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 1-5 of 5 results.

A223968 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 5 or if k-n >= 6, T(4,0) = T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 0, 0, 6, 15, 20, 15, 5, 0, 0, 6, 21, 35, 35, 20, 0, 0, 0, 0, 27, 56, 70, 55, 20, 0, 0, 0, 0, 27, 83, 126, 125, 75, 0, 0, 0, 0, 0, 0, 110, 209, 251, 200, 75, 0, 0, 0, 0, 0, 0, 110, 319, 460, 451, 275, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Mar 30 2013

Keywords

Examples

			Square array begins:
1....1....1....1....1....1....0....0....0....0....0....0
1....2....3....4....5....6....6....0....0....0....0....0
1....3....6...10...15...21...27...27....0....0....0....0
1....4...10...20...35...56...83..110..110....0....0....0
1....5...15...35...70..126..209..319..429..429....0....0
0....5...20...55..125..251..460..779.1208.1637.1637....0
0....0...20...75..200..451..911.1690.2898.4535.6172.6172
...
Square array, read by diagonals, with 0 omitted:
1, 5, 20, 75, 275, 1001, 3639, 13243, 48280, ...
1, 5, 20, 75, 275, 1001, 3639, 13243, 48280, ...
1, 4, 15, 55, 200, 726, 2638, 9604, 35037, ...
1, 3, 10, 35, 125, 451, 1637, 5965, 21794, ...
1, 2, 6, 20, 70, 251, 911, 3327, 12190, 44744, ...
1, 3, 10, 35, 126, 460, 1690, 6225, 22950, ...
1, 4, 15, 56, 209, 779, 2898, 10760, 39882, ...
1, 5, 21, 83, 319, 1208, 4535, 16932, 62986, ...
1, 6, 27, 110, 429, 1637, 6172, 23104, 86090, ...
1, 6, 27, 110, 429, 1637, 6172, 23104, 86090, ...

Crossrefs

Cf. Similar sequences: A214846, A216054, A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232, A216235, A216236, A216238, A217257, A217315, A217593, A217765, A217770

Formula

sum(T(n-k,k), 0<=k<=n) = A223940(n).
T(n,n+5) = T(n,n+4) = A221863(n).
T(n,n+3) = A221862(n).
T(n,n+2) = A221859(n).
T(n,n+1) = A216710(n).
T(n,n) = A224514(n).
T(n+1,n) = A224509(n).
T(n+2,n) = A220948(n).
T(n+3,n) = T(n+4,n) = A224422(n). - Philippe Deléham, Apr 13 2013

A217770 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=4 or if k-n >= 6, T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 0, 1, 5, 10, 10, 4, 0, 0, 6, 15, 20, 14, 0, 0, 0, 6, 21, 35, 34, 14, 0, 0, 0, 0, 27, 56, 69, 48, 0, 0, 0, 0, 0, 27, 83, 125, 117, 48, 0, 0, 0, 0, 0, 0, 110, 208, 242, 165, 0, 0, 0, 0, 0, 0, 0, 110, 318, 450, 407, 165
Offset: 0

Views

Author

Philippe Deléham, Mar 24 2013

Keywords

Comments

A hexagon arithmetic of E. Lucas.

Examples

			Square array begins:
n=0: 1, 1,  1,  1,   1,   1,   0,   0,    0,    0,    0, 0, ...
n=1: 1, 2,  3,  4,   5,   6,   6,   0,    0,    0,    0, 0, ...
n=2: 1, 3,  6, 10,  15,  21,  27,  27,    0,    0,    0, 0, ...
n=3: 1, 4, 10, 20,  35,  56,  83, 110,  110,    0,    0, 0, ...
n=4: 0, 4, 14, 34,  69, 125, 208, 318,  428,  428,    0, 0, ...
n=5: 0, 0, 14, 48, 117, 242, 450, 768, 1196, 1624, 1624, 0, ...
...
Square array, read by rows, with 0 omitted:
...1,    1,     1,     1,     1,      1
...1,    2,     3,     4,     5,      6,      6
...1,    3,     6,    10,    15,     21,     27,     27
...1,    4,    10,    20,    35,     56,     83,    110,    110
...4,   14,    34,    69,   125,    208,    318,    428,    428
..14,   48,   117,   242,   450,    768,   1196,   1624,   1624
..48,  165,   407,   857,  1625,   2821,   4445,   6069,   6069
.165,  572,  1429,  3054,  5875,  10320,  16389,  22458,  22458
.572, 2001,  5055, 10930, 21250,  37639,  60097,  82555,  82555
2001, 7056, 17986, 39236, 76875, 136972, 219527, 302082, 302082
...
Triangle begins:
1
1, 1
1, 2,  1
1, 3,  3,  1
1, 4,  6,  4,  0
1, 5, 10, 10,  4,  0
0, 6, 15, 20, 14,  0, 0
0, 6, 21, 35, 34, 14, 0, 0
...

