A157664 -id:A157664 - cp's OEIS frontend

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

Henry Bottomley, May 18 2000

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

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).

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

Different from A031698.

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

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 *)

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

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

A157663 a(n) = 8000*n + 40.

Original entry on oeis.org

8040, 16040, 24040, 32040, 40040, 48040, 56040, 64040, 72040, 80040, 88040, 96040, 104040, 112040, 120040, 128040, 136040, 144040, 152040, 160040, 168040, 176040, 184040, 192040, 200040, 208040, 216040, 224040, 232040, 240040, 248040, 256040

Offset: 1

Vincenzo Librandi, Mar 04 2009

The identity (80000*n^2 + 800*n + 1)^2 - (100*n^2 + n)*(8000*n + 40)^2 = 1 can be written as A157664(n)^2 - A055438(n)*a(n)^2 = 1 (see Bruno Berselli's comment at A157664). - Vincenzo Librandi, Feb 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A055438, A157664.

List([1..40], n -> 40*(200*n + 1)); # G. C. Greubel, Nov 17 2018

I:=[8040, 16040]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 04 2012

LinearRecurrence[{2, -1}, {8040, 16040}, 50] (* Vincenzo Librandi, Feb 04 2012 *)
8000*Range[40]+40 (* Harvey P. Dale, Dec 31 2024 *)

for(n=1, 50, print1(8000*n + 40", ")); \\ Vincenzo Librandi, Feb 04 2012

[40*(200*n + 1) for n in (1..40)] # G. C. Greubel, Nov 17 2018

G.f.: x*(8040 - 40*x)/(1-x)^2. - Vincenzo Librandi, Feb 04 2012
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Feb 04 2012
E.g.f.: 40*(-1 + (1 + 200*x)*exp(x)). - G. C. Greubel, Nov 17 2018

cp's OEIS Frontend

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

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