A177072 a(n) = (9*n+2)*(9*n+7).

14, 176, 500, 986, 1634, 2444, 3416, 4550, 5846, 7304, 8924, 10706, 12650, 14756, 17024, 19454, 22046, 24800, 27716, 30794, 34034, 37436, 41000, 44726, 48614, 52664, 56876, 61250, 65786, 70484, 75344, 80366, 85550, 90896, 96404, 102074, 107906, 113900, 120056

Offset: 0

Author

Vincenzo Librandi, May 31 2010

Keywords

• nonn
• easy

Comments

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 81*A002061(n+1) - 67. - Bruno Berselli, Aug 24 2010

Links

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

Crossrefs

Cf. A002061, A017185, A017245, A177059.

Programs

Magma

I:=[14, 176, 500]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Apr 08 2013

Mathematica

CoefficientList[Series[2(7 + 67 x + 7 x^2)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 08 2013 *)
Table[(9*n + 2)*(9*n + 7), {n, 0, 40}] (* Amiram Eldar, Feb 19 2023 *)
LinearRecurrence[{3,-3,1},{14,176,500},50] (* Harvey P. Dale, Jun 10 2023 *)

PARI

a(n)=(9*n+2)*(9*n+7) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 162*n + a(n-1) with n > 0, a(0)=14.

From Vincenzo Librandi, Apr 08 2013: (Start)

G.f.: 2*(7+67*x+7*x^2)/(1-x)^3.

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

From Amiram Eldar, Feb 19 2023: (Start)

a(n) = A017185(n)*A017245(n).

Sum_{n>=0} 1/a(n) = cot(2*Pi/9)*Pi/45.

Product_{n>=0} (1 - 1/a(n)) = cosec(2*Pi/9)*cos(sqrt(29)*Pi/18).

Product_{n>=0} (1 + 1/a(n)) = cosec(2*Pi/9)*cos(sqrt(21)*Pi/18). (End)

E.g.f.: exp(x)*(14 + 81*x*(2 + x)). - Elmo R. Oliveira, Oct 18 2024

Extensions

Edited by N. J. A. Sloane, Jun 22 2010