequences and using Euler's Transform. Thus partition 1+1+1 yields the finite sequence (3); partition 2+2+2+3 yields the finite sequence (0,3,1); and, when combined, partition 1+1+1+2+2+2+3 yields (3,3,1). - _Alford Arnold_, Sep 24 2009
%H A165188 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,1,-3,-3,1,3,0,-1).
%F A165188 a(n) = -a(n-1)+2*a(n-2)+3*a(n-3)-3*a(n-5)-2*a(n-6)+a(n-7)+a(n-8)+1 for n > 8; a(1)=1, a(2)=0, a(3)=3, a(4)=1, a(5)=6, a(6)=3, a(7)=11, a(8)=6. - _Klaus Brockhaus_, Sep 15 2009
%F A165188 G.f.: x/((1-x)^4*(1+x)^3*(1+x+x^2)). - _Klaus Brockhaus_, Sep 15 2009
%F A165188 a(n) = (2*n^3+21*n^2+63*n+49)/288-(-1)^n*(9+7*n+n^2)/32+A057078(n)/9. - _R. J. Mathar_, Sep 17 2009
%F A165188 Euler transform of length 3 sequence [ 0, 3, 1]. - _Michael Somos_, Feb 01 2015
%F A165188 G.f.: x / ((1 - x^2)^3 * (1 - x^3)). - _Michael Somos_, Feb 01 2015
%F A165188 a(n) = -a(-7 - n) for all n in Z.
%F A165188 a(n+3) - a(n) = 0 if n odd else (n+6) * (n+4) / 8. - _Michael Somos_, Feb 01 2015
%F A165188 a(2*n + 1) = a(2*n + 4) = A014125(n) for all n in Z. - _Michael Somos_, Feb 01 2015
%e A165188 A014125 begins 1,3,6,11,18,27,..., thus this sequence begins 1,0,3,1,6,3,11,6,18,11,27,18,... .
%e A165188 G.f. = x + 3*x^3 + x^4 + 6*x^5 + 3*x^6 + 11*x^7 + 6*x^8 + 18*x^9 + 11*x^10 + ...
%t A165188 a[ n_] := Module[{s = 1, m = n}, If[ n < 0, s = -1; m = -7 - n]; s SeriesCoefficient[ x / ((1 - x^2)^3 (1 - x^3)), {x, 0, m}]]; (* _Michael Somos_, Feb 01 2015 *)
%t A165188 LinearRecurrence[{0,3,1,-3,-3,1,3,0,-1},{1,0,3,1,6,3,11,6,18},60] (* _Harvey P. Dale_, Apr 14 2018 *)
%o A165188 (PARI) /* first computes u = A014125 as second bisection of A001400, then interleaves */ {m=28; u=vector(m, n, polcoeff(1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))+O(x^(2*n)), 2*n-1)); vector(2*m, k, if(k%2==1,u[(k+1)/2], if(k==2, 0, u[k/2-1])))} /* _Klaus Brockhaus_, Sep 15 2009 */
%o A165188 (PARI) {a(n) = my(s=1); if( n<0, s=-1; n=-7-n); s * polcoeff( x / ((1 - x^2)^3 * (1 - x^3)) + x * O(x^n), n)}; /* _Michael Somos_, Feb 01 2015 */
%o A165188 (Magma) I:=[1,0,3,1,6,3,11,6]; [n le 8 select I[n] else -Self(n-1)+2*Self(n-2)+3*Self(n-3)-3*Self(n-5)-2*Self(n-6)+Self(n-7)+Self(n-8)+1: n in [1..60]]; // _Vincenzo Librandi_, Jun 24 2015
%Y A165188 Cf. A014125, A000217, A164680, A001400.
%K A165188 nonn
%O A165188 1,3
%A A165188 _Alford Arnold_, Sep 13 2009
%E A165188 Edited and extended by _Klaus Brockhaus_, Sep 15 2009
%E A165188 Linear recurrence link and signature correct by _Michel Marcus_, Jun 25 2015