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a(n) = (9*n^2 + 13*n + 6)/2.
7

%I #46 Oct 27 2024 14:29:50

%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 4248,4529,4819,5118,5426,5743,6069,6404,6748,7101,7463,7834,8214,8603,9001,9408,9824

%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 American Mathematical Society, 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 From _Paul Barry_, Mar 15 2003: (Start)

%F a(n) = 3*C(n,0) + 11*C(n,1) + 9*C(n,2); binomial transform of (3, 11, 9, 0, 0, 0, ...).

%F G.f.: (3 + 5*x + x^2)/(1-x)^3.

%F a(n) = A081268(n) + 2. (End)

%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

%F E.g.f.: (3 + 11*x + 9*x^2/2)*exp(x). - _Elmo R. Oliveira_, Oct 21 2024

%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 Cf. A051682, A064225, A081267, A081268, A235332.

%K nonn,easy,changed

%O 0,1

%A _N. J. A. Sloane_, Sep 22 2001