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.

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A221216 T(n,k) = ((n+k)^2-2*(n+k)+4-(3*n+k-2)*(-1)^(n+k))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 6, 4, 3, 2, 12, 13, 14, 15, 11, 10, 9, 8, 7, 23, 24, 25, 26, 27, 28, 22, 21, 20, 19, 18, 17, 16, 38, 39, 40, 41, 42, 43, 44, 45, 37, 36, 35, 34, 33, 32, 31, 30, 29, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 80
Offset: 1

Views

Author

Boris Putievskiy, Feb 22 2013

Keywords

Comments

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). Let m be natural number. The order of the list:
T(1,1)=1;
T(3,1), T(2,2), T(1,3);
T(1,2), T(2,1);
. . .
T(2*m+1,1), T(2*m,2), T(2*m-1,3),...T(2,2*m), T(1,2*m+1);
T(1,2*m), T(2,2*m-1), T(3,2*m-2),...T(2*m-1,2),T(2*m,1);
. . .
First row contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.
Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)}, read downwards.

Examples

			The start of the sequence as table:
  1....5...4..12..11..23..22...
  6....3..13..10..24..21..39...
  2...14...9..25..20..40..35...
  15...8..26..19..41..34..60...
  7...27..18..42..33..61..52...
  28..17..43..32..62..51..85...
  16..44..31..63..50..86..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  5,6;
  4,3,2;
  12,13,14,15;
  11,10,9,8,7;
  23,24,25,26,27,28;
  22,21,20,19,18,17,16;
  . . .
Row number r consecutive contains r numbers.
If r is odd,  row is decreasing.
If r is even, row is increasing.

Links

Crossrefs

Cf. A211394, A221215, A002260, A004736, A003057;
table T(n,k) contains: in rows A084849, A096376, A014105, A014107, A168244, A033537, A100040, A100041;
in columns A130883, A000384, A014106, A033816, A100037, A091823, A071355, A100038, A100039,A130861;
main diagonal and parallel diagonals are A058331, A001844, A005893,A046092, A093328, A142463, A090288, A059993, A051890, A001105, A097080, A056220, A137882, A054000.

Programs

  • Python
    t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-2*(t+2)+4-(3*i+j-2)*(-1)**t)/2

Formula

As table
T(n,k) = ((n+k)^2-2*(n+k)+4-(3*n+k-2)*(-1)^(n+k))/2.
As linear sequence
a(n) = (A003057(n)^2-2*A003057(n)+4-(3*A002260(n)+A004736(n)-2)*(-1)^A003056(n))/2; a(n) = ((t+2)^2-2*(t+2)+4-(i+3*j-2)*(-1)^t)/2,
where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A221217 T(n,k) = ((n+k)^2-2*n+3-(n+k-1)*(1+2*(-1)^(n+k)))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 6, 5, 4, 3, 2, 15, 14, 13, 12, 11, 10, 9, 8, 7, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 91
Offset: 1

Views

Author

Boris Putievskiy, Feb 22 2013

Keywords

Comments

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). Let m be natural number. The order of the list:
T(1,1)=1;
T(3,1), T(2,2), T(1,3);
T(2,1), T(1,2);
. . .
T(2*m+1,1), T(2*m,2), T(2*m-1,3),...T(1,2*m+1);
T(2*m,1), T(2*m-1,2), T(2*m-2,3),...T(1,2*m);
. . .
First row contains antidiagonal {T(1,2*m+1), ... T(2*m+1,1)}, read upwards.
Second row contains antidiagonal {T(1,2*m), ... T(2*m,1)}, read upwards.

Examples

			The start of the sequence as table:
  1....6...4..15..11..28..22...
  5....3..14..10..27..21..44...
  2...13...9..26..20..43..35...
  12...8..25..19..42..34..63...
  7...24..18..41..33..62..52...
  23..17..40..32..61..51..86...
  16..39..31..60..50..85..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  6,5;
  4,3,2;
  15,14,13,12;
  11,10,9,8,7;
  28,27,26,25,24,23;
  22,21,20,19,18,17,16;
  . . .
Row number r consecutive contains r numbers in decreasing order.

Links

Crossrefs

Cf. A211394, A221215, A002260, A004736, A003057, A002024;
table T(n,k) contains: in rows A084849, A000384, A014106, A014105, A014107, A091823, A071355, A091823, A071355, A100040, A130861, A100041;
in columns A130883, A096376, A033816, A100037, A100038, A100039;
main diagonal and parallel diagonals are A058331, A051890, A005893, A097080, A093328, A137882, A001844, A001105, A056220, A142463, A054000, A090288, A059993.

