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.

A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 18, 6, 3, 0, 192, 48, 12, 4, 0, 2500, 500, 100, 20, 5, 0, 38880, 6480, 1080, 180, 30, 6, 0, 705894, 100842, 14406, 2058, 294, 42, 7, 0, 14680064, 1835008, 229376, 28672, 3584, 448, 56, 8, 0, 344373768, 38263752, 4251528, 472392, 52488, 5832, 648, 72, 9
Offset: 0

Views

Author

Alois P. Heinz, Aug 19 2013

Keywords

Examples

			T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,2], [2,1].
T(2,2) = 2: [1,1], [2,2].
T(3,1) = 18: [1,1,2], [1,1,3], [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3], [3,3,1], [3,3,2].
T(3,2) = 6: [1,2,2], [1,3,3], [2,1,1], [2,3,3], [3,1,1], [3,2,2].
T(3,3) = 3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
  1;
  0,        1;
  0,        2,       2;
  0,       18,       6,      3;
  0,      192,      48,     12,     4;
  0,     2500,     500,    100,    20,    5;
  0,    38880,    6480,   1080,   180,   30,   6;
  0,   705894,  100842,  14406,  2058,  294,  42,  7;
  0, 14680064, 1835008, 229376, 28672, 3584, 448, 56,  8;

Links

Crossrefs

Row sums give: A000312.
Columns k=0-4 give: A000007, A066274(n) = 2*A081131(n) for n>1, A053506(n) for n>2, A055865(n-1) = A085389(n-1) for n>3, A085390(n-1) for n>4.
Main diagonal gives: A028310.
Lower diagonals include (offsets may differ): A002378, A045991, A085537, A085538, A085539.
Cf. A228154, A228617.

Programs

  • Maple
    T:= (n, k)-> `if`(n=0 and k=0, 1, `if`(k<1 or k>n, 0,
             `if`(k=n, n, (n-1)*n^(n-k)))):
seq(seq(T(n,k), k=0..n), n=0..12);
  • Mathematica
    f[0,0]=1;
f[n_,k_]:=Which[1<=k<=n-1,n^(n-k)*(n-1),k<1,0,k==n,n,k>n,0];
Table[Table[f[n,k],{k,0,n}],{n,0,10}]//Grid (* Geoffrey Critzer, May 19 2014 *)

Formula

T(0,0) = 1, else T(n,k) = 0 for k<1 or k>n, else T(n,n) = n, else T(n,k) = (n-1)*n^(n-k).
Sum_{k=0..n} T(n,k) = A000312(n).
Sum_{k=0..n} k*T(n,k) = A031972(n).

A163285 Triangle read by rows in which row n lists n+1 terms, starting with n^5 and ending with n^6, such that the difference between successive terms is equal to n^5 - n^4.

Original entry on oeis.org

0, 1, 1, 32, 48, 64, 243, 405, 567, 729, 1024, 1792, 2560, 3328, 4096, 3125, 5625, 8125, 10625, 13125, 15625, 7776, 14256, 20736, 27216, 33696, 40176, 46656, 16807, 31213, 45619, 60025, 74431, 88837, 103243, 117649, 32768, 61440, 90112, 118784, 147456
Offset: 0

Views

Author

Omar E. Pol, Jul 24 2009

Keywords

Comments

The first term of row n is A000584(n) and the last term of row n is A001014(n).
The main entry for this sequence is A159797. See also A163282, A163283 and A163284.
Row sums give A163275. - Omar E. Pol, Mar 18 2012

Examples

			Triangle begins:
0;
1,1;
32,48,64;
243,405,567,729;
1024,1792,2560,3328,4096;
3125,5625,8125,10625,13125,15625;
7776,14256,20736,27216,33696,40176,46656;
16807,31213,45619,60025,74431,88837,103243,117649;
32768,61440,90112,118784,147456,176128,204800,233472,262144;
59049,111537,164025,216513,269001,321489,373977,426465,478953,531441;
100000,190000,280000,370000,460000,550000,640000,730000,820000,910000,1000000;

Links

Crossrefs

Cf. A000584, A001014, A085538, A159797, A162611, A162614, A162622, A163282, A163283, A163284.

