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 21-30 of 55 results. Next

A212930 T(n,k) = number of n X k 0..k-1 arrays with no column j greater than column j-1 in all rows.

Original entry on oeis.org

1, 3, 1, 10, 15, 1, 35, 568, 63, 1, 126, 39695, 18226, 255, 1, 462, 4431876, 14177855, 518320, 1023, 1, 1716, 724082352, 23124921876, 4041974015, 14230810, 4095, 1, 6435, 163050236504, 68264066143602, 85800824609376, 1075113010175, 386357608
Offset: 1

Views

Author

R. H. Hardin, May 31 2012

Keywords

Comments

Table starts
.1....3.......10............35................126......................462
.1...15......568.........39695............4431876................724082352
.1...63....18226......14177855........23124921876...........68264066143602
.1..255...518320....4041974015.....85800824609376......4051316109991426752
.1.1023.14230810.1075113010175.285912852294921876.207406617181155352354002

Examples

			Some solutions for n=3 k=4
..2..0..1..0....1..0..0..0....3..1..1..1....1..0..2..3....2..2..1..0
..2..0..1..2....0..3..1..2....3..2..3..0....2..1..1..0....0..3..0..0
..1..3..0..0....1..0..1..3....1..1..1..1....3..3..0..1....1..1..1..3

Links

Crossrefs

Column 2 is A024036.
Row 1 is A001700(n-1).
Cf. A212943.

Formula

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 5*a(n-1) -4*a(n-2)
k=3: a(n) = 37*a(n-1) -279*a(n-2) +243*a(n-3)
k=4: a(n) = 405*a(n-1) -43860*a(n-2) +1524160*a(n-3) -15636480*a(n-4) +14155776*a(n-5)
The coefficient of a(n-1) is A209671(k) (through at least k=1..7)

A366602 Number of divisors of 4^n-1.

Original entry on oeis.org

2, 4, 6, 8, 8, 24, 8, 16, 32, 48, 16, 96, 8, 64, 96, 32, 8, 512, 8, 192, 144, 128, 16, 768, 128, 128, 160, 256, 64, 4608, 8, 128, 384, 128, 512, 8192, 32, 128, 192, 768, 32, 9216, 32, 1024, 4096, 512, 64, 6144, 32, 8192, 1536, 1024, 64, 10240, 3072, 2048, 384
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(4)=8 because 4^4-1 has divisors {1, 3, 5, 15, 17, 51, 85, 255}.

Links

Crossrefs

Cf. A024036, A000005, A057957, A295501, A366603, A366604.
Cf. A046798, A046801, A366575, A366612, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

Programs

  • Maple
    a:=n->numtheory[tau](4^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[0,4^Range[100]-1] (* Paolo Xausa, Oct 14 2023 *)
  • PARI
    a(n) = numdiv(4^n-1);

Formula

a(n) = sigma0(4^n-1) = A000005(A024036(n)).
a(n) = A046801(2*n) = A046798(n) * A046801(n). - Max Alekseyev, Jan 07 2024

A366603 Sum of the divisors of 4^n-1.

Original entry on oeis.org

4, 24, 104, 432, 1536, 8736, 22528, 111456, 473600, 1999872, 5909760, 38054016, 89522176, 462274560, 2015330304, 7304603328, 22907191296, 166290432000, 366506672128, 2220409884672, 7645340651520, 29833839544320, 95821839806976, 648494317126656
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(4)=432 because 4^4-1 has divisors {1, 3, 5, 15, 17, 51, 85, 255}.

Links

Crossrefs

Cf. A024036, A000203, A057957, A069061, A075708, A295501, A366602, A366604.
Cf. A075708, A366576, A366613, A366622, A366634, A366653, A366662, A102146, A366684, A366710.

Programs

  • Maple
    a:=n->numtheory[sigma](4^n-1):
seq(a(n), n=1..100);
  • Mathematica
    DivisorSigma[1,4^Range[30]-1] (* Paolo Xausa, Oct 14 2023 *)

Formula

a(n) = sigma(4^n-1) = A000203(A024036(n)).
a(n) = A069061(n) * A075708(n). - Robert Israel, Nov 22 2023

A024140 a(n) = 12^n - 1.

Original entry on oeis.org

0, 11, 143, 1727, 20735, 248831, 2985983, 35831807, 429981695, 5159780351, 61917364223, 743008370687, 8916100448255, 106993205379071, 1283918464548863, 15407021574586367, 184884258895036415
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

In base 12 these are 0, B, BB, BBB, ... . - David Rabahy, Dec 12 2016

Links

Crossrefs

Cf. A001021, A016125, A178248.
Cf. Similar sequences of the type k^n-1: A000004 (k=1), A000225 (k=2), A024023 (k=3), A024036 (k=4), A024049 (k=5), A024062 (k=6), A024075 (k=7), A024088 (k=8), A024101 (k=9), A002283 (k=10), A024127 (k=11), this sequence (k=12).

