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A016778 a(n) = (3*n+1)^2.

%I A016778 #59 May 29 2025 15:47:48
%S A016778 1,16,49,100,169,256,361,484,625,784,961,1156,1369,1600,1849,2116,
%T A016778 2401,2704,3025,3364,3721,4096,4489,4900,5329,5776,6241,6724,7225,
%U A016778 7744,8281,8836,9409,10000,10609,11236,11881,12544,13225,13924,14641,15376,16129,16900,17689
%N A016778 a(n) = (3*n+1)^2.
%C A016778 From _Paul Curtz_, Mar 28 2019: (Start)
%C A016778 Sequence is a spoke of the hexagonal spiral built from the terms of A016777:
%C A016778 .
%C A016778         \
%C A016778        100--97--94--91
%C A016778           \           \
%C A016778           49--46--43  88
%C A016778           / \       \   \
%C A016778         52  16--13  40  85
%C A016778         /   / \   \   \   \
%C A016778       55  19   1  10  37  82
%C A016778       /   /   /   /   /   /
%C A016778     58  22   4---7  34  79
%C A016778       \   \         /   /
%C A016778       61  25--28--31  76
%C A016778         \             /
%C A016778         64--67--70--73
%C A016778 (End)
%H A016778 Shawn A. Broyles, <a href="/A016778/b016778.txt">Table of n, a(n) for n = 0..1000</a>
%H A016778 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A016778 a(n) = a(n-1) + 3*(6*n-1); a(0)=1. - _Vincenzo Librandi_, Nov 20 2010
%F A016778 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=16, a(2)=49. - _Harvey P. Dale_, Mar 03 2013
%F A016778 a(n) = A247792(n) + 6*n. - _Miquel Cerda_, Oct 23 2016
%F A016778 G.f.: (1 + 13*x + 4*x^2)/(1 - x)^3. - _Ilya Gutkovskiy_, Oct 23 2016
%F A016778 a(n) = A000212(3*n) + A000212(1+3*n) + A000212(2+3*n). - _Paul Curtz_, Mar 28 2019
%F A016778 From _Amiram Eldar_, Nov 12 2020: (Start)
%F A016778 Sum_{n>=0} 1/a(n) = A214550.
%F A016778 Sum_{n>=0} (-1)^n/a(n) = A262178. (End)
%F A016778 From _Elmo R. Oliveira_, May 29 2025: (Start)
%F A016778 E.g.f.: exp(x)*(1 + 15*x + 9*x^2).
%F A016778 a(n) = A000290(A016777(n)) = A016777(n)^2. (End)
%t A016778 (3*Range[0,50]+1)^2 (* or *) LinearRecurrence[{3,-3,1},{1,16,49},50] (* _Harvey P. Dale_, Mar 03 2013 *)
%o A016778 (Maxima) A016778(n):=(3*n+1)^2$
%o A016778 makelist(A016778(n),n,0,20); /* _Martin Ettl_, Nov 12 2012 */
%o A016778 (PARI) a(n)=(3*n+1)^2 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A016778 Cf. A000212, A000290, A016777, A016779, A016780, A016781, A214550, A247792, A262178.
%K A016778 nonn,easy
%O A016778 0,2
%A A016778 _N. J. A. Sloane_