%I #42 Mar 06 2023 02:21:50

%S 2,7,18,31,50,71,98,127,162,199,242,287,338,391,450,511,578,647,722,

%T 799,882,967,1058,1151,1250,1351,1458,1567,1682,1799,1922,2047,2178,

%U 2311,2450,2591,2738,2887,3042,3199,3362,3527,3698,3871,4050,4231,4418,4607,4802

%N a(n) = (4*n*(n+2)+(-1)^n+1)/2 + 1.

%C 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 Also A077591 (without first term) and A157914 interleaved.

%H Bruno Berselli, <a href="/A195605/b195605.txt">Table of n, a(n) for n = 0..1000</a>

%H Bruno Berselli, <a href="http://www.base5forum.it/upload/A195605.jpg">Illustration of initial terms: An origin of A195605</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%F G.f.: (2+3*x+4*x^2-x^3)/((1+x)*(1-x)^3).

%F a(n) = a(-n-2) = 2*a(n-1)-2*a(n-3)+a(n-4).

%F a(n) = A047524(A000982(n+1)).

%F 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 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 LinearRecurrence[{2,0,-2,1},{2,7,18,31},50] (* _Harvey P. Dale_, Jan 21 2017 *)

%o (Magma) [(4*n*(n+2)+(-1)^n+3)/2: n in [0..48]];

%o (PARI) for(n=0, 48, print1((4*n*(n+2)+(-1)^n+3)/2", "));

%Y Cf. A000217, A077591, A157914.

%Y Cf. A047621 (contains first differences), A016754 (contains the sum of any two consecutive terms).

%Y Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195241 [incomplete list].

%K nonn,easy

%O 0,1

%A _Bruno Berselli_, Sep 21 2011 - based on remarks and sequences by _Omar E. Pol_.