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 1-4 of 4 results.

A047673 Row 4 of square array defined in A047671.

Original entry on oeis.org

1, 13, 61, 193, 481, 1021, 1933, 3361, 5473, 8461, 12541, 17953, 24961, 33853, 44941, 58561, 75073, 94861, 118333, 145921, 178081, 215293, 258061, 306913, 362401, 425101, 495613, 574561, 662593, 760381, 868621, 988033, 1119361, 1263373, 1420861, 1592641, 1779553
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047671.

Programs

  • PARI
    my(x='x+O('x^38)); Vec(x*(1+8*x+6*x^2+8*x^3+x^4)/(1-x)^5) \\ Elmo R. Oliveira, Aug 30 2025

Formula

a(n) = n^4 - 2*n^3 + 5*n^2 - 4*n + 1.
From Elmo R. Oliveira, Aug 30 2025: (Start)
G.f.: -x*(1 + 8*x + 6*x^2 + 8*x^3 + x^4)/(x-1)^5.
E.g.f.: -1 + (1 + 6*x^2 + 4*x^3 + x^4)*exp(x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)

Extensions

More terms from Elmo R. Oliveira, Aug 30 2025

A047672 Row 3 of square array defined in A047671.

Original entry on oeis.org

1, 10, 37, 94, 193, 346, 565, 862, 1249, 1738, 2341, 3070, 3937, 4954, 6133, 7486, 9025, 10762, 12709, 14878, 17281, 19930, 22837, 26014, 29473, 33226, 37285, 41662, 46369, 51418, 56821, 62590, 68737, 75274, 82213, 89566, 97345, 105562, 114229, 123358, 132961
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Programs

Formula

a(n) = 2*n^3 - 3*n^2 + 4*n - 2.
From Elmo R. Oliveira, Aug 28 2025: (Start)
G.f.: x*(1 + 6*x + 3*x^2 + 2*x^3)/(1-x)^4.
E.g.f.: 2 + exp(x)*(-2 + 3*x + 3*x^2 + 2*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. (End)

Extensions

More terms from Elmo R. Oliveira, Aug 28 2025

A047674 Row 5 of square array defined in A047671.

Original entry on oeis.org

1, 16, 91, 346, 1021, 2524, 5479, 10774, 19609, 33544, 54547, 85042, 127957, 186772, 265567, 369070, 502705, 672640, 885835, 1150090, 1474093, 1867468, 2340823, 2905798, 3575113, 4362616, 5283331, 6353506, 7590661, 9013636, 10642639, 12499294, 14606689, 16989424
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047671.

Programs

  • PARI
    my(x='x+O('x^35)); Vec(x*(1+10*x+10*x^2+20*x^3+5*x^4+2*x^5)/(1-x)^6) \\ Elmo R. Oliveira, Aug 30 2025

Formula

a(n) = (1/5)*(2*n^5 - 5*n^4 + 20*n^3 - 25*n^2 + 23*n - 10).
From Elmo R. Oliveira, Aug 30 2025: (Start)
G.f.: x*(1 + 10*x + 10*x^2 + 20*x^3 + 5*x^4 + 2*x^5)/(x-1)^6.
E.g.f.: 2 + (-10 + 15*x + 30*x^2 + 40*x^3 + 15*x^4 + 2*x^5)*exp(x)/5.
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) for n > 6. (End)

Extensions

More terms from Elmo R. Oliveira, Aug 30 2025

A027618 c(i,j) is cost of evaluation of edit distance of two strings with lengths i and j, when you use recursion (every call has a unit cost, other computations are free); sequence gives c(n,n).

Original entry on oeis.org

1, 4, 19, 94, 481, 2524, 13483, 72958, 398593, 2193844, 12146179, 67570078, 377393953, 2114900428, 11885772379, 66963572734, 378082854913, 2138752086628, 12118975586803, 68774144872414, 390815720696161, 2223564321341884
Offset: 0

Views

Author

Bruno Petazzoni (Bruno.Petazzoni(AT)ac-idf.jussieu.fr)

Keywords

References

  • Found by 7 students: Dufour, Hermon, Lesueur, Moynot, Schabanel, Sers and Wolf.

Links

Crossrefs

Delannoy numbers A008288, A001850 are given by c'(i, j)=(3c(i, j)-1)/2.

Programs

  • Mathematica
    Table[SeriesCoefficient[(3/Sqrt[1-6*x+x^2]-1/(1-x))/2,{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
  • PARI
    x='x+O('x^66); Vec((3/sqrt(1-6*x+x^2)-1/(1-x))/2) \\ Joerg Arndt, May 04 2013

Formula

c(n, n) where c(i, 0)=c(0, j)=1 and c(i+1, j+1)=1+c(i+1, j)+c(i, j+1)+c(i, j) (c(i, j) is A047671).
G.f.: (3/sqrt(1-6*x+x^2)-1/(1-x))/2.
Recurrence: n*(2*n-3)*a(n) = (2*n-1)*(7*n-10)*a(n-1) - (2*n-3)*(7*n-4)*a(n-2) + (n-2)*(2*n-1)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ 3*sqrt(8+6*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
Showing 1-4 of 4 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.