A055438 -id:A055438 - cp's OEIS frontend

A055436 a(n) = concatenation of n^2 and n.

11, 42, 93, 164, 255, 366, 497, 648, 819, 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, 129636, 136937, 144438, 152139

Offset: 1

Henry Bottomley, May 18 2000

Cf. A053061, A055437 (10n^2+n), A055438 (100n^2+n).

[Seqint(Intseq(n) cat Intseq(n^2)): n in [1..40]]; // Vincenzo Librandi, Jan 03 2015

a:= n-> parse(cat(n*n, n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 03 2015

Table[n^2*10^IntegerLength[n] + n, {n, 36}] (* Jayanta Basu, Jul 12 2013 *)
Table[FromDigits[Join[IntegerDigits[n^2], IntegerDigits[n]]],{n, 40}] (* Vincenzo Librandi, Jan 03 2015 *)

a(n) = n^2*floor(log_10(n) + 1) + n.

a(n) = A055437(n) if 1 <= n < 10, a(n) = A055438(n) if 10 <= n < 100.

A055437 a(n) = 10*n^2+n.

Original entry on oeis.org

11, 42, 93, 164, 255, 366, 497, 648, 819, 1010, 1221, 1452, 1703, 1974, 2265, 2576, 2907, 3258, 3629, 4020, 4431, 4862, 5313, 5784, 6275, 6786, 7317, 7868, 8439, 9030, 9641, 10272, 10923, 11594, 12285, 12996, 13727, 14478, 15249, 16040, 16851, 17682, 18533

Offset: 1

Henry Bottomley, May 18 2000

a(n) = A055436(n) if 1<=n<10.

Number of edges in the join of the complete 4-partite graph of order 4n and the cycle graph of order n, K_n,n,n,n * C_n. - Roberto E. Martinez II, Jan 07 2002

			From the third formula: a(8) = 648 = 16^2 -17^2 +18^2 ... +30^2 -31^2 +32^2 = -33^2 +34^2 -35^2 ... +46^2 -47^2 +48^2.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A002378, A055436, A055438.

[10*n^2+n : n in [1..50]]; // Wesley Ivan Hurt, Apr 27 2016

A055437:=n->10*n^2+n: seq(A055437(n), n=1..50); # Wesley Ivan Hurt, Apr 27 2016

Table[10 n^2 + n, {n, 50}] (* Wesley Ivan Hurt, Apr 27 2016 *)

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

From Bruno Berselli, Nov 26 2013: (Start)
G.f.: x*(11 + 9*x) / (1 - x)^3.
a(n) = Sum_{i=0..2*n} (-1)^i*(2*n+i)^2.
a(n) = Sum_{i=1..2*n} (-1)^i*(4*n+i)^2. (End)
From Wesley Ivan Hurt, Apr 27 2016: (Start)
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3.
a(n) = (1/5) * Sum_{i=0..10*n} i. (End)
E.g.f.: x*(11 + 10*x)*exp(x). - Ilya Gutkovskiy, Apr 27 2016
a(n) = A000217(6*n) - A000217(4*n). - Bruno Berselli, Sep 21 2016

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

A157664 a(n) = 80000n^2 + 800n + 1.

Original entry on oeis.org

80801, 321601, 722401, 1283201, 2004001, 2884801, 3925601, 5126401, 6487201, 8008001, 9688801, 11529601, 13530401, 15691201, 18012001, 20492801, 23133601, 25934401, 28895201, 32016001, 35296801, 38737601, 42338401, 46099201

Offset: 1

Vincenzo Librandi, Mar 04 2009

The identity (80000n^2 + 800n + 1)^2 - (100n^2 + n)*(8000n + 40)^2 = 1 can be written as a(n)^2 - A055438(n)*A157663(n)^2 = 1. - Vincenzo Librandi, Feb 04 2012

This is the case s=10 of the identity (8*n^2*s^4 + 8*n*s^2 + 1)^2 - (n^2*s^2 + n)*(8*n*s^3 + 4*s)^2 = 1. - Bruno Berselli, 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 (3,-3,1).

Cf. A055438, A157663.

List([1..40], n -> 80000*n^2+800*n+1); # G. C. Greubel, Nov 17 2018

I:=[80801, 321601, 722401]; [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}, {80801, 321601, 722401}, 50] (* Vincenzo Librandi, Feb 04 2012 *)

for(n=1, 40, print1(80000*n^2 + 800*n + 1", ")); \\ Vincenzo Librandi, Feb 04 2012

[80000*n^2+800*n+1 for n in (1..40)] # G. C. Greubel, Nov 17 2018

G.f.: x*(80801 + 79198*x + x^2)/(1-x)^3. - Vincenzo Librandi, Feb 04 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 04 2012
E.g.f.: (1 + 80800*x + 80000*x^2)*exp(x) - 1. - G. C. Greubel, Nov 17 2018

A157956 a(n) = 200*n + 1.

Original entry on oeis.org

201, 401, 601, 801, 1001, 1201, 1401, 1601, 1801, 2001, 2201, 2401, 2601, 2801, 3001, 3201, 3401, 3601, 3801, 4001, 4201, 4401, 4601, 4801, 5001, 5201, 5401, 5601, 5801, 6001, 6201, 6401, 6601, 6801, 7001, 7201, 7401, 7601, 7801, 8001, 8201, 8401, 8601

Offset: 1

Vincenzo Librandi, Mar 10 2009

The identity (200*n + 1)^2 - (100*n^2 + n)*20^2 = 1 can be written as a(n)^2 - A055438(n)*20^2 = 1. - Vincenzo Librandi, Feb 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1
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 (2,-1).

Cf. A055438.

I:=[201, 401]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 04 2012

200Range[50]+1  (* Harvey P. Dale, Feb 24 2011 *)
LinearRecurrence[{2, -1}, {201, 401}, 50] (* Vincenzo Librandi, Feb 04 2012 *)

for(n=1, 50, print1(200*n + 1", ")); \\ Vincenzo Librandi, Feb 04 2012

G.f.: x*(201-x)/(1-x)^2. - Vincenzo Librandi, Feb 04 2012

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

A031698 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 20.

Original entry on oeis.org

101, 402, 903, 1604, 2505, 3606, 4907, 6408, 8109, 10010, 12111, 14412, 16913, 19614, 22515, 25616, 28917, 32418, 36119, 40020, 44121, 48422, 48862, 52923, 57624, 62525, 67626, 72927, 78428, 84129, 90030, 96131, 102432, 108933, 115634, 122535, 129636

Offset: 1

David W. Wilson

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Different from A055438.

Select[Range[200000],!IntegerQ[Sqrt[#]]&&Min[ContinuedFraction[Sqrt[#]][[2]]]==20&] (* Vincenzo Librandi, Feb 06 2012 *)

cp's OEIS Frontend

A055436 a(n) = concatenation of n^2 and n.

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A055437 a(n) = 10*n^2+n.

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

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A157664 a(n) = 80000*n^2 + 800*n + 1.

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A157956 a(n) = 200*n + 1.

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Magma

Mathematica

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A031698 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 20.

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Mathematica

A157664 a(n) = 80000n^2 + 800n + 1.