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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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A169940 Consider the 2^(n-1) monic polynomials f(x) with coefficients 0 or 1, degree n and f(0)=1. Sequence gives triangle read by rows, in which T(n,k) (n>=1) is the number of such polynomials of thickness k (2 <= k <= n+1).

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 3, 1, 1, 5, 4, 6, 0, 1, 7, 7, 10, 6, 1, 1, 13, 8, 27, 6, 9, 0, 1, 15, 21, 41, 23, 17, 9, 1, 1, 27, 20, 98, 34, 56, 8, 12, 0, 1, 25, 53, 148, 96, 104, 50, 22, 12, 1, 1, 45, 56, 325, 116, 294, 66, 96, 10, 15, 0, 1, 59, 89, 487, 319, 518, 262, 184, 86
Offset: 1

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Author

N. J. A. Sloane, Aug 01 2010

Keywords

Comments

The thickness of a polynomial f(x) is the magnitude of the largest coefficient in the expansion of f(x)^2.

Examples

			Triangle begins:
[1]
[1, 1]
[3, 0, 1]
[3, 3, 1, 1]
[5, 4, 6, 0, 1]
[7, 7, 10, 6, 1, 1]
[13, 8, 27, 6, 9, 0, 1]
[15, 21, 41, 23, 17, 9, 1, 1]
[27, 20, 98, 34, 56, 8, 12, 0, 1]
[25, 53, 148, 96, 104, 50, 22, 12, 1, 1]
[45, 56, 325, 116, 294, 66, 96, 10, 15, 0, 1]
[59, 89, 487, 319, 518, 262, 184, 86, 27, 15, 1, 1]
[89, 112, 942, 434, 1279, 346, 608, 112, 143, 12, 18, 0, 1]
[103, 197, 1348, 1042, 2181, 1153, 1166, 528, 291, 131, 32, 18, 1, 1]
[163, 220, 2613, 1320, 4981, 1568, 3313, 720, 1083, 168, 199, 14, 21, 0, 1]
...
For n=3 there are four polynomials x^3+1, x^3+x+1, x^3+x^2+1, x^3+x^2+x+1. Their squares are x^6+2*x^3+1, x^6+2*x^4+2*x^3+x^2+2*x+1, x^6+2*x^5+2*x^3+x^4+2*x^2+1 and x^6+2*x^5+3*x^4+4*x^3+3*x^2+2*x+1. Their thicknesses are 2,2,2,4. So T(3,2)=3, T(3,3)=0, T(3,4)=1.
The next 15 rows of the triangle are:
[187, 397, 3693, 2849, 8393, 4499, 6123, 2873, 2157, 939, 413, 185, 37, 21, 1, 1]
[281, 456, 6672, 3854, 17730, 6404, 15634, 4056, 6864, 1316, 1730, 234, 264, 16, 24, 0, 1]
[313, 711, 9458, 7940, 28938, 16432, 28534, 13398, 13488, 5906, 3568, 1514, 556, 248, 42, 24, 1, 1]
[469, 850, 16483, 10670, 58520, 23610, 67290, 19842, 37934, 8502, 12540, 2158, 2582, 310, 338, 18, 27, 0, 1]
[533, 1347, 22903, 20511, 94574, 55510, 120550, 57880, 73288, 32006, 25552, 10754, 5484, 2284, 716, 320, 47, 27, 1, 1]
[835, 1428, 39252, 27560, 183225, 80676, 267894, 86894, 189156, 48572, 78530, 15786, 20948, 3292, 3660, 396, 421, 20, 30, 0, 1]
[873, 2303, 53874, 51088, 290401, 179485, 469928, 232610, 359532, 158100, 158248, 66158, 43924, 18026, 7948, 3274, 895, 401, 52, 30, 1, 1]
[1319, 2642, 89947, 68614, 545421, 260616, 998433, 353278, 868696, 244418, 442240, 101860, 146260, 26948, 32804, 4750, 4997, 492, 513, 22, 33, 0, 1]
[1551, 3777, 123653, 121487, 853975, 549189, 1725367, 876575, 1621096, 725016, 877388, 365898, 304048, 123536, 70436, 28400, 11029, 4511, 1093, 491, 57, 33, 1, 1]
[2093, 4636, 200706, 164644, 1558400, 798552, 3526978, 1340828, 3719207, 1137278, 2280612, 580200, 912118, 192574, 251928, 43126, 48875, 6572, 6616, 598, 614, 24, 36, 0, 1]
[2347, 6693, 271092, 285484, 2403986, 1616482, 5997220, 3147524, 6830683, 3108825, 4457858, 1874174, 1873798, 754630, 537286, 213744, 107163, 42619, 14802, 6022, 1310, 590, 62, 36, 1, 1]
[3477, 7550, 438403, 379800, 4292926, 2346592, 11882630, 4821002, 15021379, 4920018, 10948081, 3008372, 5200638, 1217690, 1719966, 336912, 408989, 65534, 70061, 8794, 8546, 714, 724, 26, 39, 0, 1]
[3881, 11109, 585071, 644971, 6538688, 4594134, 19912060, 10801102, 27155069, 12640031, 21054795, 8950909, 10529720, 4248966, 3632012, 1428638, 890393, 348839, 156301, 61531, 19322, 7834, 1546, 698, 67, 39, 1, 1]
[5363, 12876, 927332, 860898, 11437031, 6656592, 38401950, 16551444, 57664535, 20086508, 49373458, 14542512, 27487209, 6959998, 10699424, 2334678, 3027695, 555714, 633348, 95568, 97301, 11454, 10814, 840, 843, 28, 42, 0, 1]
[5871, 17965, 1239392, 1419768, 17273147, 12579603, 63611068, 35500374, 102865259, 48877549, 93622166, 40321020, 54860417, 22275601, 22298854, 8743268, 6540369, 2528691, 1403386, 543422, 220305, 86061, 24650, 9974, 1801, 815, 72, 42, 1, 1]

