A033567 a(n) = (2*n-1)*(4*n-1).

1, 3, 21, 55, 105, 171, 253, 351, 465, 595, 741, 903, 1081, 1275, 1485, 1711, 1953, 2211, 2485, 2775, 3081, 3403, 3741, 4095, 4465, 4851, 5253, 5671, 6105, 6555, 7021, 7503, 8001, 8515, 9045, 9591, 10153, 10731, 11325, 11935, 12561, 13203, 13861, 14535, 15225

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n+1) = A005563(1), A061037(3), A061039(5), A061041(7), A061043(9), A061045(11), A061047(13), A061049(15). Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys, Hansen-Strong, ... spectra of hydrogen. - Paul Curtz, Oct 08 2008

Sequence found by reading the segment [1, 3] together with the line from 3, in the direction 3, 21, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 03 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A045944, A014634.

Programs

Magma

[(2*n-1)*(4*n-1): n in [0..50]]; // G. C. Greubel, Sep 19 2018

Mathematica

Table[(2*n - 1)*(4*n - 1), {n, 0, 50}] (* G. C. Greubel, Jul 06 2017 *)
LinearRecurrence[{3,-3,1},{1,3,21},50] (* Harvey P. Dale, Aug 25 2019 *)

PARI

vector(60, n, n--; (2*n-1)*(4*n-1)) \\ Michel Marcus, Apr 12 2015

Formula

a(n) = a(n-1) + 16*n - 14 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010

From G. C. Greubel, Jul 06 2017: (Start)

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

E.g.f.: (1 + 2*x + 8*x^2)*exp(x).

G.f.: (1 + 15*x^2)/(1 - x)^3. (End)

From Amiram Eldar, Jan 03 2022: (Start)

Sum_{n>=0} 1/a(n) = 1 + Pi/4 - log(2)/2.

Sum_{n>=0} (-1)^n/a(n) = 1 + (sqrt(2)-1)*Pi/4 + log(sqrt(2)-1)/sqrt(2). (End)

Extensions

More terms from Michel Marcus, Apr 12 2015