cp's OEIS Frontend

A195241 Expansion of (1-x+19*x^3-3*x^4)/(1-x)^3.

%I A195241 #48 Jul 09 2025 04:33:27
%S A195241 1,2,3,23,59,111,179,263,363,479,611,759,923,1103,1299,1511,1739,1983,
%T A195241 2243,2519,2811,3119,3443,3783,4139,4511,4899,5303,5723,6159,6611,
%U A195241 7079,7563,8063,8579,9111,9659,10223,10803,11399,12011,12639,13283,13943
%N A195241 Expansion of (1-x+19*x^3-3*x^4)/(1-x)^3.
%C A195241 Sequence found by reading the line 1, 2, 3, 23,.. in the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.
%C A195241 This is a subsequence of A110326 (without signs) and  A047838 (apart from the second term, 2).
%H A195241 Bruno Berselli, <a href="/A195241/b195241.txt">Table of n, a(n) for n = 0..1000</a>
%H A195241 Bruno Berselli, <a href="http://www.base5forum.it/upload/A195241.jpg">Illustration of initial terms: An origin of A195241</a>.
%H A195241 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A195241 G.f.: (1-x+19*x^3-3*x^4)/(1-x)^3.
%F A195241 a(n) = 8*n^2-20*n+11 for n>1; a(0)=1, a(1)=2.
%t A195241 CoefficientList[Series[(1 - x + 19 x^3 - 3 x^4)/(1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 26 2013 *)
%t A195241 LinearRecurrence[{3,-3,1},{1,2,3,23,59},50] (* _Harvey P. Dale_, Dec 04 2022 *)
%o A195241 (PARI) Vec((1-x+19*x^3-3*x^4)/(1-x)^3+O(x^44))
%o A195241 (Magma) m:=44; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+19*x^3-3*x^4)/(1-x)^3));
%o A195241 (Maxima) makelist(coeff(taylor((1-x+19*x^3-3*x^4)/(1-x)^3, x, 0, n), x, n), n, 0, 43);
%Y A195241 Cf. A000217.
%Y A195241 Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195605 [incomplete list].
%K A195241 nonn,easy
%O A195241 0,2
%A A195241 _Bruno Berselli_, Sep 13 2011 - based on remarks and sequences by _Omar E. Pol_