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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A130886 4n^4 + 3n^3 + 2n^2 + n + 1.

Original entry on oeis.org

1, 11, 99, 427, 1253, 2931, 5911, 10739, 18057, 28603, 43211, 62811, 88429, 121187, 162303, 213091, 274961, 349419, 438067, 542603, 664821, 806611, 969959, 1156947, 1369753, 1610651, 1882011, 2186299, 2526077, 2904003
Offset: 0

Views

Author

Mohammad K. Azarian, Jul 26 2007

Keywords

Links

Crossrefs

Cf. A084849, A056109, A056105.

Programs

  • Magma
    [4*n^4+3*n^3+2*n^2+n+1: n in [0..40]]; // Vincenzo Librandi, Feb 12 2013
  • Mathematica
    CoefficientList[Series[(1 + 6 x + 54 x^2 + 32 x^3 + 3 x^4)/(1 - x)^5, {x, 0, 35}], x] (* Vincenzo Librandi, Feb 12 2013 *)

Formula

G.f.: (1 + 6*x + 54*x^2 + 32*x^3 + 3*x^4)/(1 - x)^5. - Vincenzo Librandi, Feb 12 2013

A131466 a(n) = 5n^4 - 4n^3 + 3n^2 - 2n + 1.

Original entry on oeis.org

1, 3, 57, 319, 1065, 2691, 5713, 10767, 18609, 30115, 46281, 68223, 97177, 134499, 181665, 240271, 312033, 398787, 502489, 625215, 769161, 936643, 1130097, 1352079, 1605265, 1892451, 2216553, 2580607, 2987769, 3441315
Offset: 0

Views

Author

Mohammad K. Azarian, Jul 26 2007

Keywords

Links

Crossrefs

Cf. A084849, A056109, A056105, A056578, A056579.

Programs

  • Mathematica
    Table[5n^4-4n^3+3n^2-2n+1,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,3,57,319,1065},30] (* Harvey P. Dale, Nov 01 2020 *)
  • PARI
    a(n)=5*n^4-4*n^3+3*n^2-2*n+1 \\ Charles R Greathouse IV, Oct 21 2022

Formula

From Chai Wah Wu, Nov 13 2018: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: (-15*x^4 - 54*x^3 - 52*x^2 + 2*x - 1)/(x - 1)^5. (End)

A131901 2*A002024 - A131821.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 7, 7, 4, 5, 9, 9, 9, 5, 6, 11, 11, 11, 11, 6, 7, 13, 13, 13, 13, 13, 7, 8, 15, 15, 15, 15, 15, 15, 8, 9, 17, 17, 17, 17, 17, 17, 17, 9, 10, 19, 19, 19, 19, 19, 19, 19, 19, 10
Offset: 1

Views

Author

Gary W. Adamson, Jul 26 2007

Keywords

Comments

Row sums = A084849: (1, 4, 11, 22, 37, ...).

Examples

			First few rows of the triangle:
  1;
  2,  2;
  3,  5,  3;
  4,  7,  7,  4;
  5,  9,  9,  9,  5;
  6, 11, 11, 11, 11,  6;
  7, 13, 13, 13, 13, 13,  7;
  ...

Crossrefs

Cf. A002024, A131821, A084849.

Formula

2*A002024 - A131821 as infinite lower triangular matrices.

A185669 a(n) = 4*n^2 + 3*n + 2.

Original entry on oeis.org

2, 9, 24, 47, 78, 117, 164, 219, 282, 353, 432, 519, 614, 717, 828, 947, 1074, 1209, 1352, 1503, 1662, 1829, 2004, 2187, 2378, 2577, 2784, 2999, 3222, 3453, 3692, 3939, 4194, 4457, 4728, 5007, 5294, 5589, 5892, 6203, 6522, 6849, 7184, 7527, 7878, 8237, 8604, 8979, 9362, 9753, 10152, 10559, 10974, 11397, 11828
Offset: 0

