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 1-7 of 7 results.

A214297 a(0)=-1, a(1)=0, a(2)=-3; thereafter a(n+2) - 2*a(n+1) + a(n) has period 4: repeat -4, 8, -4, 2.

Original entry on oeis.org

-1, 0, -3, 2, 3, 6, 5, 12, 15, 20, 21, 30, 35, 42, 45, 56, 63, 72, 77, 90, 99, 110, 117, 132, 143, 156, 165, 182, 195, 210, 221, 240, 255, 272, 285, 306, 323, 342, 357, 380, 399, 420, 437, 462, 483, 506, 525, 552, 575, 600, 621, 650, 675, 702, 725, 756, 783, 812, 837, 870, 899, 930, 957, 992, 1023, 1056, 1085, 1122, 1155, 1190
Offset: 0

Views

Author

Paul Curtz, Jul 11 2012

Keywords

Comments

Let a(n)/A000290(n) = [-1/0, 0/1, -3/4, 2/9, 3/16, 6/25, 5/36, 12/49, 15/64, 20/81, 21/100, 30/121, ...] = a(n)/b(n) (say).
Then b(n)-4*a(n)=4, 1, 16, 1 (period of length 4).
Permutation from a(n) to A061037(n): 1, 3, 2, 7, 5, 11, 4, 15, 9, 19, 6, ... = shifted A145979 + 1.
A061037(n) - a(n) = 0, 3, -3, -3, 0, -15, 3, -33, 0 -57, 15, -87, 0, -123, ...
First 3 rows:
-1 0 -3 2 3 6 5 12 15 20 21 30 35
1 -3 5 1 3 -1 7 3 5 1 9 5 7
-4 8 -4 2 -4 8 -4 2 -4 8 -4 2 -4.
Note that the terms of a(n) increase from 12. Compare to increasing terms permutation of A061037(n): -3,-1,0,2,3,5,6,12,15, .... and A129647.
c(n) = 0, -1, 0, -1, 2, 1, 2, 1, 4, 3, 4, 3, 6, 5, 6, 5, ... (cf. A134967)
d(n) = -1, 1, 1, 3, 1, 3, 3, 5, 3, 5, 5, 7, 5, 7, 7, 9, ..., hence:
a(n) = c(n+1) * d(n+1).

Links

Programs

  • Magma
    [(2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8: n in [0..100]]; // G. C. Greubel, Sep 19 2018
  • Maple
    A214297 := proc(n)
    option remember;
    if n <=5 then
        op(n+1,[-1,0,-3,2,3,6]) ;
    else
        2*procname(n-1)-procname(n-2)+procname(n-4)-2*procname(n-5)+procname(n-6) ;
    end if;
end proc: # R. J. Mathar, Jun 28 2013
  • Mathematica
    Table[(2 n^2 - 11 - 9 (-1)^n + 6 ((-1)^((2 n + 1 - (-1)^n)/4) + (-1)^((2 n - 1 + (-1)^n)/4)))/8, {n, 0, 69}] (* or *)
CoefficientList[Series[-(1 - 2 x + 4 x^2 - 8 x^3 + 3 x^4)/((1 - x)^2*(1 - x^4)), {x, 0, 69}], x] (* Michael De Vlieger, Mar 24 2017 *)
  • PARI
    vector(100, n, n--; (2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8) \\ G. C. Greubel, Sep 19 2018

Formula

a(k+4) - a(k) = 2*k + 4.
a(k+2) - a(k-2) = 2*k.
a(k+6) - a(k-6) = 6*k.
a(k+10) - a(k-10) = 10*k.
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(2*k) = -1, -3, followed by 3, 5, 15, 21, 35, 45, ... (A142717);
a(2*k+1) = k*(k+1) (see A002378).
A198442(n) = -1,0,0,2,3,6,8,12, minus 3 at A198442(4*n+2).
G.f. -( 1-2*x+4*x^2-8*x^3+3*x^4 )/( (1-x)^2*(1-x^4) ). - R. J. Mathar, Jul 17 2012; edited by N. J. A. Sloane, Jul 22 2012
From R. J. Mathar, Jun 28 2013: (Start)
a(4*k) = A000466(k);
a(4*k+1) = A002943(k);
a(4*k+2) = A078371(k-1) for k>0;
a(4*k+3) = A002939(k+1). (End)
a(n) = (2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8. - Luce ETIENNE, Oct 27 2016

Extensions

Edited by N. J. A. Sloane, Jul 22 2012

A129648 Largest order of a permutation of n elements with exactly 3 cycles. Also the largest LCM of a 3-partition of n.

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 6, 12, 15, 30, 21, 60, 35, 84, 105, 140, 84, 210, 165, 280, 315, 360, 385, 504, 495, 630, 693, 792, 819, 990, 1001, 1170, 1287, 1430, 1365, 1716, 1683, 2002, 2145, 2310, 2431, 2730, 2805, 3120, 3315, 3570, 3705, 4080, 4199, 4560, 4845, 5168
Offset: 1

Views

Author

Nickolas Reynolds (nickels(AT)gmail.com), Apr 25 2007

Keywords

Comments

a(n) is asymptotic to (n^3)/27.

Examples

			a(9) = 15 because 9 = 5+3+1 and lcm(1,3,5) = 15 is maximal.