Links

Crossrefs

Related to thickness: A169940-A169954, A061909.

Programs

  • Mathematica
    row[n_] := Module[{dd, xx, mm}, dd = Join[{1}, PadLeft[IntegerDigits[#, 2], n-1], {1}]& /@ Range[0, 2^(n-1) - 1]; xx = (((x^Range[n, 0, -1]).#)& /@ dd)^2 // Expand; mm = Max[CoefficientList[#, x]]& /@ xx; Table[Count[mm, k], {k, 2, n+1}]]; Table[row[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Oct 10 2017 *)
  • PARI
    T(n)={ my(c=vector(n)); forstep(j=1<M. F. Hasler, Nov 12 2010

Formula

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?
From M. F. Hasler, Nov 12 2010: (Start)
T(n,n+1) = 1 = T(2m,2m), T(2m+1,2m+1) = 0,
T(n+1,n) = (3, 3, 6, 6, 9, 9, ...) = 3*[n/2-1] = A168237(n) (n>2),
T(2m+2,2m) = (3, 10, 17, 22, 27, 32, 37, ...) = 5m+2 for m>2,
T(2m+3,2m+1) = (4, 6, 8, 10, ...) = 2m+2 for m>0,
T(2m+3,2m) = (5, 27, 56, 96, 143, 199, 264, ...) = m(9m+13)/2-2 for m>3,
T(2m+4,2m+1) = (7, 23, 50, 86, 131, 185, 248, ...) = 9m(m+1)/2-4 for m>1,
... (End)

Extensions

Rows 16-30 from Nathaniel Johnston, Nov 12 2010

A274912 Square array read by antidiagonals upwards in which each new term is the least nonnegative integer distinct from its neighbors.

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 1, 2, 1, 2, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2
Offset: 0

Views

Author

Omar E. Pol, Jul 11 2016

Keywords

Comments

In the square array we have that:
Antidiagonal sums give A168237.
Odd-indexed rows give A010673.
Even-indexed rows give A010684.
Odd-indexed columns give A000035.
Even-indexed columns give A010693.
Odd-indexed antidiagonals give the initial terms of A010674.
Even-indexed antidiagonals give the initial terms of A000034.
Main diagonal gives A010674.
This is also a triangle read by rows in which each new term is the least nonnegative integer distinct from its neighbors.
In the triangle we have that:
Row sums give A168237.
Odd-indexed columns give A000035.
Even-indexed columns give A010693.
Odd-indexed diagonals give A010673.
Even-indexed diagonals give A010684.
Odd-indexed rows give the initial terms of A010674.
Even-indexed rows give the initial terms of A000034.
Odd-indexed antidiagonals give the initial terms of A010673.
Even-indexed antidiagonals give the initial terms of A010684.

