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 1-6 of 6 results.

A126885 T(n,k) = n*T(n,k-1) + k, with T(n,1) = 1, square array read by ascending antidiagonals (n >= 0, k >= 1).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 11, 10, 5, 1, 6, 18, 26, 15, 6, 1, 7, 27, 58, 57, 21, 7, 1, 8, 38, 112, 179, 120, 28, 8, 1, 9, 51, 194, 453, 543, 247, 36, 9, 1, 10, 66, 310, 975, 1818, 1636, 502, 45, 10, 1, 11, 83, 466, 1865, 4881, 7279, 4916, 1013, 55, 11
Offset: 0

Views

Author

Gary W. Adamson, Dec 30 2006

Keywords

Examples

			Square array begins:
  n\k | 1   2   3   4    5     6      7       8 ...
  -------------------------------------------------
    0 | 1   2   3   4    5     6      7       8 ... A000027
    1 | 1   3   6  10   15    21     28      36 ... A000217
    2 | 1   4  11  26   57   120    247     502 ... A000295
    3 | 1   5  18  58  179   543   1636    4916 ... A000340
    4 | 1   6  27 112  453  1818   7279   29124 ... A014825
    5 | 1   7  38 194  975  4881  24412  122068 ... A014827
    6 | 1   8  51 310 1865 11196  67183  403106 ... A014829
    7 | 1   9  66 466 3267 22875 160132 1120932 ... A014830
    8 | 1  10  83 668 5349 42798 342391 2739136 ... A014831
    ...

Crossrefs

Antidiagonal sums are A134195.
Main diagonal gives A062805.
Cf. A000027, A000217, A000295, A000340, A014825, A014827, A014829, A014830, A014831.

Programs

  • Maxima
    T(n, k) := if k = 1 then 1 else n*T(n, k - 1) + k$
create_list(T(n - k + 1, k), n, 0, 20, k, 1, n + 1);
/* Franck Maminirina Ramaharo, Jan 26 2019 */

Formula

T(1,k) = k*(k + 1)/2, and T(n,k) = (k - (k + 1)*n + n^(k + 1))/(n^2 - 2*n + 1) elsewhere.
T(n,k) = third entry in the vector M^k * (1, 0, 0), where M is the following 3 X 3 matrix:
1, 0, 0
1, 1, 0
1, 1, n.

Extensions

Edited and name clarified by Franck Maminirina Ramaharo, Jan 26 2019

A048440 Take the first n numbers written in base 8, concatenate them, then convert from base 8 to base 10.

Original entry on oeis.org

1, 10, 83, 668, 5349, 42798, 342391, 21913032, 1402434057, 89755779658, 5744369898123, 367639673479884, 23528939102712589, 1505852102573605710, 96374534564710765455, 6167970212141488989136, 394750093577055295304721, 25264005988931538899502162
Offset: 1

Views

Author

Patrick De Geest, May 15 1999

Keywords

Comments

83 is the only prime in this sequence among the first 3000 terms (email from Kurt Foster, Oct 24 2015). - N. J. A. Sloane, Oct 25 2015

Examples

			a(9): (1)(2)(3)(4)(5)(6)(7)(10)(11) = 12345671011_8 = 1402434057.

Links

Crossrefs

Cf. A014831.
Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: this sequence, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

Programs

  • Magma
    [n eq 1 select 1 else Self(n-1)*8^(1+Ilog(8, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012
  • Mathematica
    If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 8]]]; Table[AppendTo[n, IntegerDigits[w, 8]]; n=Flatten[n]; FromDigits[n, 8], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
Table[FromDigits[Flatten[IntegerDigits[#,8]&/@Range[n]],8],{n,20}] (* Harvey P. Dale, Dec 07 2012 *)
  • Python
    from functools import reduce
def A048440(n): return reduce(lambda i,j:(i<<3*(1+(j.bit_length()-1)//3))+j,range(n+1)) # Chai Wah Wu, Feb 26 2023

A353099 a(1) = 7; for n>1, a(n) = 8 * a(n-1) + 8 - n.

Original entry on oeis.org

7, 62, 501, 4012, 32099, 256794, 2054353, 16434824, 131478591, 1051828726, 8414629805, 67317038436, 538536307483, 4308290459858, 34466323678857, 275730589430848, 2205844715446775, 17646757723574190, 141174061788593509, 1129392494308748060
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2022

Keywords

Links

Crossrefs

Cf. A064617, A353094, A353095, A353096, A353097, A353098, A353100.
Cf. A014831.

