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.

A352697 a(n) = A037237(n-1) - A281434(n).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 6, 0, 3, 0, 6, 0, 4, 0, 6, 0, 5, 1, 7, 2, 6, 2, 7, 1, 8, 0, 8, 0, 8, 0, 11, 0, 9, 1, 10, 0, 11, 0, 11, 0, 11, 0, 14, 3, 12, 0, 13, 0, 13, 0, 15, 0, 15, 0, 15, 0, 16, 0, 18, 0, 16, 0, 17, 0, 17, 0, 19, 0, 18, 2, 19, 0, 19, 0, 20, 2
Offset: 1

Views

Author

Vladimir Reshetnikov, Mar 29 2022

Keywords

Comments

It appears that this sequence has an infinite number of zeros, i.e., A281434(n) agrees with A037237(n-1) at an infinite number of indices; also, it seems that a(n) = O(n).

Crossrefs

Cf. A037237, A281434.

Programs

  • Mathematica
    With[{m = 20}, Table[(4 n + 3 n^2 + 2 n^3)/3, {n, m}] - Length /@ Rest[NestList[Expand[D[#, x]] &, x^x^x, m]]]

A047662 Square array a(n,k) read by antidiagonals: a(n,1)=n, a(1,k)=k, a(n,k)=a(n-1,k-1)+a(n-1,k)+a(n,k-1)+1.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 31, 20, 5, 6, 30, 64, 64, 30, 6, 7, 42, 115, 160, 115, 42, 7, 8, 56, 188, 340, 340, 188, 56, 8, 9, 72, 287, 644, 841, 644, 287, 72, 9, 10, 90, 416, 1120, 1826, 1826, 1120, 416, 90, 10, 11, 110, 579, 1824, 3591
Offset: 1

Views

Author

Don Knuth, N. J. A. Sloane

Keywords

Examples

			The array begins:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, ...
3, 12, 31, 64, 115, 188, 287, 416, 579, 780, 1023, 1312, ...
4, 20, 64, 160, 340, 644, 1120, 1824, 2820, 4180, 5984, 8320, ...
5, 30, 115, 340, 841, 1826, 3591, 6536, 11181, 18182, 28347, 42652, ...
6, 42, 188, 644, 1826, 4494, 9912, 20040, 37758, 67122, 113652, 184652, ...
7, 56, 287, 1120, 3591, 9912, 24319, 54272, 112071, 216952, 397727, 696032, ...
8, 72, 416, 1824, 6536, 20040, 54272, 132864, 299208, 628232, 1242912, 2336672, ...
...
The first few antidiagonals are:
1,
2, 2,
3, 6, 3,
4, 12, 12, 4,
5, 20, 31, 20, 5,
6, 30, 64, 64, 30, 6,
7, 42, 115, 160, 115, 42, 7,
8, 56, 188, 340, 340, 188, 56, 8,
9, 72, 287, 644, 841, 644, 287, 72, 9,
10, 90, 416, 1120, 1826, 1826, 1120, 416, 90, 10,
...

Links

Crossrefs

Rows give A037237, 4*A006007, A047661, A047663, A047664, main diagonal is A047665 (see also A001850).
See also A008288, A048776.

Programs

  • Maple
    A047662 := proc(n,k) option remember; if n = 1 then k; elif k = 1 then n; else A047662(n-1,k-1)+A047662(n,k-1)+A047662(n-1,k)+1; fi; end;
  • Mathematica
    a[n_, 1] := n; a[1, k_] := k; a[n_, k_] := a[n, k] = a[n-1, k-1] + a[n-1, k] + a[n, k-1] + 1; Table[ a[n-k+1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 13 2013 *)

Formula

a(n, k) =(A008288(n, k)-1)/2. Sum of antidiagonals is A048776.

A185912 Accumulation array of A185910; by antidiagonals.

