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.

A055651 Table T(m,k)=m^k-k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

Original entry on oeis.org

0, 1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 2, 0, -2, -1, 1, 3, 1, -1, -3, -1, 1, 4, 0, 0, 0, -4, -1, 1, 5, -7, -17, 17, 7, -5, -1, 1, 6, -28, -118, 0, 118, 28, -6, -1, 1, 7, -79, -513, -399, 399, 513, 79, -7, -1, 1, 8, -192, -1844, -2800, 0, 2800, 1844, 192, -8, -1, 1, 9, -431
Offset: 0

Views

Author

Henry Bottomley, Jun 07 2000

Keywords

Crossrefs

Rows A000012 (offset), A023443, A024012, A024026, A024040 and diagonals A000004, A007925, A046065, A055652.

Extensions

Title corrected by Sean A. Irvine, Mar 30 2022

A062275 Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 16, 3, 0, 0, 4, 72, 72, 4, 0, 0, 5, 256, 729, 256, 5, 0, 0, 6, 800, 5184, 5184, 800, 6, 0, 0, 7, 2304, 30375, 65536, 30375, 2304, 7, 0, 0, 8, 6272, 157464, 640000, 640000, 157464, 6272, 8, 0, 0, 9, 16384, 750141, 5308416, 9765625
Offset: 0

Views

Author

Henry Bottomley, Jul 02 2001

Keywords

Comments

Here 0^0 is defined to be 1. - Wolfdieter Lang, May 27 2018

Examples

			A(3, 2) = 3^2 * 2^3 = 9*8 = 72.
The array A(n, k) begins:
n\k 0 1   2   3    4     5      6      7       8        9       10 ...
0:  1 0   0   0    0     0      0      0       0        0        0 ...
1:  0 1   2   3    4     5      6      7       8        9       10 ...
2:  0 2  16  72  256   800   2304   6272   16384    41472   102400 ...
3:  0 3  72 729 5184 30375 157464 750141 3359232 14348907 59049000 ...
...
The triangle T(n, k) begins:
n\k  0  1    2      3      4      5      6    7  8  9 ...
0:   1
1:   0  0
2:   0  1    0
3:   0  2    2      0
4:   0  3   16      3      0
5:   0  4   72     72      4      0
6:   0  5  256    729    256      5      0
7:   0  6  800   5184   5184    800      6    0
8:   0  7 2304  30375  65536  30375   2304    7  0
9:   0  8 6272 157464 640000 640000 157464 6272  8  0
... - _Wolfdieter Lang_, May 22 2018

Links

Crossrefs

Columns and rows of A, or columns and diagonals of T, include A000007, A001477, A007758, A062074, A062075 etc. Diagonals of A include A062206, A051443, A051490. Sum of rows of T are A062817(n), for n >= 1
Cf. A055651, A055652, A303990.

Programs

  • Mathematica
    {{1}}~Join~Table[(#^k k^#) &[n - k], {n, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 24 2018 *)
  • PARI
    t1(n)=n-binomial(round(sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
a(n)=t1(n)^t2(n)*t2(n)^t1(n) \\ Eric Chen, Jun 09 2018

Formula

From Wolfdieter Lang, May 22 2018: (Start)
As a sequence: a(n) = A003992(n)*A004248(n).
As a triangle: T(n, k) = (n-k)^k * k^(n-k), for n >= 1 and k = 1..n. (End)

A093898 Triangle read by rows: T(n,r) = n^r + r^n (1 <= r <= n).

Original entry on oeis.org

2, 3, 8, 4, 17, 54, 5, 32, 145, 512, 6, 57, 368, 1649, 6250, 7, 100, 945, 5392, 23401, 93312, 8, 177, 2530, 18785, 94932, 397585, 1647086, 9, 320, 7073, 69632, 423393, 1941760, 7861953, 33554432, 10, 593, 20412, 268705, 2012174, 10609137, 45136576
Offset: 1

Views

Author

Amarnath Murthy, Apr 23 2004

Keywords

Examples

			2,
3, 8,
4, 17, 54,
5, 32, 145, 512,
6, 57, 368, 1649, 6250,
7, 100, 945, 5392, 23401, 93312,

Crossrefs

Same information as A055652. - Franklin T. Adams-Watters, Oct 26 2009

Programs

  • Maple
    T:=(n,r)->n^r+r^n: for n from 1 to 10 do seq(T(n,r),r=1..n) od; # yields sequence in triangular form # Emeric Deutsch, Feb 04 2006
  • Mathematica
    Flatten[Table[n^r+r^n,{n,10},{r,n}]] (* Harvey P. Dale, Jun 19 2011 *)

Extensions

More terms from Emeric Deutsch, Feb 04 2006

A109958 Concatenate n and the sum of primes dividing n (counting multiplicity).

