OFFSET
1,2
COMMENTS
From Omar E. Pol, Oct 23 2019: (Start)
a(n) is also the sum of terms that are in the n-th finite row and in the n-th finite column of the square [1,n]x[1,n] of the natural number array A000027; e.g., the [1,3]x[1,3] square is
1..3..6
2..5..9
4..8..13,
so that a(1) = 1, a(2) = 2 + 3 + 5 = 10, a(3) = 4 + 6 + 8 + 9 + 13 = 40.
Hence the partial sums give A185505. (End)
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), pages 132, 416.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1000
Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn, Combinatorics on bounded free Motzkin paths and its applications, arXiv:2205.15554 [math.CO], 2022. (See p. 14).
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
G.f.: x*(1+x)*(1+5*x+x^2)/(1-x)^4. - Colin Barker, Mar 02 2012
a(n) = Sum_{k = n^2-2*n+2..n^2} A064788(k). - Lior Manor, Jan 13 2013
From G. C. Greubel, Dec 01 2017: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
E.g.f.: (-6 + 12*x + 21*x^2 + 14*x^3)*exp(x)/6 + 1. (End)
MATHEMATICA
Table[(2*n-1)*(7*n^2-7*n+6)/6, {n, 1, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 10, 40, 105}, 50] (* G. C. Greubel, Dec 01 2017 *)
PROG
(PARI) a(n) = { (2*n - 1)*(7*n^2 - 7*n + 6)/6 } \\ Harry J. Smith, Aug 23 2009
(PARI) my(x='x+O('x^30)); Vec(serlaplace((-6 + 12*x + 21*x^2 + 14*x^3 )*exp(x)/6 + 1)) \\ G. C. Greubel, Dec 01 2017
(Magma) [(2*n-1)*(7*n^2-7*n+6)/6: n in [1..30]]; // G. C. Greubel, Dec 01 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Aug 01 2001
STATUS
approved