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 11-17 of 17 results.

A168096 a(n) = number of natural numbers m such that n - 6 <= m <= n + 6.

Original entry on oeis.org

6, 7, 8, 9, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13
Offset: 0

Views

Author

Jaroslav Krizek, Nov 18 2009

Keywords

Comments

Generalization: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see, e.g., A158799).

Links

Crossrefs

Cf. A000027.

Programs

  • Mathematica
    CoefficientList[Series[(6 - 5*x - x^8)/(1 - x)^2, {x, 0, 25}], x] (* G. C. Greubel, Jul 12 2016 *)
PadRight[{6,7,8,9,10,11,12},120,{13}] (* Harvey P. Dale, May 24 2022 *)

Formula

a(n) = 6 + n for 0 <= n <= 6, a(n) = 13 for n >= 7.
G.f.: (6 - 5*x - x^8)/(1-x)^2. - G. C. Greubel, Jul 12 2016

A168097 a(n) = number of natural numbers m such that n - 7 <= m <= n + 7.

Original entry on oeis.org

7, 8, 9, 10, 11, 12, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15
Offset: 0

Views

Author

Jaroslav Krizek, Nov 18 2009

Keywords

Comments

Generalization: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see, e.g., A158799).

Links

Crossrefs

Cf. A000027.

Programs

  • Mathematica
    CoefficientList[Series[(7 - 6*x - x^9)/(1 - x)^2, {x, 0, 25}], x] (* G. C. Greubel, Jul 12 2016 *)

Formula

a(n) = 7 + n for 0 <= n <= 7, a(n) = 15 for n >= 8.
G.f.: (7 - 6*x - x^9)/(1-x)^2. - G. C. Greubel, Jul 12 2016

A168098 a(n) = number of natural numbers m such that n - 8 <= m <= n + 8.

Original entry on oeis.org

8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17
Offset: 0

Views

Author

Jaroslav Krizek, Nov 18 2009

Keywords

Comments

Generalization: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see, e.g., A158799).

Links

Crossrefs

Cf. A000027.

Programs

  • Mathematica
    CoefficientList[Series[(8 - 7*x - x^10)/(1 - x)^2, {x, 0, 25}], x] (* G. C. Greubel, Jul 12 2016 *)

Formula

a(n) = 8 + n for 0 <= n <= 8, a(n) = 17 for n >= 9.
G.f.: (8 - 7*x - x^10)/(1 - x)^2. - G. C. Greubel, Jul 12 2016

A168099 a(n) = number of natural numbers m such that n - 9 <= m <= n + 9.

Original entry on oeis.org

9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19
Offset: 0

Views

Author

Jaroslav Krizek, Nov 18 2009

Keywords

Comments

Generalization: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see, e.g., A158799). a(n) = 9 + n for 0 <= n <= 9, a(n) = 19 for n >= 10.

Links

Programs

  • Mathematica
    CoefficientList[Series[(9 - 8*x - x^11)/(1 - x)^2, {x, 0, 25}], x] (* G. C. Greubel, Jul 12 2016 *)

Formula

G.f.: (9 - 8*x - x^11)/(1 - x)^2. - G. C. Greubel, Jul 12 2016

A168100 a(n) = number of natural numbers m such that n - 10 <= m <= n + 10.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21
Offset: 0

Views

Author

Jaroslav Krizek, Nov 18 2009

Keywords

Comments

Generalization: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see, e.g., A158799). a(n) = 10 + n for 0 <= n <= 10, a(n) = 21 for n >= 11.

Links

Programs

Formula

G.f.: (10 - 9*x - x^12)/(1 - x)^2. - G. C. Greubel, Jul 12 2016

A179995 Generating function A(5,t)(1+t+t^2)/(1-t)^6, where A(5,t) is an Eulerian polynomial.

Original entry on oeis.org

1, 33, 276, 1299, 4392, 11925, 27708, 57351, 108624, 191817, 320100, 509883, 781176, 1157949, 1668492, 2345775, 3227808, 4358001, 5785524, 7565667, 9760200, 12437733, 15674076, 19552599, 24164592, 29609625, 35995908
Offset: 0

Views

Author

Peter Luschny, Aug 05 2010

Keywords

Comments

The Eulerian polynomials A(n,t) are here defined in accordance with the Digital Library of Mathematical Functions, Table 26.14.1.
Sums of 3 consecutive fifth powers: a(n) = (n-1)^5+n^5+(n+1)^5. - Bruno Berselli, Jun 24 2013

Links

Crossrefs

Cf. A158799, A008486, A005918, A027602, A160827 which have generating functions of type A(n, t)(1+t+t^2)/(1-t)^(n+1).

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+27*x+93*x^2+118*x^3+93*x^4+27*x^5+x^6)/(1-x)^6)); // Bruno Berselli, Jun 24 2013
  • Maple
    gfA179995 := proc(t) local i;
add([1,27,93,118,93,27,1][i+1]*t^i,i=0..5)/(1-t)^6 end:
seq(coeff(series(gfA179995(t),t,24),t,j),j=0..16);
  • Mathematica
    Join[{1}, Table[n (3 n^4 + 20 n^2 + 10), {n, 30}]] (* Bruno Berselli, Jun 24 2013 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 33, 276, 1299, 4392, 11925, 27708}, 30] (* Harvey P. Dale, Apr 10 2015 *)

Formula

From Bruno Berselli, Jun 24 2013: (Start)
G.f.: (1 + 27*x + 93*x^2 + 118*x^3 + 93*x^4 + 27*x^5 + x^6) / (1 - x)^6.
a(n) = n*(3*n^4 + 20*n^2 + 10) for n>0, a(0)=1. (End)
a(0)=1, a(1)=33, a(2)=276, a(3)=1299, a(4)=4392, a(5)=11925, a(6)=27708; for n>6, a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, Apr 10 2015

A278597 One half of A278481.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2
Offset: 1

Views

Author

Omar E. Pol, Nov 23 2016

Keywords

Comments

Apart from the left border and the right border, the rest of the elements are 3's.

Examples

			The sequence written as a triangle begins:
                      1;
                    2,  2;
                  2,  3,  2;
                2,  3,  3,  2;
              2,  3,  3,  3,  2;
            2,  3,  3,  3,  3,  2;
          2,  3,  3,  3,  3,  3,  2;
        2,  3,  3,  3,  3,  3,  3,  2;
      2,  3,  3,  3,  3,  3,  3,  3,  2;
    2,  3,  3,  3,  3,  3,  3,  3,  3,  2;
  ...

Crossrefs

Row sums give A016777.
Left border gives A040000, the same as the right border.
Middle column gives A122553.
Every diagonal that is parallel to any of the borders gives the elements greater than 1 of A158799.
Cf. A278481.

Formula

a(n) = A278481(n)/2.
Previous Showing 11-17 of 17 results.

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

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