Programs

  • Python
    t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-2*i+3-(t+1)*(1+2*(-1)**t))/2

Formula

As table
T(n,k) = ((n+k)^2-2*n+3-(n+k-1)*(1+2*(-1)^(n+k)))/2.
As linear sequence
a(n) = (A003057(n)^2-2*A002260(n)+3-A002024(n)*(1+2*(-1)^A003056(n)))/2;
a(n) = ((t+2)^2-2*i+3-(t+1)*(1+2*(-1)**t))/2, where i=n-t*(t+1)/2,
j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A274136 a(n) = (n+1)*(2*n+2)!/(n+2).

Original entry on oeis.org

1, 16, 540, 32256, 3024000, 410572800, 76281004800, 18598035456000, 5762136335155200, 2211729098342400000, 1030334000462807040000, 572721601599913328640000, 374484928188990947328000000, 284562454970932936468070400000
Offset: 0

Views

Author

Hong-Chang Wang, Jun 10 2016

Keywords

Comments

Sequence is inspired by A273889, A274119 and A273983.
Since Product_{i=0..n}(i*k+a) - Product_{i=0..n}(-i*k-b) ≡ 0 mod (n*k+a+b), then define B(n,k,a,b) = (Product_{i=0..n}(i*k+a) - Product_{i=0..n}(-i*k-b))/(n*k+a+b), with n*k+a+b <> 0, n >= 0 and k,a,b are integers, such that B(1,n,2,1) = (Product_{i=0..1}(i*n+2) - Product_{i=0..1}(-i*n-1))/(n+3) = A000012(n) with n >= 0;
B(3,n,2,1) = (Product_{i=0..3}(i*n+2) - Product_{i=0..3}(-i*n-1))/(3*n+3) = A100040(n+2) with n >= 0;
B(2*n+1,0,2,1) = (Product_{i=0..2*n+1}(2) - Product_{i=0..2*n+1}(-1))/3 = A002450(n+1) with n >= 0;
B(2*n+1,2,2,1) = (Product_{i=0..2*n+1}(2*i+2) - Product_{i=0..2*n+1}(-2*i-1))/(4*n+5) = A273983(n+1) with n >= 0;
and a(n) is B(2*n+1,1,2,1). - Hong-Chang Wang, Jun 17 2016

Examples

			a(0) = (2*3 - 1*2)/4 = 1.
a(1) = (2*3*4*5 - 1*2*3*4)/6 = 16.
a(2) = (2*3*4*5*6*7 - 1*2*3*4*5*6)/8 = 540.

Links

Crossrefs

Cf. A000012, A002450, A062779, A100040, A273889, A273983, A274119.

Programs

  • Mathematica
    a[n_] := (Product[i + 2, {i, 0, 2*n + 1}] - Product[-i - 1, {i, 0, 2*n + 1}])/(2*n + 4); Table[a[n], {n, 0, 10}] (* G. C. Greubel, Jun 19 2016 *)
Table[((n+1)(2n+2)!)/(n+2),{n,0,30}] (* Harvey P. Dale, Oct 24 2020 *)
  • PARI
    a(n) = ((2*n+1)!-(2*n)!)/(2*(n+1)) \\ Felix Fröhlich, Jun 11 2016
  • PARI
    a(n) = (prod(i=0, 2*n+1, i+2)-prod(i=0, 2*n+1, -i-1))/(2*n+4) \\ Felix Fröhlich, Jul 05 2016
  • Python
    # subroutine
def B (n, k, a, b):
    pa = pb = 1
    for i in range(n+1):
        pa *= (i*k+a)
        pb *= (-i*k-b)
    m = n*k+a+b
    p = pa-pb
    if m == 0:
        return "NaN"
    else:
        return p/m
# main program
for j in range(101):
    print(str(j)+" "+str(B(2*j+1, 1, 2, 1)))  # Hong-Chang Wang, Jun 14 2016

Formula

a(n) = B(2*n+1,1,2,1) = (Product_{i=0..2*n+1}(i+2) - Product_{i=0..2*n+1}(-i-1))/(2*n+4), n >= 0.
a(n) = A062779(n)/(2*(n+1)). - Michel Marcus, Jun 11 2016
a(n) ~ sqrt(Pi)*exp(-2*n)*(48*n*(24*n + 13) + 1177)*4^(n-2)*n^(2*n+1/2)/9. - Ilya Gutkovskiy, Jul 07 2016
Previous Showing 11-13 of 13 results.

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