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%I #40 Sep 18 2023 18:12:32
%S 5,9,17,29,45,65,89,117,149,185,225,269,317,369,425,485,549,617,689,
%T 765,845,929,1017,1109,1205,1305,1409,1517,1629,1745,1865,1989,2117,
%U 2249,2385,2525,2669,2817,2969,3125,3285,3449,3617,3789,3965,4145,4329,4517,4709,4905
%N a(n) = 2*n^2 + 2*n + 5.
%C This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:
%C 2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.
%H Colin Barker, <a href="/A294774/b294774.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
%F E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
%F a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
%F n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
%F a(n) = Integral_{x=0..2n+4} |3-x| dx. - _Pedro Caceres_, Dec 29 2020
%p seq(2*n^2 + 2*n + 5, n=0..100); # _Robert Israel_, Nov 10 2017
%t Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* _Harvey P. Dale_, Sep 18 2023 *)
%o (PARI) Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ _Colin Barker_, Nov 13 2017
%Y 1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
%Y Second column of Mathar's array in A016813 (Comments section).
%Y Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
%Y Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).
%K nonn,easy
%O 0,1
%A _Bruno Berselli_, Nov 08 2017
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