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A064225 a(n) = (9*n^2 + 5*n + 2)/2.

%I A064225 #63 Mar 05 2025 20:28:09
%S A064225 1,8,24,49,83,126,178,239,309,388,476,573,679,794,918,1051,1193,1344,
%T A064225 1504,1673,1851,2038,2234,2439,2653,2876,3108,3349,3599,3858,4126,
%U A064225 4403,4689,4984,5288,5601,5923,6254,6594,6943,7301,7668,8044,8429,8823,9226
%N A064225 a(n) = (9*n^2 + 5*n + 2)/2.
%C A064225 Diagonal of triangular spiral in A051682. - _Michael Somos_, Jul 22 2006
%C A064225 Ehrhart polynomial of closed quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - _Michael Somos_, Jul 22 2006
%C A064225 In the natural number array A000027 this sequence is the first knight moves diagonal (A081267 is the second, A001844 is the main diagonal). It can be used to define this diagonal for any array: A007318(A064225-1)=A005809 (Subtraction by 1 because A007318 is defined with offset 0.) - _Tilman Piesk_, Mar 24 2012
%C A064225 Or positions of pentagonal numbers, such that p(a(n)) = p(a(n)-1) + p(3*n+1), where p=A000326. - _Vladimir Shevelev_, Jan 21 2014
%H A064225 Harry J. Smith, <a href="/A064225/b064225.txt">Table of n, a(n) for n = 0..1000</a>
%H A064225 National Security Agency, <a href="http://www.ams.org/notices/200202/rev-dauben.pdf">Intrigued? (advertisement)</a>, Notices of the Amer. Math. Soc., vol. 49 (2002), p. 216.
%H A064225 J. A. Siehler, <a href="https://arxiv.org/abs/1409.3869">Selections without adjacency on a rectangular grid</a>, arXiv:1409.3869, Table 3, k=2 (different offset)
%H A064225 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A064225 a(n) = 9*n+a(n-1)-2, with n>0, a(0) = 1. - _Vincenzo Librandi_, Aug 07 2010
%F A064225 a(0)=1, a(1)=8, a(2)=24, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Sep 13 2011
%F A064225 G.f.: (1+5*x+3*x^2)/(1-x)^3. - _Colin Barker_, Feb 23 2012
%F A064225 A064226(n) = a(-1-n). - _Michael Somos_, Jul 22 2006 (While the sequence itself is only one-way infinite, this identity works, as the defining formula (in the Name-field) produces integers also for the negative values of n, -1, -2, -3, etc.) - _Antti Karttunen_, Mar 24 2012
%F A064225 E.g.f.: exp(x)*(2 + 14*x + 9*x^2)/2. - _Stefano Spezia_, Dec 25 2022
%e A064225 Illustration of initial terms:
%e A064225 .
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%e A064225 .        o o      o o o     o o o o o
%e A064225 .   o     o o    o o o o     o o o o o
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%e A064225 .                   o o     o o o o o
%e A064225 .                      o       o o o o
%e A064225 .                                 o o
%e A064225 .                                    o
%e A064225 .
%e A064225 .   1     8        24           49
%e A064225 - _Aaron David Fairbanks_, Feb 23 2025
%t A064225 Table[(9n^2+5n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,8,24},51] (* _Harvey P. Dale_, Sep 13 2011 *)
%o A064225 (PARI) {a(n) = 1 + n * (9*n + 5) / 2}; /* _Michael Somos_, Jul 22 2006 */
%o A064225 (Scheme) (define (A064225 n) (/ (+ (* 9 n n) (* 5 n) 2) 2))
%Y A064225 Cf. A000027, A000326, A001844, A005809, A007318, A051682, A064226, A081267, A235332.
%Y A064225 Column w=2 of A371967.
%K A064225 nonn,easy
%O A064225 0,2
%A A064225 _N. J. A. Sloane_, Sep 22 2001