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.

A083075 Square array read by antidiagonals: T(n,k) = (k*(2*k+3)^n + 1)/(k+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 33, 63, 1, 1, 9, 61, 229, 313, 1, 1, 11, 97, 547, 1601, 1563, 1, 1, 13, 141, 1065, 4921, 11205, 7813, 1, 1, 15, 193, 1831, 11713, 44287, 78433, 39063, 1, 1, 17, 253, 2893, 23801, 128841, 398581, 549029, 195313, 1, 1, 19, 321
Offset: 0

Views

Author

Paul Barry, Apr 23 2003

Keywords

Examples

			Array begins:
  1     1     1     1     1 ...
  1     3    13    63   313 ...
  1     5    33   229  1601 ...
  1     7    61   547  4921 ...
  1     9    97  1065 11713 ...
  ...

Links

Crossrefs

Rows include A034478, A083076, A066443, A083077, A083078.
Columns include odds, A082109, A083079.
Diagonals include A083079, A083080, A083081, A083082.

Programs

  • Maple
    T := proc(n,k) return (k*(2*k+3)^n+1)/(k+1): end: seq(seq(T(k,n-k),k=0..n),n=0..10); # Nathaniel Johnston, Jun 26 2011

A082412 a(n) = (2*8^n + 2^n)/3.

Original entry on oeis.org

1, 6, 44, 344, 2736, 21856, 174784, 1398144, 11184896, 89478656, 715828224, 5726623744, 45812985856, 366503878656, 2932031012864, 23456248070144, 187649984495616, 1501199875833856, 12009599006408704, 96076792050745344, 768614336404914176
Offset: 0

Views

Author

Paul Barry, Apr 23 2003

Keywords

Comments

Binomial transform of A083076.

Links

Crossrefs

Cf. A082413.

Programs

  • Maple
    seq((2*8^n+2^n)/3,n=0..20); # Nathaniel Johnston, Jun 26 2011
  • Mathematica
    Table[(2*8^n+2^n)/3,{n,0,30}] (* or *) LinearRecurrence[{10,-16},{1,6},30] (* Harvey P. Dale, Sep 30 2018 *)
  • PARI
    a(n)=(2*8^n+2^n)/3 \\ Charles R Greathouse IV, Oct 07 2015
  • Python
    def A082412(n): return (2<<(n<<1)|1)//3<Chai Wah Wu, Apr 25 2025

Formula

G.f.: (1-4*x)/((1-2*x)*(1-8*x));
E.g.f.: (2*exp(8*x) + exp(2*x))/3.
a(n) = (2*8^n + 2^n)/3.
a(n) = 2^n*A001045(2n+1). - Paul Barry, Sep 10 2007

A083077 Fifth row of number array A083075.

Original entry on oeis.org

1, 9, 97, 1065, 11713, 128841, 1417249, 15589737, 171487105, 1886358153, 20749939681, 228249336489, 2510742701377, 27618169715145, 303799866866593, 3341798535532521, 36759783890857729, 404357622799435017
Offset: 0

Views

Author

Paul Barry, Apr 23 2003

Keywords

Links

Crossrefs

Cf. A083076, A066443, A083078.

Programs

Formula

a(n) = (4*11^n + 1)/5.
G.f.: (1-3*x)/((1-x)*(1-11*x)).
E.g.f.: (4*exp(11*x) + exp(x))/5.
a(n) = 11*a(n-1) - 2, a(n) = 12*a(n-1) - 11*a(n-2). - Vincenzo Librandi, Nov 10 2011

A208532 Mirror image of triangle in A125185; unsigned version of A120058.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 9, 10, 4, 5, 16, 28, 24, 8, 6, 25, 60, 80, 56, 16, 7, 36, 110, 200, 216, 128, 32, 8, 49, 182, 420, 616, 560, 288, 64, 9, 64, 280, 784, 1456, 1792, 1408, 640, 128, 10, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256
Offset: 0

Views

Author

Philippe Deléham, Feb 27 2012

Keywords

Comments

Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Equals A007318*A134309*A097806 as infinite lower triangular matrix.
Row sums are powers of 3 (A000244).
Diagonal sums are powers of 2 (A000079).

Examples

			Triangle begins :
1
2, 1
3, 4, 2
4, 9, 10, 4
5, 16, 28, 24, 8
6, 25, 60, 80, 56, 16
7, 36, 110, 200, 216, 128, 32
8, 49, 182, 420, 616, 560, 288, 64
9, 64, 280, 784, 1456, 1792, 1408, 640, 128
10, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256
Triangle (1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
3, 4, 2, 0
4, 9, 10, 4, 0
5, 16, 28, 24, 8, 0
6, 25, 60, 80, 56, 16, 0

Crossrefs

Cf. Columns: A000027, A000290, A006331, A112742.
Cf. Diagonals: A011782, 2*A045623,

Formula

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-1,k-1), T(0,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1-y*x)/((1-x)*(1-(1+2*y)*x)).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A083085(n), A084567(n), A000012(n), A000027(n+1), A000244(n), A083065(n), A083076(n) for x = -3, -2, -1, 0, 1, 2, 3 respectively.

A270471 Expansion of g.f. (1-3*x)/(1-7*x).

Original entry on oeis.org

1, 4, 28, 196, 1372, 9604, 67228, 470596, 3294172, 23059204, 161414428, 1129900996, 7909306972, 55365148804, 387556041628, 2712892291396, 18990246039772, 132931722278404, 930522055948828, 6513654391641796, 45595580741492572, 319169065190448004, 2234183456333136028
Offset: 0

Views

Author

Colin Barker, Mar 17 2016

Keywords

Comments

After 1, is this A208704?

Links

Crossrefs

Cf. A208704.
Cf. A000420 (powers of 7), A083076 (partial sums).
Cf. A193577: (1-2*x)/(1-7*x); A169634: (1-4*x)/(1-7*x).

Programs

  • Mathematica
    CoefficientList[Series[(1 - 3 x)/(1 - 7 x), {x, 0, 21}], x] (* Michael De Vlieger, Mar 18 2016 *)
Join[{1},NestList[7#&,4,20]] (* Harvey P. Dale, Dec 21 2019 *)
  • PARI
    Vec((1-3*x)/(1-7*x) + O(x^30))

Formula

G.f.: (1-3*x)/(1-7*x).
a(n) = 7*a(n-1) for n>1.
a(n) = 4*7^(n-1) for n>0.
E.g.f.: (4*exp(7*x) + 3)/7. - Elmo R. Oliveira, Mar 25 2025

A199416 2*7^n+1.

Original entry on oeis.org

3, 15, 99, 687, 4803, 33615, 235299, 1647087, 11529603, 80707215, 564950499, 3954653487, 27682574403, 193778020815, 1356446145699, 9495123019887, 66465861139203, 465261027974415, 3256827195820899, 22797790370746287
Offset: 0

Views

Author

Vincenzo Librandi, Nov 06 2011

Keywords

Links

Programs

  • Magma
    [2*7^n+1: n in [0..30]];

Formula

a(n) = 3*A083076(n).
a(n) = 7*a(n-1)-6.
a(n) = 8*a(n-1)-7*a(n-2).
G.f.: 3*(1-3*x)/((1-x)*(1-7*x)).
Showing 1-6 of 6 results.

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