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.

A253397 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with every 2X2 subblock antidiagonal maximum minus diagonal minimum nondecreasing horizontally and diagonal maximum minus antidiagonal minimum nondecreasing vertically.

Original entry on oeis.org

16, 44, 44, 96, 102, 96, 180, 143, 143, 180, 304, 197, 174, 197, 304, 476, 250, 246, 246, 250, 476, 704, 320, 316, 346, 316, 320, 704, 996, 391, 419, 465, 465, 419, 391, 996, 1360, 477, 520, 632, 666, 632, 520, 477, 1360, 1804, 564, 651, 823, 932, 932, 823, 651
Offset: 1

Views

Author

R. H. Hardin, Dec 31 2014

Keywords

Comments

Table starts
...16..44..96..180..304..476..704...996..1360..1804..2336..2964..3696..4540
...44.102.143..197..250..320..391...477...564...666...769...887..1006..1140
...96.143.174..246..316..419..520...651...780...939..1096..1283..1468..1683
..180.197.246..346..465..632..823..1071..1351..1695..2079..2535..3039..3623
..304.250.316..465..666..932.1269..1693..2201..2814..3527..4360..5309..6394
..476.320.419..632..932.1318.1855..2528..3408..4498..5864..7521..9542.11949
..704.391.520..823.1269.1855.2726..3810..5311..7163..9569.12493.16140.20493
..996.477.651.1071.1693.2528.3810..5396..7717.10593.14543.19463.25921.33918
.1360.564.780.1351.2201.3408.5311..7717.11392.15966.22500.30675.41701.55452
.1804.666.939.1695.2814.4498.7163.10593.15966.22634.32533.44959.62402.84560

Examples

			Some solutions for n=4 k=4
..1..1..1..1..1....0..0..0..0..0....0..0..0..1..1....0..1..0..1..1
..1..1..1..1..1....0..0..0..0..1....0..0..0..0..0....1..1..0..0..0
..1..1..1..1..1....0..0..0..0..1....0..0..0..0..1....1..1..1..1..1
..1..1..1..1..0....0..0..0..0..1....0..0..1..0..1....1..0..0..0..0
..0..1..1..1..1....0..0..0..0..1....1..0..1..0..1....1..1..1..1..1

Links

Crossrefs

Column 1 is A217873(n+1)

Formula

Empirical for column k:
k=1: a(n) = (4/3)*n^3 + 4*n^2 + (20/3)*n + 4
k=2: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4) for n>8
k=3: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4) for n>8
k=4: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5) for n>11
k=5: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5) for n>13
k=6: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6) for n>16
k=7: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6) for n>18
Empirical quasipolynomials for column k:
k=2: polynomial of degree 2 plus a quasipolynomial of degree 0 with period 2 for n>4
k=3: polynomial of degree 2 plus a quasipolynomial of degree 0 with period 2 for n>4
k=4: polynomial of degree 3 plus a quasipolynomial of degree 0 with period 2 for n>6
k=5: polynomial of degree 3 plus a quasipolynomial of degree 0 with period 2 for n>8
k=6: polynomial of degree 4 plus a quasipolynomial of degree 0 with period 2 for n>10
k=7: polynomial of degree 4 plus a quasipolynomial of degree 0 with period 2 for n>12

A210440 a(n) = 2*n*(n+1)*(n+2)/3.

Original entry on oeis.org

0, 4, 16, 40, 80, 140, 224, 336, 480, 660, 880, 1144, 1456, 1820, 2240, 2720, 3264, 3876, 4560, 5320, 6160, 7084, 8096, 9200, 10400, 11700, 13104, 14616, 16240, 17980, 19840, 21824, 23936, 26180, 28560, 31080, 33744, 36556, 39520, 42640, 45920, 49364, 52976
Offset: 0

Views

Author

Michel Marcus, Jan 20 2013

Keywords

Comments

Number of tin boxes necessary to build a tetrahedron with side length n, as shown in the link.
If "0" is prepended, a(n) is the convolution of 2n with itself. - Wesley Ivan Hurt, Mar 14 2015

Links

Crossrefs

Cf. A000292, A028552, A033488 (partial sums), A046092, A130809.