Programs

  • Mathematica
    LinearRecurrence[{10, -17, 8}, {7, 62, 501}, 20] (* Amiram Eldar, Apr 23 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(x*(7-8*x)/((1-x)^2*(1-8*x)))
  • PARI
    a(n) = (6*8^(n+1)+7*n-48)/49;
  • PARI
    b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
a(n) = b(n, 8);

Formula

G.f.: x * (7 - 8 * x)/((1 - x)^2 * (1 - 8 * x)).
a(n) = 10*a(n-1) - 17*a(n-2) + 8*a(n-3).
a(n) = 6 * A014831(n) + n.
a(n) = (6*8^(n+1) + 7*n - 48)/49.
a(n) = Sum_{k=0..n-1} (8 - n + k)*8^k.
E.g.f.: exp(x)*(48*(exp(7*x) - 1) + 7*x)/49. - Stefano Spezia, May 29 2023

A145729 Partial sums of A052379.

Original entry on oeis.org

0, 8, 80, 664, 5344, 42792, 342384, 2739128, 21913088, 175304776, 1402438288, 11219506392, 89756051232, 718048409960, 5744387279792, 45955098238456, 367640785907776, 2941126287262344, 23529010298098896, 188232082384791320
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Oct 17 2008

Keywords

Links

Crossrefs

Cf. A052379.

Programs

  • Magma
    [8*(8^(n+1)-7*n-8)/49 : n in [0..20]]; // Vincenzo Librandi, Jul 04 2011
  • Mathematica
    lst={};s=0;Do[s+=(s+=(s+=(s+=n)));AppendTo[lst,s],{n,0,5!}];lst
  • PARI
    concat(0, Vec(-8*x/((x-1)^2*(8*x-1)) + O(x^100))) \\ Colin Barker, Oct 27 2014

Formula

a(n) = sum_{i=0..n-1} A052379(i).
a(n) = 8*(8^(n+1)-7*n-8)/49 = 8*A014831(n) = 2*A145730(n). - R. J. Mathar, Oct 21 2008
a(n) = 10*a(n-1)-17*a(n-2)+8*a(n-3). G.f.: -8*x / ((x-1)^2*(8*x-1)). - Colin Barker, Oct 27 2014

Extensions

Edited by R. J. Mathar, Oct 21 2008

A145730 Partial sums of A108019.

Original entry on oeis.org

0, 4, 40, 332, 2672, 21396, 171192, 1369564, 10956544, 87652388, 701219144, 5609753196, 44878025616, 359024204980, 2872193639896, 22977549119228, 183820392953888, 1470563143631172, 11764505149049448, 94116041192395660
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Oct 17 2008

Keywords

Links

Crossrefs

Cf. A108019, A145729.

Programs

  • Mathematica
    lst={};s=0;Do[s+=(s+=(s+=n+s));AppendTo[lst,s],{n,0,5!}];lst
Accumulate[NestList[8#+4&,0,20]] (* or *) LinearRecurrence[{10,-17,8},{0,4,40},20] (* Harvey P. Dale, Aug 08 2013 *)
  • PARI
    concat(0, Vec(-4*x/((x-1)^2*(8*x-1)) + O(x^100))) \\ Colin Barker, Oct 28 2014

Formula

a(n) = Sum_{i=0..n} A108019(i).
a(n) = 4*(8^(n+1)-7n-8)/49 = 4*A014831(n). - R. J. Mathar, Oct 21 2008
a(0)=0, a(1)=4, a(2)=40, a(n)=10*a(n-1)-17*a(n-2)+8*a(n-3). - Harvey P. Dale, Aug 08 2013
a(n) = A145729(n)/2. G.f.: -4*x / ((x-1)^2*(8*x-1)). - Colin Barker, Oct 28 2014

Extensions

Edited by R. J. Mathar, Oct 21 2008

A014855 Numbers k that divide s(k), where s(1)=1, s(j)=8*s(j-1)+j.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 12, 14, 18, 20, 24, 28, 36, 40, 42, 49, 52, 54, 56, 60, 72, 84, 98, 100, 104, 108, 114, 120, 126, 136, 140, 156, 162, 168, 180, 196, 200, 216, 220, 228, 252, 260, 280, 294, 300, 312, 324, 342, 343, 360, 364, 378, 392, 408, 420, 438, 440, 444
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Crossrefs

s(n) = A014831(n).
Showing 1-6 of 6 results.

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

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