A134594 a(n) = n^2 + 10*n + 5: coefficients of the irrational part of (1 + sqrt(n))^5.

5, 16, 29, 44, 61, 80, 101, 124, 149, 176, 205, 236, 269, 304, 341, 380, 421, 464, 509, 556, 605, 656, 709, 764, 821, 880, 941, 1004, 1069, 1136, 1205, 1276, 1349, 1424, 1501, 1580, 1661, 1744, 1829, 1916, 2005, 2096, 2189, 2284, 2381, 2480, 2581, 2684

Offset: 0

Author

Artur Jasinski, Nov 04 2007

Keywords

• nonn
• easy

Comments

(1+sqrt(n))^5 = (5*n^2 + 10*n + 1) + (n^2 + 10*n + 5)*sqrt(n). For coefficients of the rational part see A134593.

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. A134593.

Programs

GAP

List([0..50],n->n^2+10*n+5); # Muniru A Asiru, Nov 24 2018

Magma

[n^2 +10*n +5: n in [0..50]]; // G. C. Greubel, Nov 23 2018

Mathematica

Table[(n^2 + 10n + 5), {n, 0, 50}]
LinearRecurrence[{3,-3,1}, {5,16,29}, 50] (* G. C. Greubel, Nov 23 2018 *)

PARI

a(n)=n^2+10*n+5 \\ Charles R Greathouse IV, Jun 17 2017

Sage

[n^2 +10*n +5 for n in range(50)] # G. C. Greubel, Nov 23 2018

Formula

a(n) = ((1+sqrt(n))^5 - (5*n^2 + 10*n + 1))/sqrt(n), for n > 0. [corrected by Jon E. Schoenfield, Nov 23 2018]

G.f.: (1+x)*(5-4*x)/(1-x)^3. - R. J. Mathar, Nov 14 2007

a(n) = 2*n + a(n-1) + 9 (with a(0)=5). - Vincenzo Librandi, Nov 23 2010

E.g.f.: (5 +11*x +x^2)*exp(x). - G. C. Greubel, Nov 23 2018