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.

User: Mark R. Diamond

Mark R. Diamond's wiki page.

Mark R. Diamond has authored 3 sequences.

A053013 Positions in decimal expansion of Pi where next prime begins.

Original entry on oeis.org

1, 2, 7, 8, 11, 12, 14, 15, 19, 3076, 3077, 3078, 3081, 3082, 3083, 3831, 3833, 3835, 3836, 3838, 3921, 3926, 3927, 3929, 4000, 4010, 4011, 4012, 4014, 4016, 4018, 4019, 4020, 4023, 4024, 4038, 4040, 4045, 4096, 4097, 4103, 4104, 4110, 4113, 4115, 4124
Offset: 1

Views

Author

Mark R. Diamond, Feb 22 2000

Keywords

Comments

This sequence refers to a variant of A047777 in which one does not require that all primes are distinct. See A229181 for first differences of this sequence. - M. F. Hasler, Sep 15 2013

Programs

  • Mathematica
    A229181 = {1}; digits = Join[{{1}}, RealDigits[Pi, 10, 5000] // First // Rest]; digits //. {a:({A229181,%20lg%5D;%20%7B%7B1%7D,%20c%7D)%20;%20Join%5B%7B1%7D,%20Accumulate%5BA229181%5D%20+%201%5D%20(*%20_Jean-Fran%C3%A7ois%20Alcover">Integer..}..), b__Integer /; PrimeQ[FromDigits[{b}]], c___Integer} :> (Print[lg = {b} // Length]; AppendTo[A229181, lg]; {{1}, c}) ; Join[{1}, Accumulate[A229181] + 1] (* _Jean-François Alcover, Oct 17 2013 *)

Formula

Partial sums of A229181: a(n)=sum_{k=1...n} A229181(k). - M. F. Hasler, Sep 15 2013

A010551 Multiply successively by 1,1,2,2,3,3,4,4,..., n >= 1, a(0) = 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 36, 144, 576, 2880, 14400, 86400, 518400, 3628800, 25401600, 203212800, 1625702400, 14631321600, 131681894400, 1316818944000, 13168189440000, 144850083840000, 1593350922240000, 19120211066880000, 229442532802560000, 2982752926433280000
Offset: 0

Views

Author

Mark R. Diamond

Keywords

Comments

From Emeric Deutsch, Dec 14 2008: (Start)
Number of permutations of {1,2,...,n-1} having a single run of odd entries. Example: a(5)=12 because we have 1324,1342,3124,3142,2134,4132,2314,4312, 2413, 4213, 2431 and 4231.
a(n) = A152666(n-1,1). (End)
a(n+1) gives the permanent of the n X n matrix whose (i,j)-element is i+j-1 modulo 2. - John W. Layman, Jan 03 2011
From Daniel Forgues, May 20 2011: (Start)
a(0) = 1 since it is the empty product.
A010551(n-2), n >= 2, equal to (ceiling((n-2)/2))! * (floor((n-2)/2))!, gives the number of arrangements of n-2 entries from 2 to n-1, starting with an even entry and where the parity of adjacent entries alternates. This is the number of arrangements to investigate for row n of a prime pyramid (A051237). (End)
Partial products of A008619. - Reinhard Zumkeller, Apr 02 2012
Also size of the equivalence class of S_n containing the identity permutation under transformations of positionally adjacent elements of the form abc <--> acb where a < b < c, cf. A210667 (equivalently under such transformations of the form abc <--> bac where a < b < c.) - Tom Roby, May 15 2012
Row sums of A246117. - Peter Bala, Aug 15 2014
a(n) is the number of parity-alternating permutations of size n. A permutation is parity-alternating if it sends even integers to even, and odd to odd. - Per W. Alexandersson, Jun 06 2022
n divides a(n) if and only if n is not prime. Since a(n) = floor(n/2)!*floor((n+1)/2)!, if n is prime then n is not a factor of a(n). All the prime factors of a(n) are in fact less than or equal to (n+1)/2. If n is composite, then it's possible to write it as p*q with p and q less than or equal to n/2. So p and q are factors of a(n). - Davide Oliveri, Apr 01 2023
Number of permutations of {1, 2, ..., n-1} where each entry is not greater than twice the previous entry. - Dewangga Putra Sheradhien, Jul 13 2024

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 4*x^4 + 12*x^5 + 36*x^6 + 144*x^7 + 576*x^8 + ...
For n = 7, a(n) = 1*1*2*2*3*3*4 (7 factors), which is 144. - _Michael B. Porter_, Jul 03 2016

Links

Crossrefs

Cf. A008619, A064044, A246117, A130820, A229020.
Column k=2 of A275062.