Programs

  • Mathematica
    12^Range[0,20]-1 (* or *) LinearRecurrence[{13,-12},{0,11},20] (* Harvey P. Dale, Feb 01 2019 *)

Formula

From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-12*x) - 1/(1-x).
E.g.f.: exp(12*x) - exp(x). (End)
a(n) = 12*a(n-1) + 11 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = Sum_{i=1..n} 11^i*binomial(n,n-i) for n>0, a(0)=0. - Bruno Berselli, Nov 11 2015
From Elmo R. Oliveira, Dec 15 2023: (Start)
a(n) = 13*a(n-1) - 12*a(n-2) for n>1.
a(n) = A001021(n)-1 = A178248(n)-2.
a(n) = 11*(A016125(n) - 1)/12. (End)

A166920 a(n) = 2^n - (1 + (-1)^n)/2.

Original entry on oeis.org

0, 2, 3, 8, 15, 32, 63, 128, 255, 512, 1023, 2048, 4095, 8192, 16383, 32768, 65535, 131072, 262143, 524288, 1048575, 2097152, 4194303, 8388608, 16777215, 33554432, 67108863, 134217728, 268435455, 536870912, 1073741823, 2147483648, 4294967295
Offset: 0

Views

Author

Paul Curtz, Oct 23 2009

Keywords

Comments

Partial sums of A014551. The inverse binomial transform yields a sequence 0,2,-1,5,-7,17,...: zero followed by a sign alternating A014551.
The table of a(n) plus higher order differences in successive rows shows A131577 on the main diagonal.
a(n) = 2^n when n is odd and 2^n-1 when n is even. - Wesley Ivan Hurt, Nov 15 2013

Links

Crossrefs

Cf. A004171, A014551, A024036, A131577, A168361.

Programs

Formula

G.f.: x*(2-x)/((1-x)*(1-2*x)*(1+x)).
a(n) = 2^n - (1+(-1)^n)/2.
a(2*n) = A024036(n); a(2*n+1) = A004171(n).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n+1) - 2*a(n) = A168361(n).
a(n) = A000225(n+1) - A051049(n) = A014551(n) - A168361(n).
E.g.f.: exp(2*x) - cosh(x). - G. C. Greubel, May 28 2016
a(n) = Sum_{k=1..n+1} Sum_{i=0..n+1} C(n-k,i). - Wesley Ivan Hurt, Sep 22 2017
a(n) = 2*A001045(n) + A000975(n-1) for n>0. - Yuchun Ji, Aug 30 2018

Extensions

Edited and extended by R. J. Mathar, Mar 02 2010

A366604 Number of distinct prime divisors of 4^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 3, 6, 6, 5, 3, 8, 3, 7, 6, 7, 4, 9, 7, 7, 6, 8, 6, 11, 3, 7, 8, 7, 9, 12, 5, 7, 7, 9, 5, 12, 5, 10, 11, 9, 6, 12, 5, 12, 10, 10, 6, 12, 11, 11, 8, 9, 6, 15, 3, 8, 11, 9, 9, 14, 5, 10, 8, 15, 6, 17, 6, 10, 13, 11, 10, 16, 5
Offset: 1

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A024036, A001221, A057957, A133801, A366602, A366603.
Cf. A046800, A133801, A366611, A366620, A366632, A366651, A366660, A102347, A366681, A366707.

Programs

  • Mathematica
    PrimeNu[4^Range[100]-1] (* Paolo Xausa, Oct 14 2023 *)
  • PARI
    for(n = 1, 100, print1(omega(4^n - 1), ", "))
  • Python
    from sympy import primenu
def A366604(n): return primenu((1<<(n<<1))-1) # Chai Wah Wu, Oct 15 2023

Formula

a(n) = omega(4^n-1) = A001221(A024036(n)).
a(n) = A046800(2*n) = A046799(n) + A046800(n). - Max Alekseyev, Jan 07 2024

A142705 Numerator of 1/4 - 1/(2n)^2.

Original entry on oeis.org

0, 3, 2, 15, 6, 35, 12, 63, 20, 99, 30, 143, 42, 195, 56, 255, 72, 323, 90, 399, 110, 483, 132, 575, 156, 675, 182, 783, 210, 899, 240, 1023, 272, 1155, 306, 1295, 342, 1443, 380, 1599, 420, 1763, 462, 1935, 506, 2115, 552, 2303, 600, 2499, 650, 2703, 702
Offset: 1

Views

Author

Paul Curtz, Sep 24 2008

Keywords

Comments

Read modulo 10 (the last digits), a sequence with period length 10 results: 0, 3, 2, 5, 6, 5, 2, 3, 0, 9. Read modulo 9, a sequence with period length 18 results.
Denominators are in A154615.
a(n) is the numerator of (n-1)*(n+1)/4. - Altug Alkan, Apr 19 2018

Links

Crossrefs

Essentially the same as A070260. Cf. A078371 (second bisection of A061037), A142888 (first differences), A154615 (denominators), A225948.

