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

1, 8, 24, 49, 83, 126, 178, 239, 309, 388, 476, 573, 679, 794, 918, 1051, 1193, 1344, 1504, 1673, 1851, 2038, 2234, 2439, 2653, 2876, 3108, 3349, 3599, 3858, 4126, 4403, 4689, 4984, 5288, 5601, 5923, 6254, 6594, 6943, 7301, 7668, 8044, 8429, 8823, 9226

Offset: 0

Author

N. J. A. Sloane, Sep 22 2001

Keywords

• nonn
• easy

Comments

Diagonal of triangular spiral in A051682. - Michael Somos, Jul 22 2006

Ehrhart polynomial of closed quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - Michael Somos, Jul 22 2006

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

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

Examples

			Illustration of initial terms:
.
.                                    o
.                                 o o
.                      o       o o o o
.                   o o     o o o o o
.           o    o o o o     o o o o o
.        o o      o o o     o o o o o
.   o     o o    o o o o     o o o o o
.        o o      o o o     o o o o o
.           o    o o o o     o o o o o
.                   o o     o o o o o
.                      o       o o o o
.                                 o o
.                                    o
.
.   1     8        24           49
- _Aaron David Fairbanks_, Feb 23 2025
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• National Security Agency, Intrigued? (advertisement), Notices of the Amer. Math. Soc., vol. 49 (2002), p. 216.
• J. A. Siehler, Selections without adjacency on a rectangular grid, arXiv:1409.3869, Table 3, k=2 (different offset)
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000027, A000326, A001844, A005809, A007318, A051682, A064226, A081267, A235332.

Column w=2 of A371967.

Programs

Mathematica

Table[(9n^2+5n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,8,24},51] (* Harvey P. Dale, Sep 13 2011 *)

PARI

{a(n) = 1 + n * (9*n + 5) / 2}; /* Michael Somos, Jul 22 2006 */

Scheme

(define (A064225 n) (/ (+ (* 9 n n) (* 5 n) 2) 2))

Formula

a(n) = 9*n+a(n-1)-2, with n>0, a(0) = 1. - Vincenzo Librandi, Aug 07 2010

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

G.f.: (1+5*x+3*x^2)/(1-x)^3. - Colin Barker, Feb 23 2012

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

E.g.f.: exp(x)*(2 + 14*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022