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%I #42 Apr 16 2023 21:30:24
%S 3,14,34,63,101,148,204,269,343,426,518,619,729,848,976,1113,1259,
%T 1414,1578,1751,1933,2124,2324,2533,2751,2978,3214,3459,3713,3976,
%U 4248,4529,4819,5118,5426,5743,6069,6404,6748,7101,7463,7834,8214,8603,9001,9408
%N a(n) = (9*n^2 + 13*n + 6) / 2.
%C Diagonal of triangular spiral in A051682. - _Paul Barry_, Mar 15 2003
%C Ehrhart polynomial of open quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - _Michael Somos_, Jul 22 2006
%H Harry J. Smith, <a href="/A064226/b064226.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 3*C(n, 0)+11*C(n, 1)+9*C(n, 2); binomial transform of (3, 11, 9, 0, 0, 0, .....). G.f.:(3+5*x+x^2)/(1-x)^3. - _Paul Barry_, Mar 15 2003
%F A064225(n) = a(-1-n). - _Michael Somos_, Jul 22 2006
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Wesley Ivan Hurt_, Apr 16 2023
%p A064226:=n-> (9*n^2 + 13*n + 6) / 2; seq(A064226(n), n=0..50); # _Wesley Ivan Hurt_, May 08 2014
%t Table[(9 n^2 + 13 n + 6)/2, {n, 0, 50}] (* _Wesley Ivan Hurt_, May 08 2014 *)
%t LinearRecurrence[{3, -3, 1}, {3, 14, 34}, 50] (* _Vincenzo Librandi_, Jul 19 2015 *)
%o (PARI) {a(n) = 3 + n * (9*n + 13) / 2}; /* _Michael Somos_, Jul 22 2006 */
%o (PARI) for (n=0, 1000, write("b064226.txt", n, " ", 3 + n*(9*n + 13)/2) ) \\ _Harry J. Smith_, Sep 10 2009
%o (Magma) I:=[3,14,34]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Jul 19 2015
%Y a(n) = A081268(n)+2. Cf. A051682, A081267.
%Y Cf. A064225.
%Y Cf. A235332.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_, Sep 22 2001
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