Crossrefs

Cf. Similar sequences: A214846, A216054, A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232, A216235, A216236, A216238, A217257, A217315, A217593, A217765.

Formula

T(n,n+4) = T(n,n+5) = A094788(n+2).
T(n,n+3) = A217783(n).
T(n,n+2) = A217779(n).
T(n,n+1) = A081567(n).
T(n,n) = A217782(n).
T(n+1,n) = A217778(n).
T(n+3,n) = T(n+2,n) = A094667(n+1).
Sum(T(n-k,k), k=0..n) = A217777(n).

A216241 Number of n-step walks (each step +-1 starting from 0) which are never more than 5 or less than -5.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 62, 124, 236, 472, 890, 1780, 3340, 6680, 12502, 25004, 46732, 93464, 174554, 349108, 651740, 1303480, 2432918, 4865836, 9080956, 18161912, 33892954, 67785908, 126494956, 252989912, 472095062, 944190124, 1761901676, 3523803352, 6575544410, 13151088820
Offset: 0

Views

Author

Philippe Deléham, Mar 15 2013

Keywords

Crossrefs

Cf. Rows of A068913: A000007, A016116 (without initial term), A068911, A068912, A214846, A216212.

Programs

  • Mathematica
    nn=35;CoefficientList[Series[(1+2x)(1-x^2)^2/(1-6x^2+9x^4-2x^6),{x,0,nn}],x] (* Geoffrey Critzer, Jan 14 2014 *)

Formula

a(n) = A068913(5,n).
a(n) = 6*a(n-2) - 9*a(n-4) + 2*a(n-6).
a(n) = 2^n for n < 6.
G.f.: ((1-x)^2*(1+x)^2*(1+2*x)) / ((1-2*x^2)*(1-4*x^2+x^4)).
a(2*n+1) = 2*a(2*n).
a(n) = Sum_{k=0..n} A214846(n-k, k). - Philippe Deléham, Mar 25 2013

Extensions

a(34) corrected by Sean A. Irvine, May 19 2019

A216263 Expansion of 1 / ((1-2*x)*(1-4*x+x^2)).

Original entry on oeis.org

1, 6, 27, 110, 429, 1638, 6187, 23238, 87021, 325358, 1215435, 4538430, 16942381, 63239286, 236031147, 880918070, 3287706669, 12270039678, 45792714187, 170901341358, 637813699821, 2380355555078, 8883612714795, 33154103692710, 123732818833261, 461777205194766, 1723376069054667
Offset: 0

Views

Author

Philippe Deléham, Mar 15 2013

Keywords

Links

Crossrefs

A diagonal of A214846.
Cf. A001075.

Programs

  • Mathematica
    CoefficientList[Series[1/((1 - 2 x)*(1 - 4 x + x^2)), {x, 0, 26}], x] (* Michael De Vlieger, Aug 05 2021 *)
  • PARI
    Vec(1/((1-2*x)*(1-4*x+x^2)) + O(x^30)) \\ Colin Barker, Feb 05 2017

Formula

G.f.: 1/((1-2*x)*(1-4*x+x^2)).
a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3), a(0) = 1, a(1) = 6, a(2) = 27.
3*a(n) = -2^(n+2) + A001075(n+2). - R. J. Mathar, Mar 29 2013
a(n) = (-2^(3+n) + (7-4*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(7+4*sqrt(3))) / 6. - Colin Barker, Feb 05 2017

A216271 Expansion of (1-x)/((1-2x)(1-4x+x^2)).

Original entry on oeis.org

1, 5, 21, 83, 319, 1209, 4549, 17051, 63783, 238337, 890077, 3322995, 12403951, 46296905, 172791861, 644886923, 2406788599, 8982333009, 33522674509, 125108627171, 466912358463, 1742541855257, 6503257159717, 24270490977915, 90578715140551, 338044386361505, 1261598863859901
Offset: 0

Views

Author

Philippe Deléham, Mar 16 2013

Keywords

Comments

Partial sums are in A216263.
Diagonal of square array A214846.

Links

Crossrefs

Cf. A001353, A087946, A214846, A216263.

Programs

  • Mathematica
    CoefficientList[Series[(1-x)/((1-2x)(1-4x+x^2)),{x,0,30}],x] (* Harvey P. Dale, Oct 05 2019 *)

Formula

a(n) = A001353(n+2) - A087946(n+1).
G.f.: (1-x)/(1-6x+9x^2-2x^3).
a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3), a(0) = 1, a(1) = 5, a(2) = 21.
Sum_{k=0..n} a(k) = A216263(n).
Showing 1-5 of 5 results.

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