cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Henry Bottomley, May 18 2000

Keywords

Crossrefs

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

Programs

  • Magma
    [Seqint(Intseq(n) cat Intseq(n^2)): n in [1..40]]; // Vincenzo Librandi, Jan 03 2015
  • Maple
    a:= n-> parse(cat(n*n, n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 03 2015
  • Mathematica
    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 *)

Formula

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.

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

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, 52923, 57624, 62525, 67626, 72927, 78428, 84129, 90030, 96131, 102432, 108933, 115634, 122535
Offset: 1

Views

Author

Henry Bottomley, May 18 2000

Keywords

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

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

A023111 Squares that remain square when the digit 1 is appended.

Original entry on oeis.org

0, 36, 51984, 74960964, 108093658176, 155870980128900, 224765845252215696, 324112192982714904804, 467369557515229640511744, 673946577824768158903030116, 971830497853758169908528915600, 1401378903958541456239939793265156, 2020787407677718926139823273359439424
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

The terms of the sequence are the squares of the y-values in the solution to the Pellian equation x^2-10*y^2=1. - Colin Barker, Sep 28 2013
After 0, the sequence lists the numbers k for which A055437(k) is a perfect square. - Bruno Berselli, Jan 16 2018

Examples

			36 is a term because both 36 and 361 are squares.

Links

Crossrefs

Cf. A023110.

Programs

  • Mathematica
    LinearRecurrence[{1443,-1443,1},{0,36,51984},20] (* Harvey P. Dale, Dec 23 2013 *)
  • PARI
    concat(0, Vec(36*x^2*(1 + x) / ((1 - x)*(1 - 1442*x + x^2)) + O(x^15))) \\ Colin Barker, Dec 29 2017

Formula

G.f.: 36*x^2*(1 + x) / ((1 - x)*(1 - 1442*x + x^2)). - Colin Barker, Jan 31 2013
a(0)=0, a(1)=36, a(2)=51984, a(n) = 1443*a(n-1)-1443*a(n-2)+a(n-3). - Harvey P. Dale, Dec 23 2013
a(n) = (721 + 228*sqrt(10))^(-n)*(721+228*sqrt(10) - 2*(721+228*sqrt(10))^n + (721-228*sqrt(10))*(721+228*sqrt(10))^(2*n)) / 40. - Colin Barker, Dec 29 2017

A225144 a(n) = Sum_{i=n..2*n} i^2*(-1)^i.

Original entry on oeis.org

0, 3, 11, 18, 42, 45, 93, 84, 164, 135, 255, 198, 366, 273, 497, 360, 648, 459, 819, 570, 1010, 693, 1221, 828, 1452, 975, 1703, 1134, 1974, 1305, 2265, 1488, 2576, 1683, 2907, 1890, 3258, 2109, 3629, 2340, 4020, 2583, 4431, 2838, 4862, 3105, 5313, 3384
Offset: 0

Views

Author

Bruno Berselli, Jun 06 2013

Keywords

Comments

3 and 11 are the only primes in the sequence.

Examples

			a(6) = 6^2-7^2+8^2-9^2+10^2-11^2+12^2 = 93.
a(7) = -7^2+8^2-9^2+10^2-11^2+12^2-13^2+14^2 = 84.

Links

Crossrefs

Cf. A050409: sum(i^2, i=n..2n); A064455: sum(i*(-1)^i, i=n..2n); A065679: A000217(n)+(-1)^n*A000217(n-1); A089594: sum(i^2*(-1)^i, i=1..n).

Programs

  • Magma
    [&+[i^2*(-1)^i: i in [n..2*n]]: n in [0..50]];
  • Mathematica
    Table[Sum[i^2 (-1)^i, {i, n, 2 n}], {n, 0, 50}]

Formula

G.f.: x*(3+11*x+9*x^2+9*x^3)/(1-x^2)^3.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6).
a(n) = n*(4*n+(n-1)*(-1)^n+2)/2.
a(n) = A000217(2n) +(-1)^n*A000217(n-1) with A000217(-1)=0.
a(2n-1) = A094159(n) for n>0; a(2n) = A055437(n) for A055437(0)=0.

A257144 Numbers n not of the form x+y*x^2 for x>1 and y>0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 40, 41, 43, 44, 45, 47, 49, 51, 53, 59, 60, 61, 63, 64, 65, 67, 69, 71, 73, 76, 77, 79, 81, 83, 85, 87, 88, 89, 91, 92, 95, 96, 97, 99, 101, 103, 104, 107, 108, 109, 112, 113, 115, 117, 119, 121
Offset: 1

Views

Author

Gionata Neri, Apr 16 2015

Keywords

Comments

Number n such that (d*k+1) /= (n/d), for k>0 and each value of d, where d is a divisor >1 of n.

Crossrefs

For numbers of the form x+y*x^2 with 0A002378 (y=1), A014105 (y=2), A049451 (y=3), A007742 (y=4), A202803 (y=5), A049453 (y=6), A092277 (y=7), A139275 (y=8), A154517 (y=9), A055437 (y=10). - Danny Rorabaugh, Apr 20 2015

Programs

  • Mathematica
    n = 71; Take[Complement[Range[n^2], DeleteDuplicates@ Sort@ Flatten@ Table[x + y x^2, {x, 2, n}, {y, 1, n}]], n] (* Michael De Vlieger, Apr 17 2015 *)
Showing 1-5 of 5 results.

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