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 A195605 #44 Jul 09 2025 04:33:33
%S A195605 2,7,18,31,50,71,98,127,162,199,242,287,338,391,450,511,578,647,722,
%T A195605 799,882,967,1058,1151,1250,1351,1458,1567,1682,1799,1922,2047,2178,
%U A195605 2311,2450,2591,2738,2887,3042,3199,3362,3527,3698,3871,4050,4231,4418,4607,4802
%N A195605 a(n) = (4*n*(n+2)+(-1)^n+1)/2 + 1.
%C A195605 Sequence found by reading the numbers in increasing order on the vertical line containing 2 of the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.
%C A195605 Also A077591 (without first term) and A157914 interleaved.
%H A195605 Bruno Berselli, <a href="/A195605/b195605.txt">Table of n, a(n) for n = 0..1000</a>
%H A195605 Bruno Berselli, <a href="http://www.base5forum.it/upload/A195605.jpg">Illustration of initial terms: An origin of A195605</a>.
%H A195605 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A195605 G.f.: (2+3*x+4*x^2-x^3)/((1+x)*(1-x)^3).
%F A195605 a(n) = a(-n-2) = 2*a(n-1)-2*a(n-3)+a(n-4).
%F A195605 a(n) = A047524(A000982(n+1)).
%F A195605 Sum_{n>=0} 1/a(n) = 1/2 + Pi^2/16 - cot(Pi/(2*sqrt(2)))*Pi/(4*sqrt(2)). - _Amiram Eldar_, Mar 06 2023
%t A195605 CoefficientList[Series[(2 + 3 x + 4 x^2 - x^3) / ((1 + x) (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 19 2013 *)
%t A195605 LinearRecurrence[{2,0,-2,1},{2,7,18,31},50] (* _Harvey P. Dale_, Jan 21 2017 *)
%o A195605 (Magma) [(4*n*(n+2)+(-1)^n+3)/2: n in [0..48]];
%o A195605 (PARI) for(n=0, 48, print1((4*n*(n+2)+(-1)^n+3)/2", "));
%Y A195605 Cf. A000217, A077591, A157914.
%Y A195605 Cf. A047621 (contains first differences), A016754 (contains the sum of any two consecutive terms).
%Y A195605 Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195241 [incomplete list].
%K A195605 nonn,easy
%O A195605 0,1
%A A195605 _Bruno Berselli_, Sep 21 2011 - based on remarks and sequences by _Omar E. Pol_