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.

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A137215 a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182
Offset: 0

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Author

Ctibor O. Zizka, Mar 06 2008

Keywords

Comments

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

Examples

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

Links

Crossrefs

Cf. A002275, A011557.

Programs

  • Mathematica
    Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)
  • PARI
    a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020
  • Sage
    [3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

Formula

a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.
O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

Extensions

More terms from R. J. Mathar, Mar 16 2008
More terms from Jinyuan Wang, Feb 27 2020

A245584 Let f(m) put the leftmost digit of the positive integer m at its end; a(n) is the sequence of all positive integers m with f^2(m)=f(m^2).

Original entry on oeis.org

1, 2, 3, 12, 122, 1222, 12222, 122222, 1222222, 12222222, 122222222
Offset: 1

Views

Author

Reiner Moewald, Jul 26 2014

Keywords

Examples

			122^2=14884 and 221^2=48841.

Crossrefs

Cf. A045878, A090843.

Programs

  • Mathematica
    f[m_Integer] := Module[{w}, w := IntegerDigits[m]; FromDigits[Rest[AppendTo[w, First[w]]]]]; a245584[n_Integer] :=
Select[Range[n], If[f[#]^2 == f[#^2] && ! Mod[#, 10] == 0, True, False] &]; a245584[10^5] (* Michael De Vlieger, Aug 17 2014 *)
  • Python
    import math
max = 10000
print('los')
for n in range(1, max):
   nst = str(n*n)
   nnewst = nst[1:] + nst[0]
   d = int(nnewst)
   e = int(math.sqrt(d))
   est = str(e)
   enewst = est[len(est)-1] + est[:len(est)-1]
   if (e * e == d) and (nnewst[0] != "0") and (str(n) == enewst):
      print(n, '  ',  e)
print('End.')

Formula

One can easily prove that all integers of the form 12...2 are elements of the sequence.
Previous Showing 11-12 of 12 results.

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