Views

Author

Paul Curtz, Feb 09 2011

Keywords

Comments

Natural numbers A000027 written clockwise as a square spiral:
.
43--44--45--46--47--48--49
|
42 21--22--23--24--25--26
| | |
41 20 7---8---9--10 27
| | | | |
40 19 6 1---2 11 28
| | | | | |
39 18 5---4---3 12 29
| | | |
38 17--16--15--14--13 30
| |
37--36--35--34--33--32--31
.
Walking in straight lines away from the center:
1, 2, 11, ... = A054552(n) = 1 -3*n+4*n^2,
1, 8, 23, ... = A033951(n) = 1 +3*n+4*n^2,
1, 3, 13, ... = A054554(n+1) = 1 -2*n-4*n^2,
1, 7, 21, ... = A054559(n+1) = 1 +2*n+4*n^2,
1, 4, 15, ... = A054556(n+1) = 1 -n+4*n^2,
1, 6, 19, ... = A054567(n+1) = 1 +n+4*n^2,
1, 5, 17, ... = A053755(n) = 1 +4*n^2,
1, 9, 25, ... = A016754(n) = 1 +4*n+4*n^2 = (1+2*n)^2,
2, 8, 22, ... = 2*A084849(n) = 2 +2*n+4*n^2,
2, 12, 30, ... = A002939(n+1) = 2 +6*n+4*n^2,
2, 9, 24, ... = a(n) = 2 +3*n+4*n^2,
2, 10, 26, ... = A069894(n) = 2 +4*n+4*n^2,
3, 11, 27, ... = A164897(n) = 3 +4*n+4*n^2,
3, 12, 29, ... = A054552(n+1)+1 = 3 +5*n+4*n^2,
3, 14, 33, ... = A033991(n+1) = 3 +7*n+4*n^2,
3, 15, 35, ... = A000466(n+1) = 3 +8*n+4*n^2,
4, 14, 32, ... = 2*A130883(n+1) = 4 +6*n+4*n^2,
4, 16, 36, ... = A016742(n+1) = 4 +8*n+4*n^2 = (2+2*n)^2,
5, 18, 39, ... = A007742(n+1) = 5 +9*n+4*n^2,
5, 19, 41, ... = A125202(n+2) = 5+10*n+4*n^2.

Links

Programs

Formula

a(n) = a(n-1) + 8*n - 1.
a(n) = 2*a(n-1) - a(n-2) + 8.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (2 +3*x +3*x^2)/(1-x)^3 . - R. J. Mathar, Feb 11 2011
a(n) = A033954(n) + 2. - Bruno Berselli, Apr 10 2011
E.g.f.: (4*x^2 + 7*x + 2)*exp(x). - G. C. Greubel, Jul 09 2017

A199855 Inverse permutation to A210521.

Original entry on oeis.org

1, 4, 2, 5, 3, 6, 11, 7, 12, 8, 13, 9, 14, 10, 15, 22, 16, 23, 17, 24, 18, 25, 19, 26, 20, 27, 21, 28, 37, 29, 38, 30, 39, 31, 40, 32, 41, 33, 42, 34, 43, 35, 44, 36, 45, 56, 46, 57, 47, 58, 48, 59, 49, 60, 50, 61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66, 79
Offset: 1

