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 1-6 of 6 results.

A075352 Group the natural numbers so that the product of the terms of the n-th group is just >= n!: (1), (2), (3, 4), (5, 6), (7, 8, 9), (10, 11, 12), (13, 14, 15, 16), (17, 18, 19, 20), (21, 22, 23, 24, 25), (26, 27, 28, 29, 30), ... Sequence gives product of numbers in each group.

Original entry on oeis.org

1, 2, 12, 30, 504, 1320, 43680, 116280, 6375600, 17100720, 1402410240, 3776965920, 8835488640, 1022755734000, 2478652606080, 380634949094400, 945378254620800, 186127248627129600, 470489622136934400, 115590003914312928000
Offset: 1

Views

Author

Amarnath Murthy, Sep 19 2002

Keywords

Crossrefs

Cf. A075353, A075354, A075355, A075356.

Extensions

More terms from David Wasserman, Jan 16 2005

A075353 Initial term of n-th group in A075352.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 17, 21, 26, 31, 37, 43, 49, 56, 63, 71, 79, 88, 97, 107, 117, 128, 139, 151, 163, 175, 188, 201, 215, 229, 244, 259, 275, 291, 308, 325, 343, 361, 380, 399, 418, 438, 458, 479, 500, 522, 544, 567, 590, 614, 638, 663, 688, 714, 740, 767, 794
Offset: 1

Views

Author

Amarnath Murthy, Sep 19 2002

Keywords

Crossrefs

Cf. A075352, A075354, A075355, A075356.

Formula

a(n) = A075354(n-1)+1 = a(n-1)+A075355(n-1). - M. F. Hasler, Jul 19 2012

Extensions

More terms from David Wasserman, Jan 16 2005

A075354 Final term of n-th group in A075352.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 48, 55, 62, 70, 78, 87, 96, 106, 116, 127, 138, 150, 162, 174, 187, 200, 214, 228, 243, 258, 274, 290, 307, 324, 342, 360, 379, 398, 417, 437, 457, 478, 499, 521, 543, 566, 589, 613, 637, 662, 687, 713, 739, 766, 793
Offset: 1

Views

Author

Amarnath Murthy, Sep 19 2002

Keywords

Crossrefs

Cf. A075352, A075353, A075355, A075356.

Formula

a(n) = A075353(n+1)-1 = a(n-1)+A075355(n). - M. F. Hasler, Jul 19 2012

Extensions

More terms from David Wasserman, Jan 16 2005

A075355 Number of terms in n-th group in A075352.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 35, 36
Offset: 1

Views

Author

Amarnath Murthy, Sep 19 2002

Keywords

Comments

Differs from A008619 in the offset and from the 13th term a(13)=6 on: Integers that appear three times in a row are {6,12,19,27,35,...}. - M. F. Hasler, Jul 19 2012

Crossrefs

Cf. A075352, A075353, A075354, A075356.

Formula

a(n) = A075354(n)-A075353(n)+1. - David Wasserman, Jan 16 2005

Extensions

More terms from David Wasserman, Jan 16 2005

A235355 0 followed by the sum of (1),(2), (3,4),(5,6), (7,8,9),(10,11,12) from the natural numbers.

Original entry on oeis.org

0, 1, 2, 7, 11, 24, 33, 58, 74, 115, 140, 201, 237, 322, 371, 484, 548, 693, 774, 955, 1055, 1276, 1397, 1662, 1806, 2119, 2288, 2653, 2849, 3270, 3495, 3976, 4232, 4777, 5066, 5679, 6003, 6688, 7049, 7810, 8210, 9051, 9492, 10417, 10901, 11914, 12443, 13548
Offset: 0