Links

Crossrefs

Cf. A000793, A129651.
Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), this sequence (k=3), A129649 (k=4), A129650 (k=5), A355367 (k=6), A355403 (k=7).

Programs

  • Mathematica
    Max[LCM @@@ Compositions[ #, 3]] & /@ Range[1, n]

A129649 Largest order of a permutation of n elements with exactly 4 cycles. Also the largest LCM of a 4-partition of n.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 6, 6, 12, 15, 30, 30, 60, 60, 84, 105, 210, 140, 420, 210, 330, 420, 840, 420, 1260, 1155, 1540, 1365, 2520, 1320, 3080, 3465, 3960, 4095, 5544, 5005, 6930, 6435, 8190, 9009, 10296, 8415, 12870, 11781, 13464, 15015, 18018, 17017, 20592, 21879
Offset: 1

Views

Author

Nickolas Reynolds (nickels(AT)gmail.com), Apr 25 2007

Keywords

Comments

a(n) is asymptotic to (n^4)/256.

Examples

			a(18)=140 because 18 = 7+5+2+2 and lcm(2,2,5,7) = 140 is maximal.

Links

Crossrefs

Cf. A000793, A129651.
Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), A129648 (k=3), this sequence (k=4), A129650 (k=5), A355367 (k=6), A355403 (k=7).

Programs

  • Mathematica
    Max[LCM @@@ Compositions[ #, 4]] & /@ Range[1, n] (* needs Combinatorica *)
Join[{0,0,0},Table[Max[LCM@@#&/@IntegerPartitions[n,{4}]],{n,4,50}]] (* Harvey P. Dale, Feb 25 2012 *)

A129650 Largest order of a permutation of n elements with exactly 5 cycles. Also the largest LCM of a 5-partition of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 6, 12, 15, 30, 30, 60, 60, 84, 105, 210, 210, 420, 420, 420, 420, 840, 840, 1260, 1260, 2310, 1540, 4620, 2520, 5460, 4620, 9240, 5460, 13860, 9240, 16380, 15015, 27720, 13860, 32760, 19635, 40040, 45045, 51480, 32760, 72072, 58905
Offset: 1

Views

Author

Nickolas Reynolds (nickels(AT)gmail.com), Apr 25 2007

Keywords

Comments

a(n) is asymptotic to n^5/3125.

Examples

			a(29)=1540 because 29 = 11+7+5+4+2 and lcm(2,4,5,7,11) = 1540 is maximal.

Links

Crossrefs

Cf. A000793, A129651.
Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), A129648 (k=3), A129649 (k=4), this sequence (k=5), A355367 (k=6), A355403 (k=7).

Programs

  • Mathematica
    Max[LCM @@@ Compositions[ #, 5]] & /@ Range[1, n]

A355367 Maximal LCM of six positive integers with sum n.

Original entry on oeis.org

1, 2, 3, 6, 6, 12, 15, 30, 30, 60, 60, 84, 105, 210, 210, 420, 420, 420, 420, 840, 840, 1260, 1260, 2310, 2310, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 13860, 16380, 16380, 30030, 27720, 60060, 32760, 40040, 60060, 120120, 60060, 180180, 120120, 157080, 120120, 360360
Offset: 6

Views

Author

Wesley Ivan Hurt, Jun 29 2022

Keywords

Links

Crossrefs

Cf. A008881.
Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), A129648 (k=3), A129649 (k=4), A129650 (k=5), this sequence (k=6), A355403 (k=7).

Programs

  • Mathematica
    Table[Max[LCM@@@IntegerPartitions[n,{6}]],{n,6,60}] (* Harvey P. Dale, Jun 23 2023 *)
  • PARI
    a(n) = { my (v=0); forpart(p=n, v=max(v, lcm(Vec(p))),, [6,6]); v } \\ Rémy Sigrist, Jul 01 2022

A355403 Maximal LCM of seven positive integers with sum n.

Original entry on oeis.org

1, 2, 3, 6, 6, 12, 15, 30, 30, 60, 60, 84, 105, 210, 210, 420, 420, 420, 420, 840, 840, 1260, 1260, 2310, 2310, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 13860, 16380, 16380, 30030, 30030, 60060, 60060, 60060, 60060, 120120, 120120, 180180, 180180, 180180, 180180
Offset: 7

Views

Author

Wesley Ivan Hurt, Jun 30 2022

Keywords

Links

Crossrefs

Cf. A009641, A355368, A355402.
Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), A129648 (k=3), A129649 (k=4), A129650 (k=5), A355367 (k=6), this sequence (k=7).

A129651 a(n) is the smallest position k at which b_n(i)=k, where b_n(m) is the largest order of a permutation of m elements with exactly n cycles.

Original entry on oeis.org

1, 6, 37, 126, 287, 540, 895
Offset: 1

Views

Author

Nickolas Reynolds (nickels(AT)gmail.com), Apr 25 2007, Apr 26 2007, Apr 27 2007

Keywords

Examples

			a(2)=6 because b_2(6)=5 and b_2(i)<b_2(i+1) for all i>=6. (That is, the largest order of a permutation of i elements with exactly 2 cycles is monotonic increasing starting at i=6.)

Crossrefs

Cf. A000793, A129647, A129648, A129649, A129650.
Showing 1-7 of 7 results.

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

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