Programs

  • Mathematica
    rw[n_]:=Range[n^5,n^6,n^5-n^4]; Join[{0,1},Flatten[Array[rw,10]]] (* Harvey P. Dale, Mar 18 2012 *)
  • PARI
    A163285(n, k)=n^5 +k*(n^5 -n^4) \\ G. C. Greubel, Dec 17 2016

A240930 a(n) = n^7 - n^6.

Original entry on oeis.org

0, 0, 64, 1458, 12288, 62500, 233280, 705894, 1835008, 4251528, 9000000, 17715610, 32845824, 57921708, 97883968, 159468750, 251658240, 386201104, 578207808, 846825858, 1216000000, 1715322420, 2380977984, 3256789558, 4395368448, 5859375000, 7722894400, 10072932714
Offset: 0

Views

Author

Martin Renner, Aug 03 2014

Keywords

Comments

For n>1 number of 7-digit positive integers in base n.

Links

Crossrefs

Cf. A001014, A001015.
Cf. A002378, A045991, A085537, A085538, A085539, A240931, A240932, A240933.

Programs

  • Magma
    [n^7-n^6 : n in [0..30]]; // Wesley Ivan Hurt, Aug 03 2014
  • Maple
    A240930:=n->n^7-n^6: seq(A240930(n), n=0..30); # Wesley Ivan Hurt, Aug 03 2014
  • Mathematica
    Table[n^7 - n^6, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 03 2014 *)
CoefficientList[Series[2 (32*x^2 + 473*x^3 + 1208*x^4 + 718*x^5 + 88*x^6 + x^7)/(x - 1)^8, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 03 2014 *)
  • PARI
    vector(100, n, (n-1)^7 - (n-1)^6) \\ Derek Orr, Aug 03 2014

Formula

a(n) = n^6*(n-1) = n^7 - n^6.
a(n) = A001015(n) - A001014(n).
G.f.: 2*(32*x^2 + 473*x^3 + 1208*x^4 + 718*x^5 + 88*x^6 + x^7)/(x - 1)^8. - Wesley Ivan Hurt, Aug 03 2014
Recurrence: a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*(n-6)+8*a(n-7)-a(n-8). - Wesley Ivan Hurt, Aug 03 2014
Sum_{n>=2} 1/a(n) = 6 - Sum_{k=2..6} zeta(k). - Amiram Eldar, Jul 05 2020

A240931 a(n) = n^8 - n^7.

Original entry on oeis.org

0, 0, 128, 4374, 49152, 312500, 1399680, 4941258, 14680064, 38263752, 90000000, 194871710, 394149888, 752982204, 1370375552, 2392031250, 4026531840, 6565418768, 10407740544, 16089691302, 24320000000, 36021770820, 52381515648, 74906159834, 105488842752, 146484375000
Offset: 0

Views

Author

Martin Renner, Aug 03 2014

Keywords

Comments

For n>1 number of 8-digit positive integers in base n.

Links

Crossrefs

Cf. A001015, A001016.
Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240932, A240933.

