A033816 a(n) = 2*n^2 + 3*n + 3.

3, 8, 17, 30, 47, 68, 93, 122, 155, 192, 233, 278, 327, 380, 437, 498, 563, 632, 705, 782, 863, 948, 1037, 1130, 1227, 1328, 1433, 1542, 1655, 1772, 1893, 2018, 2147, 2280, 2417, 2558, 2703, 2852, 3005, 3162, 3323, 3488, 3657, 3830, 4007, 4188, 4373, 4562

Offset: 0

Author

Olivier Gorin (gorin(AT)roazhon.inra.fr)

Keywords

• easy
• nonn

Comments

For n >= 1, a(n) is the (1,1)-th entry in a (2*n+1) X (2*n+1) magic square built using the "north-east" technique. [Most probably the author is referring to the "Siamese method" described in the Wikipedia article. - Petros Hadjicostas, Jul 26 2020~]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• Wikipedia, Siamese method.
• Index entries for sequences related to magic squares
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Magma

I:=[3, 8, 17]; [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 07 2012

Mathematica

a[n_]:=2*n^2+3*n+3;...and/or...Array[ -#*(1-#*2)+2&,5!,0] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
CoefficientList[Series[(3-x+2*x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 07 2012 *)
LinearRecurrence[{3,-3,1},{3,8,17},50] (* Harvey P. Dale, May 07 2015 *)

PARI

a(n)=2*n^2+3*n+3 \\ Charles R Greathouse IV, Sep 24 2015

Formula

From Colin Barker, Jun 27 2012: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: (3 - x + 2*x^2)/(1 - x)^3. (End)

E.g.f.: (2*x^2 + 5*x + 3)*exp(x). - G. C. Greubel, Jul 15 2017

Extensions

More terms from James Sellers, Jun 20 2000

Offset corrected by Charles R Greathouse IV, Jul 25 2010