A152813 a(n) = 2*n^2 + 10*n + 3.

3, 15, 31, 51, 75, 103, 135, 171, 211, 255, 303, 355, 411, 471, 535, 603, 675, 751, 831, 915, 1003, 1095, 1191, 1291, 1395, 1503, 1615, 1731, 1851, 1975, 2103, 2235, 2371, 2511, 2655, 2803, 2955, 3111, 3271, 3435, 3603, 3775, 3951, 4131, 4315, 4503, 4695

Offset: 0

Author

Vincenzo Librandi, Dec 17 2008

Keywords

• nonn
• easy

Comments

Nonzero x values at solutions of Diophantine equation 2*x^3 + 19*x^2 = y^2.

Nonnegative integers m such that 2*m + 19 is a square. - Vincenzo Librandi, Mar 01 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A001844.

Programs

Magma

I:=[3, 15, 31]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 22 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {3, 15, 31}, 50] (* Vincenzo Librandi, Feb 22 2012 *)
CoefficientList[Series[(3 + 6*x - 5*x^2)/(1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Jan 04 2013 *)

PARI

for(n=0, 40, print1(2*n^2+10*n+3", ")); \\ Vincenzo Librandi, Feb 22 2012

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 22 2012

G.f.: (3 + 6*x - 5*x^2)/(1 -x)^3. - Vincenzo Librandi, Jan 04 2013

a(n-1) = 2*n^2 + 2*n - 9. - Avi Friedlich, Mar 27 2015

a(n) = A001844(n+2) - 10. - Robert Israel, Mar 27 2015

Sum_{n>=0} 1/a(n) = 14/45 + tan(sqrt(19)*Pi/2)*Pi/(2*sqrt(19)). - Amiram Eldar, Mar 02 2023

E.g.f.: (3 + 12*x + 2*x^2)*exp(x). - Elmo R. Oliveira, Oct 18 2024

Extensions

Edited and extended by Jon E. Schoenfield, Jun 17 2010