Programs

  • Magma
    [n^8-n^7 : n in [0..30]]; // Wesley Ivan Hurt, Aug 09 2014
  • Maple
    A240931:=n->n^8-n^7: seq(A240931(n), n=0..30); # Wesley Ivan Hurt, Aug 09 2014
  • Mathematica
    Table[n^8 - n^7, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 09 2014 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,0,128,4374,49152,312500,1399680,4941258,14680064},30] (* Harvey P. Dale, Apr 29 2016 *)
  • PARI
    vector(100, n, (n-1)^8 - (n-1)^7) \\ Derek Orr, Aug 03 2014
  • PARI
    concat([0,0], Vec(-2*x^2*(x^6+183*x^5+2682*x^4+8422*x^3+7197*x^2+1611*x+64) / (x-1)^9 + O(x^100))) \\ Colin Barker, Aug 08 2014

Formula

a(n) = n^7*(n-1) = n^8 - n^7.
a(n) = A001016(n) - A001015(n).
G.f.: -2*x^2*(x^6+183*x^5+2682*x^4+8422*x^3+7197*x^2+1611*x+64) / (x-1)^9. - Colin Barker, Aug 08 2014
Sum_{n>=2} 1/a(n) = 7 - Sum_{k=2..7} zeta(k). - Amiram Eldar, Jul 05 2020

A240932 a(n) = n^9 - n^8.

Original entry on oeis.org

0, 0, 256, 13122, 196608, 1562500, 8398080, 34588806, 117440512, 344373768, 900000000, 2143588810, 4729798656, 9788768652, 19185257728, 35880468750, 64424509440, 111612119056, 187339329792, 305704134738, 486400000000, 756457187220, 1152393344256, 1722841676182
Offset: 0

Views

Author

Martin Renner, Aug 03 2014

Keywords

Comments

For n>1 number of 9-digit positive integers in base n.

Links

Crossrefs

Cf. A001016, A001017.
Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240931, A240933.

Programs

Formula

a(n) = n^8*(n-1) = n^9 - n^8.
a(n) = A001017(n) - A001016(n).
G.f.: 2*x^2*(x^7+374*x^6+9327*x^5+49780*x^4+78095*x^3+38454*x^2+5281*x+128) / (x-1)^10. - Colin Barker, Aug 08 2014
Sum_{n>=2} 1/a(n) = 8 - Sum_{k=2..8} zeta(k). - Amiram Eldar, Jul 05 2020

A240933 a(n) = n^10 - n^9.

Original entry on oeis.org

0, 0, 512, 39366, 786432, 7812500, 50388480, 242121642, 939524096, 3099363912, 9000000000, 23579476910, 56757583872, 127253992476, 268593608192, 538207031250, 1030792151040, 1897406023952, 3372107936256, 5808378560022, 9728000000000, 15885600931620, 25352653573632
Offset: 0

Views

Author

Martin Renner, Aug 03 2014

Keywords

Comments

For n>1 number of 10-digit positive integers in base n.

Links

Crossrefs

Cf. A001017, A008454.
Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240931, A240932.

Programs

  • Magma
    [n^10-n^9 : n in [0..30]]; // Wesley Ivan Hurt, Aug 03 2014
  • Maple
    A240933:=n->n^10-n^9: seq(A240933(n), n=0..30); # Wesley Ivan Hurt, Aug 03 2014
  • Mathematica
    Table[n^10 - n^9, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 03 2014 *)
CoefficientList[Series[2 (256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 03 2014 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,0,512,39366,786432,7812500,50388480,242121642,939524096,3099363912,9000000000},40] (* Harvey P. Dale, Oct 19 2022 *)
  • PARI
    vector(100, n, (n-1)^10 - (n-1)^9) \\ Derek Orr, Aug 03 2014

Formula

a(n) = n^9*(n-1) = n^10 - n^9.
a(n) = A008454(n) - A001017(n). - Michel Marcus, Aug 03 2014
G.f.: 2*(256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11. - Wesley Ivan Hurt, Aug 03 2014
Recurrence: a(n) = 11*a(n-1)-55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+330*a(n-7)-165*a(n-8)+55*a(n-9)-11*a(n-10)+a(n-11). - Wesley Ivan Hurt, Aug 03 2014
Sum_{n>=2} 1/a(n) = 9 - Sum_{k=2..9} zeta(k). - Amiram Eldar, Jul 05 2020
Showing 1-6 of 6 results.

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