Examples

			The corner of the square array begins:
0, 2, 0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, ...
0, 2, 0, 2, ...
1, 3, 1, ...
0, 2, ...
1, ...
...
The sequence written as a triangle begins:
0;
1, 2;
0, 3, 0;
1, 2, 1, 2;
0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2, 1, 2;
...

Crossrefs

Cf. A000034, A000035, A001477, A010673, A010674, A010684, A010693, A168237, A274913, A274920.

Programs

  • Maple
    ListTools:-Flatten([seq([[0,3]$i,0,[1,2]$(i+1)],i=0..10)]); # Robert Israel, Nov 14 2016
  • Mathematica
    Table[Boole@ EvenQ@ # + 2 Boole@ EvenQ@ k &[n - k + 1], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Nov 14 2016 *)

Formula

a(n) = A274913(n) - 1.
From Robert Israel, Nov 14 2016: (Start)
G.f.: 3*x/(1-x^2) - Sum_{k>=0} (2*x^(2*k^2+3*k+1)-x^(2*k^2+5*k+3))/(1+x).
G.f. as triangle: x*(1+2*y+3*x*y)/((1-x^2*y^2)*(1-x^2)). (End)

A333119 Triangle T read by rows: T(n, k) = (n - k)*(1 - (-1)^k + 2*k)/4, with 0 <= k < n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 3, 2, 2, 0, 4, 3, 4, 2, 0, 5, 4, 6, 4, 3, 0, 6, 5, 8, 6, 6, 3, 0, 7, 6, 10, 8, 9, 6, 4, 0, 8, 7, 12, 10, 12, 9, 8, 4, 0, 9, 8, 14, 12, 15, 12, 12, 8, 5, 0, 10, 9, 16, 14, 18, 15, 16, 12, 10, 5, 0, 11, 10, 18, 16, 21, 18, 20, 16, 15, 10, 6
Offset: 1

Views

Author

Stefano Spezia, Mar 08 2020

Keywords

Comments

T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A332566(n).
The h-th subdiagonal of the triangle T gives 0 followed by the multiples of h+1 repeated.
For k > 0, the (2*k-1)-th and (2*k)-th columns of the triangle T give the multiples of k.

Examples

			n\k| 0 1 2 3 4 5
---+------------
1  | 0
2  | 0 1
3  | 0 2 1
4  | 0 3 2 2
5  | 0 4 3 4 2
6  | 0 5 4 6 4 3
...
For n = 4 the matrix M(4) is
      0 1 1 2
      1 0 1 1
      1 1 0 1
      2 1 1 0
and therefore T(4, 0) = 0, T(4, 1) = 3, T(4, 2) = 2 and T(4, 3) = 2.

Crossrefs

Cf. A332566.
Cf. A000004: 1st column; A000027: 2nd and 3rd column; A004526: diagonal; A005843: 4th and 5th column; A052928: 1st subdiagonal; A168237: 2nd subdiagonal; A168273: 3rd subdiagonal; A173196: row sums.

Programs

  • Mathematica
    T[n_,k_]:=(n-k)(1-(-1)^k+2k)/4; Flatten[Table[T[n,k],{n,1,12},{k,0,n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[y*(x*(2 + y + y^2) - (1 + y + 2*y^2))/((1 - x)^2 *(1 - y)^3 (1 + y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 12]]

Formula

O.g.f.: y*(x*(2 + y + y^2) - (1 + y + 2*y^2))/((1 - x)^2*(1 - y)^3*(1 + y)^2).
T(n, k) = k*(n - k)/2 for k even.
T(n, k) = (1 + k)*(n - k)/2 for k odd.
Showing 1-3 of 3 results.

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

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