Views

Author

Boris Putievskiy, Feb 04 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). The order of the list:
T(1,1)=1;
T(2,1), T(2,2), T(1,2), T(1,3), T(3,1),
...
T(2,n-1), T(4,n-3), T(6,n-5), ..., T(n,1),
T(2,n), T(4,n-2), T(6,n-4), ..., T(n,2),
T(1,n), T(3,n-2), T(5,n-4), ..., T(n-1,2),
T(1,n+1), T(3,n-1), T(5,n-3), ..., T(n+1,1),
...
The order of the list elements of adjacent antidiagonals. Let m be a positive integer.
Movement by antidiagonal {T(1,2*m), T(2*m,1)} from T(2,2*m-1) to T(2*m,1) length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(2,2*m) to T(2*m,2) length of step is 2,
movement by antidiagonal {T(1,2*m), T(2*m,1)} from T(1,2*m) to T(2*m-1,2) length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(1,2*m+1) to T(2*m+1,1) length of step is 2.
Table contains:
row 1 is alternation of elements A001844 and A084849,
row 2 is alternation of elements A130883 and A058331,
row 3 is alternation of elements A051890 and A096376,
row 4 is alternation of elements A033816 and A005893,
row 6 is alternation of elements A100037 and A093328;
row 5 accommodates elements A097080 in odd places,
row 7 accommodates elements A137882 in odd places,
row 10 accommodates elements A100038 in odd places,
row 14 accommodates elements A100039 in odd places;
column 1 is A093005 and alternation of elements A000384 and A001105,
column 2 is alternation of elements A046092 and A014105,
column 3 is A105638 and alternation of elements A014106 and A056220,
column 4 is alternation of elements A142463 and A014107,
column 5 is alternation of elements A091823 and A054000,
column 6 is alternation of elements A090288 and |A168244|,
column 8 is alternation of elements A059993 and A033537;
column 7 accommodates elements A071355 in odd places,
column 9 accommodates elements |A147973| in even places,
column 10 accommodates elements A139570 in odd places,
column 13 accommodates elements A130861 in odd places.

Examples

			The start of the sequence as table:
   1,  4,  5,  11,  13,  22,  25,  37,  41,  56,  61, ...
   2,  3,  7,   9,  16,  19,  29,  33,  46,  51,  67, ...
   6, 12, 14,  23,  26,  38,  42,  57,  62,  80,  86, ...
   8, 10, 17,  20,  30,  34,  47,  52,  68,  74,  93, ...
  15, 24, 27,  39,  43,  58,  63,  81,  87, 108, 115, ...
  18, 21, 31,  35,  48,  53,  69,  75,  94, 101. 123, ...
  28, 40, 44,  59,  64,  82,  88, 109, 116, 140, 148, ...
  32, 36, 49,  54,  70,  76,  95, 102, 124, 132, 157, ...
  45, 60, 65,  83,  89, 110, 117, 141, 149, 176, 185, ...
  50, 55, 71,  77,  96, 103, 125, 133, 158, 167, 195, ...
  66, 84, 90, 111, 118, 142, 150, 177, 186, 216, 226, ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   4,  2;
   5,  3,  6;
  11,  7, 12,  8;
  13,  9, 14, 10, 15;
  22, 16, 23, 17, 24, 18;
  25, 19, 26, 20, 27, 21, 28;
  37, 29, 38, 30, 39, 31, 40, 32;
  41, 33, 42, 34, 43, 35, 44, 36, 45;
  56, 46, 57, 47, 58, 48, 59, 49, 60, 50;
  61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66;
  ...
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of  triangle array, located above.
Last  2*r-1 numbers are from the row number 2*r-1 of  triangle array, located above.
   1;
   4, 2, 5, 3, 6;
  11, 7,12, 8,13, 9,14,10,15;
  22,16,23,17,24,18,25,19,26,20,27,21,28;
  37,29,38,30,39,31,40,32,41,33,42,34,43,35,44,36,45;
  56,46,57,47,58,48,59,49,60,50,61,51,62,52,63,53,64,54,65,55,66;
  ...
Row number r contains permutation numbers 4*r-3 from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-3*r+2,2*r*r-5*r+4, 2*r*r-3*r+3, 2*r*r-5*r+5, 2*r*r-3*r+4, 2*r*r-5*r+6, ..., 2*r*r-3*r+1, 2*r*r-r.
...

Links

Crossrefs

Cf. A210521, A001844, A084849, A130883, A058331, A051890, A096376, A033816, A005893, A100037, A093328, A097080, A137882, A100038, A100039, A093005, A000384, A001105, A046092, A014105, A105638, A014106, A056220, A142463, A014107, A091823, A054000, A090288, A168244, A059993, A033537, A071355, A147973, A139570, A130861.

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=(2*j**2+(4*i-5)*j+2*i**2-3*i+2+(2+(-1)**j)*((1-(t+1)*(-1)**i)))/4

Formula

T(n,k) = (2*k^2+(4*n-5)*k+2*n^2-3*n+2+(2+(-1)^k)*((1-(k+n-1)*(-1)^i)))/4.
a(n) = (2*j^2+(4*i-5)*j+2*i^2-3*i+2+(2+(-1)^j)*((1-(t+1)*(-1)^i)))/4, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((sqrt(8*n-7) - 1)/2).