Programs

  • Haskell
    a010551 n = a010551_list !! n
a010551_list = scanl (*) 1 a008619_list
-- Reinhard Zumkeller, Apr 02 2012
  • Magma
    [Factorial(n div 2)*Factorial((n+1) div 2): n in [0..25]]; // Vincenzo Librandi Jan 17 2018
  • Maple
    A010551 := proc(n)
    option remember;
    if n <= 1 then
        1
    else
        procname(n-1) *trunc( (n+1)/2 );
    fi;
end:
  • Mathematica
    FoldList[ Times, 1, Flatten@ Array[ {#, #} &, 11]] (* Robert G. Wilson v, Jul 14 2010 *)
  • PARI
    {a(n)=local(X=x+x*O(x^n)); 1/polcoeff(besseli(0,2*X)+X*besseli(1,2*X),n,x)} \\ Paul D. Hanna, Apr 07 2005
  • PARI
    A010551(n)=(n\2)!*((n+1)\2)! \\ Michael Somos, Dec 29 2012, edited by M. F. Hasler, Nov 26 2017
  • Python
    def O(f):
    c = 1
    while len(f) > 1:
        f.sort()
        m = abs(f[0] - f[1])
        c *= m
        f[0] = m
        f.pop(1)
    return c
a = lambda n: O(list(range(1, n+1)))
print([a(n) for n in range(0, 26)]) # Darío Clavijo, Aug 24 2024

Formula

a(n) = floor(n/2)!*floor((n+1)/2)! is the number of permutations p of {1, 2, 3, ..., n} such that for every i, i and p(i) have the same parity, i.e., p(i) - i is even. - Avi Peretz (njk(AT)netvision.net.il), Feb 22 2001
a(n) = n!/binomial(n, floor(n/2)). - Paul Barry, Sep 12 2004
G.f.: Sum_{n>=0} x^n/a(n) = besseli(0, 2*x) + x*besseli(1, 2*x). - Paul D. Hanna, Apr 07 2005
E.g.f.: 1/(1-x/2) + (1/2)/(1-x/2)*arccos(1-x^2/2)/sqrt(1-x^2/4). - Paul D. Hanna, Aug 28 2005
G.f.: G(0) where G(k) = 1 + (k+1)*x/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1) )); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 28 2012
D-finite with recurrence: 4*a(n) - 2*a(n-1) - n*(n-1)*a(n-2) = 0. - R. J. Mathar, Dec 03 2012
a(n) = a(n-1) * (a(n-2) + a(n-3)) / a(n-3) for all n >= 3. - Michael Somos, Dec 29 2012
G.f.: 1 + x + x^2*(1 + x*(G(0) - 1)/(x-1)) where G(k) = 1 - (k+2)/(1-x/(x - 1/(1 - (k+2)/(1-x/(x - 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (k+1)/(1-x/(x - 1/(1 - (k+1)/(1-x/(x - 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: 1 + x*G(0), where G(k) = 1 + x*(k+1)/(1 - x*(k+2)/(x*(k+2) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 08 2013
G.f.: Q(0), where Q(k) = 1 + x*(k+1)/(1 - x*(k+1)/(x*(k+1) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 08 2013
Sum_{n >= 1} 1/a(n) = A130820. - Peter Bala, Jul 02 2016
a(n) ~ sqrt(Pi*n) * n! / 2^(n + 1/2). - Vaclav Kotesovec, Oct 02 2018
Sum_{n>=0} (-1)^n/a(n) = A229020. - Amiram Eldar, Apr 12 2021

A010552 Multiply successively by 1 (once), 2 (twice), 3 (thrice), etc.

Original entry on oeis.org

1, 1, 2, 4, 12, 36, 108, 432, 1728, 6912, 27648, 138240, 691200, 3456000, 17280000, 86400000, 518400000, 3110400000, 18662400000, 111974400000, 671846400000, 4031078400000, 28217548800000, 197522841600000, 1382659891200000, 9678619238400000, 67750334668800000
Offset: 0

Views

Author

Mark R. Diamond

Keywords

Links

Crossrefs

A002109 is a subsequence.
Cf. A002024.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1) * floor(sqrt(2*n)+1/2))
    end:
seq(a(n), n=0..30); # Alois P. Heinz, May 20 2013
  • Mathematica
    t = Table[n, {n, 9}, {n}] // Flatten; b[n_] := t[[n]]; a[0] = 1; a[n_] := a[n] = b[n] a[n-1]; Table[a[n], {n, 0, t // Length}] (* Jean-François Alcover, Feb 21 2016 *)
  • PARI
    lista(nn) = my(na = 1, i = 0, im = 1, list=List()); listput(list, na); while (i < nn, for (j=1, im, na *= im; listput(list, na); i++; ); im++; ); Vec(list); \\ Michel Marcus, May 20 2013; Jun 19 2025
  • Python
    from math import isqrt, comb, prod
def A010552(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))**(n-comb(a,2))*prod(i**i for i in range(2,a)) # Chai Wah Wu, Jun 19 2025

Formula

a(0)=1, a(n) = a(n-1)*floor(sqrt(2*n)+1/2) for n>0. - Alois P. Heinz, May 20 2013

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.