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.

Previous Showing 11-16 of 16 results.

A139705 Number of entries in A139704 ("nearly palindromic numbers") with n digits.

Original entry on oeis.org

0, 81, 153, 2268, 3150, 38718
Offset: 1

Views

Author

Jono Henshaw (jjono(AT)hotmail.com), Apr 30 2008

Keywords

Crossrefs

Cf. A139704, A050683.

A217789 Least difference between 2 palindromic numbers of length n.

Original entry on oeis.org

1, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11
Offset: 1

Views

Author

Michel Marcus, Mar 25 2013

Keywords

Comments

In his video, Fields medallist Villani asks about the number of palindromes of length n (cf. A050683 and A070252), and the minimal difference among any two of these (this sequence). Except for the 1 and 3-digits case (where e.g. 111-101=10), the minimal difference of 11 appears as 20...02 - 19...91 and similar patterns (1st and last digits increased by 1,...,7). - M. F. Hasler, Mar 25 2013
Also, continued fraction expansion of (2695-5*sqrt(5))/2462. [Bruno Berselli, Mar 25 2013]

Examples

			a(1)=1 for instance 8-7.
a(2)=11 for instance 22-11.
a(3)=10 for instance 111-101.
a(n)=11 for n >= 4, for instance 2002-1991, resp. generalization to n digits (cf. comment).

Links

Crossrefs

Cf. A050683, A070252.

Programs

Formula

G.f.: x*(1+10*x-x^2+x^3)/(1-x). [Bruno Berselli, Mar 25 2013]

A117858 Number of palindromes of length n (in base 6).

Original entry on oeis.org

5, 5, 30, 30, 180, 180, 1080, 1080, 6480, 6480, 38880, 38880, 233280, 233280, 1399680, 1399680, 8398080, 8398080, 50388480, 50388480, 302330880, 302330880, 1813985280, 1813985280, 10883911680, 10883911680, 65303470080, 65303470080, 391820820480, 391820820480
Offset: 1

Views

Author

Martin Renner, May 02 2006

Keywords

Links

Crossrefs

Cf. A050683.

Programs

  • Mathematica
    LinearRecurrence[{0,6},{5,5},30] (* Harvey P. Dale, Dec 09 2023 *)

Formula

a(n) = 5*6^floor((n-1)/2).
a(n) = 6*a(n-2). G.f. -5*x*(x+1)/(6*x^2-1). [Colin Barker, Feb 15 2013]

Extensions

More terms from Colin Barker, Feb 15 2013

A117859 Number of palindromes of length n (in base 7).

Original entry on oeis.org

6, 6, 42, 42, 294, 294, 2058, 2058, 14406, 14406, 100842, 100842, 705894, 705894, 4941258, 4941258, 34588806, 34588806, 242121642, 242121642, 1694851494, 1694851494, 11863960458, 11863960458, 83047723206, 83047723206, 581334062442, 581334062442
Offset: 1

Views

Author

Martin Renner, May 02 2006

Keywords

Links

Crossrefs

Cf. A050683.

Programs

  • Mathematica
    6*7^Floor[Range[0,40]/2] (* Harvey P. Dale, Mar 23 2012 *)

Formula

a(n) = 6*7^floor((n-1)/2).
a(n) = 7*a(n-2). G.f.: 6*x*(1+x)/(1-7*x^2). [Colin Barker, Jun 30 2012]

Extensions

More terms from Harvey P. Dale, Mar 23 2012

A117860 Number of palindromes of length n (in base 8).

Original entry on oeis.org

7, 7, 56, 56, 448, 448, 3584, 3584, 28672, 28672, 229376, 229376, 1835008, 1835008, 14680064, 14680064, 117440512, 117440512, 939524096, 939524096, 7516192768, 7516192768, 60129542144, 60129542144, 481036337152, 481036337152, 3848290697216, 3848290697216
Offset: 1

Views

Author

Martin Renner, May 02 2006

Keywords

Links

Crossrefs

Cf. A050683.

Programs

  • Mathematica
    LinearRecurrence[{0,8},{7,7},30] (* Harvey P. Dale, Apr 09 2021 *)

Formula

a(n) = 7*8^floor((n-1)/2).
a(n) = 8*a(n-2). G.f.: 7*x*(1+x)/(1-8*x^2). [Colin Barker, Jun 30 2012]

Extensions

More terms from Colin Barker, Jun 30 2012

A295319 a(n) is the sum of all n-digit palindromes.

Original entry on oeis.org

45, 495, 49500, 495000, 49500000, 495000000, 49500000000, 495000000000, 49500000000000, 495000000000000, 49500000000000000, 495000000000000000, 49500000000000000000, 495000000000000000000, 49500000000000000000000, 495000000000000000000000
Offset: 1

Views

Author

Melvin Peralta, Nov 19 2017

Keywords

Comments

For n > 1, the sum of the digits of a(n) is always 18 (see AoPS link).
Conjecture: All terms after a(1) are of the form 495*10^x where x is any nonnegative integer k such that k != 1 (mod 3). - Harvey P. Dale, Mar 05 2023

Examples

			The sum of all nine two-digit palindromes is 11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 = 495, and so a(2) = 495.
The sum of all three-digit palindromes is (101 + 999) + (111 + 989) + (121 + 979) + ... (545 + 565) + 555 = 49500, and so a(3) = 49500.

Links

Crossrefs

Cf. A002113, A050683, A261675.

Programs

  • Mathematica
    palSum[n_] := 99/2*10^(n - 1) * 10^Floor[(n - 1)/2]; palSum[1] = 45; Array[ palSum, 16] (* Robert G. Wilson v, Nov 21 2017 *)
  • PARI
    a(n) = if (n==1, 45, 9*10^floor((n-1)/2)*11*10^(n-1)/2); \\ Michel Marcus, Dec 26 2017

Formula

a(n) = A050683(n)*(5*10^(n-1) + (9/2)*10^(n-2) + ... + (9/2)*10 + 5) (calculates the sum by multiplying the expected value of a randomly selected n-digit palindrome with the number of n-digit palindromes).
For n > 1, a(n) = (A050683(n)/2)*11*10^(n-1).
For n > 3, a(n) = 1000 * a(n - 2). - David A. Corneth, Dec 26 2017
G.f.: x*(45 + 495*x + 4500*x^2)/(1 - 1000*x^2). - Chai Wah Wu, Jan 22 2018
E.g.f.: 9*(11*cosh(10*sqrt(10)*x) + 11*sqrt(10)*sinh(10*sqrt(10)*x) - 11 - 100*x)/200. - Stefano Spezia, Sep 19 2024
Previous Showing 11-16 of 16 results.

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