A055438 a(n) = 100*n^2 + n.

101, 402, 903, 1604, 2505, 3606, 4907, 6408, 8109, 10010, 12111, 14412, 16913, 19614, 22515, 25616, 28917, 32418, 36119, 40020, 44121, 48422, 52923, 57624, 62525, 67626, 72927, 78428, 84129, 90030, 96131, 102432, 108933, 115634, 122535

Offset: 1

Author

Henry Bottomley, May 18 2000

Keywords

• nonn
• easy

Comments

The identity (200n+1)^2 - (100n^2+n)*20^2 = 1 can be written as A157956(n)^2 - a(n)*20^2 = 1 (see Barbeau's paper). Also, the identity (80000n^2 + 800n + 1)^2 - (100n^2 + n)*(8000n + 40)^2 = 1 can be written as A157664(n)^2 - a(n)*A157663(n)^2 = 1 (see the comment from Bruno Berselli in A157664). - Vincenzo Librandi, Feb 04 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (row 14 in the first table at p. 85, case d(t) = t*(10^2*t+1)).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A157956, A157663, A157664, A002378, A055437; a(n) = A055436(n) if 10 <= n < 100.

Different from A031698.

Programs

Magma

I:=[101, 402, 903]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 04 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {101, 402, 903}, 50] (* Vincenzo Librandi, Feb 04 2012 *)
Table[100n^2+n,{n,40}] (* Harvey P. Dale, May 15 2018 *)

PARI

for(n=1, 50, print1(100*n^2+n", ")); \\ Vincenzo Librandi, Feb 04 2012

Formula

G.f.: x*(-101-99*x)/(x-1)^3. - Vincenzo Librandi, Feb 04 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 04 2012