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A195029 a(n) = n*(14*n + 13) + 3.

%I A195029 #34 Dec 29 2024 18:38:49
%S A195029 3,30,85,168,279,418,585,780,1003,1254,1533,1840,2175,2538,2929,3348,
%T A195029 3795,4270,4773,5304,5863,6450,7065,7708,8379,9078,9805,10560,11343,
%U A195029 12154,12993,13860,14755,15678,16629,17608,18615,19650,20713,21804,22923,24070,25245
%N A195029 a(n) = n*(14*n + 13) + 3.
%C A195029 Sequence found by reading the line from 3, in the direction 3, 30, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the semi-diagonal parallel to A195024 and also parallel to A195028 in the same square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].
%C A195029 56*a(n) + 1 is a perfect square. - _Bruno Berselli_, Feb 14 2017
%H A195029 Matthew House, <a href="/A195029/b195029.txt">Table of n, a(n) for n = 0..10000</a>
%H A195029 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A195029 a(n) = 14*n^2 + 13*n + 3 = A195028(n) + 3 = (2*n + 1)*(7*n + 3).
%F A195029 From _Colin Barker_, Apr 09 2012: (Start)
%F A195029 G.f.: (3 + 21*x + 4*x^2)/(1 - x)^3.
%F A195029 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F A195029 From _Elmo R. Oliveira_, Dec 29 2024: (Start)
%F A195029 E.g.f.: exp(x)*(3 + 27*x + 14*x^2).
%F A195029 a(n) = A005408(n)*A017017(n) = A022264(2*n+1). (End)
%t A195029 Table[n (14 n + 13) + 3, {n, 0, 40}] (* _Bruno Berselli_, Feb 14 2017 *)
%t A195029 LinearRecurrence[{3,-3,1},{3,30,85},50] (* _Harvey P. Dale_, May 03 2018 *)
%o A195029 (PARI) a(n)=n*(14*n+13)+3 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A195029 Bisection of A022264.
%Y A195029 Cf. A144555, A152760, A195019, A195020, A195021, A195023, A195024, A195025, A195026, A195027, A195028, A195320.
%Y A195029 Cf. A005408, A017017, A185019, A193053, A198017.
%K A195029 nonn,easy
%O A195029 0,1
%A A195029 _Omar E. Pol_, Sep 07 2011
%E A195029 Edited by _Bruno Berselli_, Feb 14 2017