A202606 a(n) = ceiling(((10^n - 1)^2/9 + 10^n)/18).

Original entry on oeis.org

1, 2, 67, 6217, 617717, 61732717, 6172882717, 617284382717, 61728399382717, 6172839549382717, 617283951049382717, 61728395066049382717, 6172839506216049382717, 617283950617716049382717, 61728395061732716049382717, 6172839506172882716049382717
Offset: 0

Views

Author

Arkadiusz Wesolowski, Dec 21 2011

Keywords

Comments

a(n) are distinct primes for n = 1 to 8.

Examples

			a(2) = 67 because (99^2/9 + 100)/18 = 66.05555....

Links

Crossrefs

Cf. A000217, A002275, A084849.

Programs

  • Magma
    [ ((10^n-1)^2/9+10^n-1)/18+1 : n in [0..15]];
  • Maple
    seq(((10^n-1)^2/9+10^n-1)/18+1, n=0..15);
  • Mathematica
    Table[a = (10^n - 1)/18; 2*a^2 + a + 1, {n, 0, 15}]
LinearRecurrence[{111,-1110,1000},{1,2,67},20] (* Harvey P. Dale, Jul 07 2017 *)
  • PARI
    for(n=0, 15, print1(((10^n-1)^2/9+10^n-1)/18+1, ", "))

Formula

a(n) = ceiling(((10^n - 1)^2/9 + 10^n)/18).
a(n) = (10^n - 1)*((10^n - 1)/9 + 1)/18 + 1.
G.f.: (1 - 109*x + 955*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)).

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).

A131465 a(n)=4n^4-3n^3+2n^2-n+1.

Original entry on oeis.org

1, 3, 47, 259, 861, 2171, 4603, 8667, 14969, 24211, 37191, 54803, 78037, 107979, 145811, 192811, 250353, 319907, 403039, 501411, 616781, 751003, 906027, 1083899, 1286761, 1516851, 1776503, 2068147, 2394309, 2757611, 3160771
Offset: 0

Views

Author

Mohammad K. Azarian, Jul 26 2007

Keywords

Links

Crossrefs

Cf. A084849, A056109, A056105.

Programs

  • Mathematica
    Table[4n^4-3n^3+2n^2-n+1,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,3,47,259,861},40] (* Harvey P. Dale, May 27 2017 *)
  • PARI
    a(n)=4*n^4-3*n^3+2*n^2-n+1 \\ Charles R Greathouse IV, Oct 21 2022

A278072 Riordan array(1/(1+x), (1-sqrt(1-4*x))/(2*x)).

Original entry on oeis.org

1, -1, 1, 1, 1, 1, -1, 4, 3, 1, 1, 10, 11, 5, 1, -1, 32, 37, 22, 7, 1, 1, 100, 128, 88, 37, 9, 1, -1, 329, 444, 341, 171, 56, 11, 1, 1, 1101, 1558, 1297, 739, 294, 79, 13, 1, -1, 3761, 5514, 4891, 3069, 1406, 465, 106, 15, 1, 1, 13035, 19680, 18365, 12435, 6346, 2442, 692, 137, 17, 1
Offset: 0

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Author

Peter Luschny, Nov 22 2016

Keywords

Examples

			[   1]
[  -1,   1]
[   1,   1,   1]
[  -1,   4,   3,   1]
[   1,  10,  11,   5,   1]
[  -1,  32,  37,  22,   7,  1]
[   1, 100, 128,  88,  37,  9,  1]
[  -1, 329, 444, 341, 171, 56, 11, 1]

Crossrefs

Cf. A033297, A084849.

Programs

  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1+#)&, (1-Sqrt[1-4#])/(2#)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)
  • Sage
    # uses[riordan_array from A256893]
riordan_array(1/(1+x), (1-sqrt(1-4*x))/(2*x), 8)
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