Programs

  • Magma
    [-(3/4)*(-1)^n*n-(3/8)*(-1)^n*n^2+(5/8)*n^2+(5/4)*n: n in [0..60]]; // Vincenzo Librandi, Jul 02 2011
  • Mathematica
    Numerator[Table[(1/4)*(1 - 1/n^2), {n,1,50}]] (* G. C. Greubel, Jul 20 2017 *)
  • PARI
    for(n=1, 50, print1(numerator((1/4)*(1 - 1/n^2)), ", ")) \\ G. C. Greubel, Jul 20 2017
  • PARI
    a(n) = if(n%2,(n^2-1)/4,n^2-1); \\ Altug Alkan, Apr 19 2018

Formula

a(n) = A061037(2*n).
a(n) = A070260(n-1), n>1.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
a(2^(n-1)) = a(1+A000225(n-1)) = 4^(n-1)-1 = A024036(n-1).
G.f.: x^2*(3+2x+6x^2-x^4)/(1-x^2)^3. - R. J. Mathar, Oct 24 2008
E.g.f.: 1 + (1/4)*((4*x^2 + x - 4)*cosh(x) + (x^2 + 4*x -1)*sinh(x)). - G. C. Greubel, Jul 20 2017
Sum_{n>=2} 1/a(n) = 3/2. - Amiram Eldar, Aug 11 2022

Extensions

Edited by R. J. Mathar, Oct 24 2008

A058896 a(n) = 4^n - 4.

Original entry on oeis.org

-3, 0, 12, 60, 252, 1020, 4092, 16380, 65532, 262140, 1048572, 4194300, 16777212, 67108860, 268435452, 1073741820, 4294967292, 17179869180, 68719476732, 274877906940, 1099511627772, 4398046511100, 17592186044412, 70368744177660, 281474976710652, 1125899906842620
Offset: 0

Views

Author

Henry Bottomley, Jan 08 2001

Keywords

Links

Crossrefs

Cf. A000302, A000918, A002450, A024036, A052548, A058809, A141725, A277799.

Programs

Formula

a(n) = A000302(n) - 4 = 4*a(n-1) + 12 = 4*A024036(n-1) = 12*A002450(n-1).
G.f.: 3*(5*x - 1)/(1 - x)/(1 - 4*x).
a(n) = A000918(n)*A052548(n). - Reinhard Zumkeller, Feb 14 2009
From Elmo R. Oliveira, Nov 16 2023 (Start)
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
E.g.f.: exp(4*x) - 4*exp(x). (End)

A198693 a(n) = 3*4^n-1.

Original entry on oeis.org

2, 11, 47, 191, 767, 3071, 12287, 49151, 196607, 786431, 3145727, 12582911, 50331647, 201326591, 805306367, 3221225471, 12884901887, 51539607551, 206158430207, 824633720831, 3298534883327, 13194139533311, 52776558133247
Offset: 0

Views

Author

Vincenzo Librandi, Oct 29 2011

Keywords

Links

Crossrefs

Cf. A024036, A097743, A114569, A156760.

Programs

  • Magma
    [3*4^n-1: n in [0..30]]
  • Mathematica
    3*4^Range[0,30]-1 (* or *) LinearRecurrence[{5,-4},{2,11},30] (* Harvey P. Dale, Jul 04 2017 *)

Formula

a(n) = 4*a(n-1)+3.
a(n) = 5*a(n-1) - 4*a(n-2), for n > 1.
G.f.: (2+x)/((4*x-1)*(x-1)). - R. J. Mathar, Oct 30 2011

A304336 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/(k!)^2, triangle read by rows, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 15, 10, 0, 1, 63, 140, 35, 0, 1, 255, 1470, 1050, 126, 0, 1, 1023, 14080, 21945, 6930, 462, 0, 1, 4095, 130130, 400400, 252252, 42042, 1716, 0, 1, 16383, 1184820, 6861855, 7747740, 2438436, 240240, 6435
Offset: 0

Views

Author

Peter Luschny, May 11 2018

Keywords

Examples

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,     3;
[3] 0, 1,    15,      10;
[4] 0, 1,    63,     140,      35;
[5] 0, 1,   255,    1470,    1050,     126;
[6] 0, 1,  1023,   14080,   21945,    6930,     462;
[7] 0, 1,  4095,  130130,  400400,  252252,   42042,   1716;
[8] 0, 1, 16383, 1184820, 6861855, 7747740, 2438436, 240240, 6435;

Crossrefs

Row sums are A304338, T(n,n) = A088218 and A001700, T(n,n-1) ~ A002803, T(n,2) ~ A024036, T(n,3) ~ bisection of A174395.
Cf. A304330, A304334.

Programs

  • Maple
    A304336 := (n, k) -> add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k)/(k!)^2:
for n from 0 to 8 do seq(A304336(n, k), k=0..n) od;
  • PARI
    T(n, k) = sum(j=0, k, (-1)^j*binomial(2*k, j)*(k - j)^(2*n))/(k!)^2;
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 11 2018

Formula

T(n, k) = A304330(n, k)/(k!)^2.
T(n, k) = A304334(n, k)/k!.
Previous Showing 21-30 of 55 results. Next

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

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