Views

Author

Paul Curtz, Jan 07 2014

Keywords

Comments

Difference table for 0 followed by a(n):
0, 0, 1, 2, 7, 11, 24, 33,...
0, 1, 1, 5, 4, 13, 9, 25,... =A147685(n)
1, 0, 4, -1, 9, -4, 16, -9,... =interleave A000290(n+1),-A000290(n)
-1, 4, -5, 10, -13, 20, -25, 34,...
5, -9, 15, -23, 33, -45, 59, -75,... =(-1)^n*A027688(n+2).
a(-n) = -a(n-1).
From the second row, signature (0,3,0,-3,0,1).
Consider a(n+2k+1)+a(2k-n):
1, 2, 6, 9, 17, 22, 34,...
9, 12, 24, 33, 57, 72, 108,...
35, 40, 60, 75, 115, 140, 200,...
91, 98, 126, 147, 203, 238, 322,...
189, 198, 234, 261, 333, 378, 486,... .
The first column is A005898(n).
The rows are successively divisible by 2*k+1. Hence
1, 2, 6, 9, 17, 22, 34,...
3, 4, 8, 11, 19, 24, 36,...
7, 8, 12, 15, 23, 28, 40,...
13, 14, 18, 21, 29, 34, 46,...
21, 22, 26, 29, 37, 42, 54,...
The first column is A002061(n+1).
The main diagonal is A212965(n).
The first difference of every row is A022998(n+1).
Compare to the (2k+1)-sections of A061037 in A165943.

Examples

			a(1)=1, a(2)=2, a(3)=3+4=7, a(4)=5+6=11, a(5)=7+8+9=24, a(6)=10+11+12=33.

Links

Crossrefs

Cf. A075356, A234305.

Programs

  • Mathematica
    LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,1,2,7,11,24,33},50] (* Harvey P. Dale, Nov 22 2014 *)
  • PARI
    Vec(x*(x^2+1)*(x^2+x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Jan 20 2014

Formula

a(n) = 4*a(n-2) -6*a(n-4) +4*a(n-6) -a(n-8), n>7.
a(2n) = 0 followed by A085786(n). a(2n+1) = A081436(n).
a(2n) + a(2n+1) = A005898(n).
a(2n-1) + a(2n) = A061317(n).
a(n) = (-1)*((-1+(-1)^n-2*n)*(2+n+n^2))/16. a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7). G.f.: x*(x^2+1)*(x^2+x+1) / ((x-1)^4*(x+1)^3). - Colin Barker, Jan 20 2014

Extensions

More terms from Colin Barker, Jan 20 2014

A271647 Irregular triangle read by rows: the natural numbers from right to left.

Original entry on oeis.org

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

Views

Author

Paul Curtz, Apr 11 2016

Keywords

Comments

A permutation of the natural numbers. Mentioned as d(n) in A269837.
Difference table:
1, 2, 4, 3, 6, 5, 9, 8, 7, 12, 11, 10, 16, 15, 14, 13, 20, 19, 18, ...
1, 2, -1, 3, -1, 4, -1, -1, 5, -1, -1, 6, -1, -1, -1, 7, -1, -1, -1, ...
1, -3, 4, -4, 5, -5, 0, 6, -6, 0, 7, -7, 0, 0, 8, -8, 0, 0, 9, ...
etc.

Examples

			Irregular triangle:
1,
2,
4,   3,
6,   5,
9,   8,  7,
12, 11, 10,
16, 15, 14, 13,
20, 19, 18, 17,
25, 24, 23, 22, 21,
30, 29, 28, 27, 26,
etc.

Crossrefs

Cf. A000027, A002620, A024206, A033638, A075356, A235355, A269837, A271584

Programs

  • Maple
    count:= 0:
for r from 1 to 20 do
  d:= ceil(r/2);
  for i from 0 to d-1 do A[r,i]:= count+ d-i od;
  count:= count+d;
od:
seq(seq(A[r,i],i=0..ceil(r/2)-1),r=1..20); # Robert Israel, Apr 11 2016
  • Mathematica
    Table[Reverse@ Range[Floor[n/2]] + Floor[(n - 1)^2/4], {n, 16}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

Formula

With offset=0, a(n) = A271584(n) + A269837(n)
Empirical g.f. as triangle: (1-y*x^3+y^2*x^4-2*y*x^4-y^2*x^5+y*x^5+y^2*x^7)*x/((1+x)*(1-x)^3*(1-y*x^2)^3). - Robert Israel, Apr 11 2016
Showing 1-6 of 6 results.

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