Programs

Formula

a(n) = 4*A000292(n).
a(n+1)-a(n) = A046092(n+1).
From Bruno Berselli, Jan 20 2013: (Start)
G.f.: 4*x/(1-x)^4.
a(n) = -a(-n-2) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).
a(n)-a(-n) = A217873(n).
a(n)+a(-n) = A016742(n).
(n-1)*a(n)-n*a(n-1) = A130809(n+1) with n>1. (End)
From Bruno Berselli, Jan 21 2013: (Start)
a(n) = n*A028552(n) - Sum_{i=0..n-1} A028552(i) for n>0.
4*A001296(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n>0. (End)
G.f.: 2*x*W(0), where W(k) = 1 + 1/(1 - x*(k+2)*(k+4)/(x*(k+2)*(k+4) + (k+1)*(k+2)/W(k+1))); (continued fraction). - Sergei N. Gladkovskii, Aug 24 2013
a(n) = Sum_{i=1..n} i*(2*n-i+3). - Wesley Ivan Hurt, Oct 03 2013
From Amiram Eldar, Apr 30 2023: (Start)
Sum_{n>=1} 1/a(n) = 3/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2) - 15/8. (End)
E.g.f.: 2*exp(x)*x*(6 + 6*x + x^2)/3. - Stefano Spezia, Jul 11 2025

A289522 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=0} ((1 + x^(2*j+1))/(1 - x^(2*j+1)))^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 8, 4, 0, 1, 8, 18, 16, 6, 0, 1, 10, 32, 44, 32, 8, 0, 1, 12, 50, 96, 102, 56, 12, 0, 1, 14, 72, 180, 256, 216, 96, 16, 0, 1, 16, 98, 304, 550, 624, 428, 160, 22, 0, 1, 18, 128, 476, 1056, 1512, 1408, 816, 256, 30, 0, 1, 20, 162, 704, 1862, 3240, 3820, 3008, 1494, 404, 40, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 07 2017

Keywords

Examples

			Square array begins:
1,  1,   1,    1,    1,     1,  ...
0,  2,   4,    6,    8,    10,  ...
0,  2,   8,   18,   32,    50,  ...
0,  4,  16,   44,   96,   180,  ...
0,  6,  32,  102,  256,   550,  ...
0,  8,  56,  216,  624,  1512,  ...

Crossrefs

Columns k=0-6 give: A000007, A080054, A007096, A261647, A014969, A261648, A014970.
Rows n=0-3 give: A000012, A005843, A001105, A217873.
Main diagonal gives A291697.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i + 1))/(1 - x^(2 i + 1)))^k, {i, 0, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

G.f. of column k: Product_{j>=0} ((1 + x^(2*j+1))/(1 - x^(2*j+1)))^k.
G.f. of column 2k: (theta_3(x)/theta_4(x))^k, where theta_() is the Jacobi theta function.
For asymptotics of column k see comment from Vaclav Kotesovec in A261648.

A292022 a(n) = 4*n*(n^2 + 2).

Original entry on oeis.org

12, 48, 132, 288, 540, 912, 1428, 2112, 2988, 4080, 5412, 7008, 8892, 11088, 13620, 16512, 19788, 23472, 27588, 32160, 37212, 42768, 48852, 55488, 62700, 70512, 78948, 88032, 97788, 108240, 119412, 131328, 144012, 157488, 171780, 186912, 202908, 219792, 237588
Offset: 1

Views

Author

Eric W. Weisstein, Sep 07 2017

Keywords

Comments

For n > 1, Wiener index of the 2n-crossed prism graph.

Links

Crossrefs

Cf. A006527, A054602, A217873.

Programs

  • Mathematica
    Table[4 n (n^2 + 2), {n, 50}]
LinearRecurrence[{4, -6, 4, -1}, {12, 48, 132, 288}, 20]
CoefficientList[Series[(12 (1 + x^2))/(-1 + x)^4, {x, 0, 20}], x]

Formula

a(n) = 4*n*(n^2 + 2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: 12*x*(1 + x^2)/(-1 + x)^4.
From Elmo R. Oliveira, Aug 09 2025: (Start)
E.g.f.: 4*x*(3 + 3*x + x^2)*exp(x).
a(n) = 12*A006527(n) = 4*A054602(n) = 3*A217873(n). (End)
Showing 1-4 of 4 results.

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