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 1-3 of 3 results.

A110449 Triangle read by rows: T(n,k) = n*((2*k+1)*n+1)/2, 0<=k<=n.

Original entry on oeis.org

0, 1, 2, 3, 7, 11, 6, 15, 24, 33, 10, 26, 42, 58, 74, 15, 40, 65, 90, 115, 140, 21, 57, 93, 129, 165, 201, 237, 28, 77, 126, 175, 224, 273, 322, 371, 36, 100, 164, 228, 292, 356, 420, 484, 548, 45, 126, 207, 288, 369, 450, 531, 612, 693, 774, 55, 155, 255, 355, 455, 555, 655, 755, 855, 955, 1055
Offset: 0

Views

Author

Reinhard Zumkeller, Jul 21 2005

Keywords

Comments

Row sums give A110450; central terms give A110451;
T(n,0) = A000217(n);
T(n,1) = A005449(n) for n>0;
T(n,2) = A005475(n) for n>1;
T(n,3) = A022265(n) for n>2;
T(n,4) = A022267(n) for n>3;
T(n,5) = A022269(n) for n>4;
T(n,6) = A022271(n) for n>5;
T(n,7) = A022263(n) for n>6;
T(n+1,n-1) = A059270(n) for n>1;
T(n,n-1) = A081436(n) for n>1;
T(n,n) = A085786(n).

Examples

			Triangle starts:
0;
1, 2;
3, 7, 11;
6, 15, 24, 33;
10, 26, 42, 58, 74;
...

Links

Crossrefs

Cf. A126890.

Programs

  • Mathematica
    Table[n*((2*k + 1)*n + 1)/2, {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 23 2017 *)
  • PARI
    tabl(nn) = {for (n=0, nn, for (k=0, n, print1(n*((2*k+1)*n+1)/2, ", ");); print(););} \\ Michel Marcus, Jun 22 2015

Formula

T(n,k) = n*((2*k + 1)*n + 1)/2, 0 <= k <= n.

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

A102296 a(n) = (1/6)*(n+1)*(10*n^2 + 17*n + 12).

Original entry on oeis.org

2, 13, 43, 102, 200, 347, 553, 828, 1182, 1625, 2167, 2818, 3588, 4487, 5525, 6712, 8058, 9573, 11267, 13150, 15232, 17523, 20033, 22772, 25750, 28977, 32463, 36218, 40252, 44575, 49197, 54128, 59378, 64957, 70875, 77142, 83768, 90763, 98137
Offset: 0

Views

Author

Creighton Dement, Feb 19 2005

Keywords

Comments

A floretion-generated sequence which arises from a particular transform of the centered square numbers: A001844. Specifically, (a(n)) is the jesfor-transform of the sequence A001844 with respect to the floretion given in the program code. The sequence relates centered square numbers, triangular numbers and centered octahedral numbers to (n+1)^3. Note: this was made possible in part by the formula already given for A085786.
Floretion Algebra Multiplication Program, FAMP Code: 4jesforseq[ + .25'j + .25'k + .25j' + .25k' + .25'ij' + .25'ik' + .25'ji' + .25'ki' + e ], vesforseq = A001844, ForType: 1A, LoopType: tes.

Links

Crossrefs

Cf. A001844, A001845, A085786, A000217.

Programs

  • Magma
    [(1/6)*(n+1)*(10*n^2+17*n+12): n in [0..50]]; // Vincenzo Librandi, May 30 2011
  • Mathematica
    LinearRecurrence[{4, -6, 4, -1}, {2, 13, 43, 102}, 50] (* Paolo Xausa, Mar 09 2024 *)
  • PARI
    a(n) = (n+1)*(10*n^2+17*n+12)/6

Formula

G.f.: (x+1)*(3x+2)/(x-1)^4;
a(n) = 2*A001844(n+1)A001845(n+1)_0%20+%20A085786(n+1)_1%20(%20%22">0 - 2*A001845(n+1)_0 + A085786(n+1)_1 ( "" denotes offset ) (n+1)^3 = 2*A001845(n+1) - 2*A001844(n+1) - A000217(n+1) - a(n).
Showing 1-3 of 3 results.

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