Original entry on oeis.org

1, 3, 5, 6, 12, 14, 10, 21, 31, 30, 15, 32, 51, 64, 55, 21, 45, 74, 102, 115, 91, 28, 60, 100, 144, 180, 188, 140, 36, 77, 129, 190, 250, 291, 287, 204, 45, 96, 161, 240, 325, 400, 441, 416, 285, 55, 117, 196, 294, 405, 515, 602, 636, 579, 385, 66, 140, 234, 352, 490, 636, 770, 864, 882, 780, 506, 78, 165, 275, 414, 580, 763, 945, 1100, 1194, 1185, 1023, 650, 91, 192, 319, 480, 675, 896, 1127, 1344, 1515, 1600, 1551, 1312, 819, 105
Offset: 1

Views

Author

Clark Kimberling, Feb 06 2011

Keywords

Comments

A member of the accumulation chain ... < A185910 < A185911 < A185912 < A185913 < ...
(See A144112 for definitions of weight array and accumulation array.)

Examples

			Northwest corner:
   1,   3,   6,  10,  15
   5,  12,  21,  32,  45
  14,  31,  51,  74, 100
  30,  64, 102, 144, 190

Links

Crossrefs

Cf. A144112, A185910, A185913.
Row 1 to 2: A000217, A028347.
Column 1 to 3: A000330, A037237, 3*A145066.

Programs

  • Mathematica
    f[n_, 0] := 0; f[0, k_] := 0;
f[n_, k_] := n^2 + k - 1;
s[n_, k_] := Sum[f[i, j], {i, 1, n}, {j, 1, k}];(*accumulation array of {f(n,k)}*)
FullSimplify[s[n, k]]  (*formula for A185812*)
Table[s[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten
T[n_, k_] := (k*n/6)*(2*n^2 + 3*n + 3*k - 2) ; Table[T[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 22 2017 *)

Formula

T(n,k) = (k*n/6)*(2*n^2 + 3*n + 3*k - 2), k >= 1, n >= 1.

A100504 a(n) = (4*n^3 + 6*n^2 + 8*n + 6)/3.

Original entry on oeis.org

2, 8, 26, 64, 130, 232, 378, 576, 834, 1160, 1562, 2048, 2626, 3304, 4090, 4992, 6018, 7176, 8474, 9920, 11522, 13288, 15226, 17344, 19650, 22152, 24858, 27776, 30914, 34280, 37882, 41728, 45826, 50184, 54810, 59712, 64898, 70376, 76154, 82240
Offset: 0

Views

Author

N. J. A. Sloane, Nov 24 2004

Keywords

Comments

Bisection of A000125.
This sequence is related to A002061 by a(n) = (n+1)*A002061(n+1) + Sum_{i=0..n} A002061(i). - Bruno Berselli, Dec 19 2013

Links

Crossrefs

Cf. A037237.

Programs

  • Magma
    I:=[2, 8, 26, 64]; [n le 4 select I[n] else 4*Self(n-1) -6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 26 2012
  • Mathematica
    CoefficientList[Series[2*(1+3x^2)/((1-x)^4),{x,0,40}],x] (* Vincenzo Librandi, Jun 26 2012 *)
LinearRecurrence[{4,-6,4,-1},{2,8,26,64},40] (* Harvey P. Dale, Dec 27 2015 *)
  • PARI
    a(n)=n*(4*n^2+6*n+8)/3+2 \\ Charles R Greathouse IV, Jan 18 2012
  • SageMath
    [2 + 2*n*(2*n^2+3*n+4)/3 for n in range(41)] # G. C. Greubel, Apr 03 2023

Formula

a(n) = a(n-1) + (2*n)^2 + 2. - Philippe Deléham, Jan 18 2012
From Vincenzo Librandi, Jun 26 2012: (Start)
G.f.: 2*(1+3*x^2)/(1-x)^4;
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
From G. C. Greubel, Apr 03 2023: (Start)
a(n) = 2 + 2*A037237(n-1).
E.g.f.: (2/3)*(3 + 9*x + 9*x^2 + 2*x^3)*exp(x). (End)

Extensions

More terms from Hugo Pfoertner, Nov 25 2004
New name based on formula from Ralf Stephan
Showing 1-4 of 4 results.

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

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