Original entry on oeis.org

10, 22, 33, 44, 55, 65, 77, 86, 96, 107, 1111, 127, 1313, 149, 158, 168, 1717, 188, 1919, 209, 2110, 2213, 2323, 249, 2510, 2615, 279, 2811, 2929, 3010, 3131, 3210, 3314, 3419, 3512, 3610, 3737, 3821, 3916, 4011, 4141, 4212, 4343, 4415, 4511, 4625, 4747
Offset: 1

Views

Author

Jason Earls, Jul 06 2005

Keywords

Links

Crossrefs

Cf. A001414, A055652, A109959.

Programs

  • Maple
    f:= proc(n) local q;
q:= add(t[1]*t[2],t=ifactors(n)[2]);
10^(1+ilog10(q))*n+q
end proc:
f(1):= 10:
map(f, [$1..100]); # Robert Israel, Jun 26 2018
  • Mathematica
    pr[{a_,b_}]:=a*b;Join[{10},Table[FromDigits[Flatten[IntegerDigits[Join[{n},{Total[pr/@FactorInteger[n]]}]]]],{n,2,47}]] (* James C. McMahon, Apr 02 2024 *)

Formula

a(n) = 10^(A055652(A001414(n)))*n+A001414(n). - Robert Israel, Jun 26 2018

A156354 Triangle T(n, k) = k^(n-k) + (n-k)^k with T(0, 0) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 17, 17, 5, 1, 1, 6, 32, 54, 32, 6, 1, 1, 7, 57, 145, 145, 57, 7, 1, 1, 8, 100, 368, 512, 368, 100, 8, 1, 1, 9, 177, 945, 1649, 1649, 945, 177, 9, 1, 1, 10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10, 1, 1, 11, 593, 7073, 18785, 23401, 23401, 18785, 7073, 593, 11, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 08 2009

Keywords

Comments

This sequence is an approximation of Pascal's triangle with interior Kurtosis.
Essentially the same as A055652. - R. J. Mathar, Feb 19 2009

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,    1;
  1,  4,   8,    4,     1;
  1,  5,  17,   17,     5,     1;
  1,  6,  32,   54,    32,     6,     1;
  1,  7,  57,  145,   145,    57,     7,     1;
  1,  8, 100,  368,   512,   368,   100,     8,    1;
  1,  9, 177,  945,  1649,  1649,   945,   177,    9,   1;
  1, 10, 320, 2530,  5392,  6250,  5392,  2530,  320,  10,  1;
  1, 11, 593, 7073, 18785, 23401, 23401, 18785, 7073, 593, 11, 1;
The interior Kurtosis, T(n,k) - binomial(n, k), is:
  0;
  0, 0;
  0, 0,   0;
  0, 0,   0,    0;
  0, 0,   2,    0,     0;
  0, 0,   7,    7,     0,     0;
  0, 0,  17,   34,    17,     0,     0;
  0, 0,  36,  110,   110,    36,     0,     0;
  0, 0,  72,  312,   442,   312,    72,     0,    0;
  0, 0, 141,  861,  1523,  1523,   861,   141,    0,   0;
  0, 0, 275, 2410,  5182,  5998,  5182,  2410,  275,   0, 0;
  0, 0, 538, 6908, 18455, 22939, 22939, 18455, 6908, 538, 0, 0;

Links

Crossrefs

Cf. A026898.

Programs

  • Magma
    [k eq 0 select 1 else k^(n-k) + (n-k)^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 07 2021
  • Mathematica
    T[n_, k_]:= If[n==0, 1, (k^(n-k) + (n-k)^k)];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
  • Sage
    flatten([[1 if k==n else k^(n-k) + (n-k)^k for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 07 2021

Formula

T(n, k) = k^(n-k) + (n-k)^k with T(0, 0) = 1.
T(n, k) = T(n, n-k).
Sum_{k=0..n} T(n,k) = [n=0] + 2*A026898(n-1). - G. C. Greubel, Mar 07 2021

Extensions

Edited by G. C. Greubel, Mar 07 2021

A160814 a(1) = 1; a(n+1) = a(n)^n + n^a(n).

Original entry on oeis.org

1, 2, 8, 7073
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 26 2009

Keywords

Comments

Next term is too large to display: 2.3459495195697547514*10^4258
The next term (a(5)) has 4259 digits. - Harvey P. Dale, Jul 18 2021

Crossrefs

Cf. A093898, A055652, A076980. - R. J. Mathar, May 29 2009

Programs

  • Mathematica
    a=1;lst={};Do[a=a^n+n^a;AppendTo[lst,IntegerPart[a]],{n,0,4}];lst
nxt[{n_, a_}] := {n + 1, a^n + n^a}; NestList[nxt, {1, 1}, 4][[All, 2]] (* Harvey P. Dale, Jul 18 2021 *)

Extensions

Edited by N. J. A. Sloane, May 29 2009
Showing 1-6 of 6 results.

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

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