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.

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A087423 a(n) = S(3*n,3)/S(n,3) where S(n,m) = Sum_{k=0..n} binomial(n,k)*Fibonacci(m*k).

Original entry on oeis.org

32, 768, 20672, 565248, 15491072, 424685568, 11643256832, 319215894528, 8751751626752, 239941585993728, 6578336360824832, 180354352643309568, 4944668491903926272, 135565048129674805248, 3716706651755063017472, 101898745479045492768768
Offset: 1

Views

Author

Benoit Cloitre, Oct 22 2003

Keywords

Links

Crossrefs

Cf. A020876.

Programs

  • Mathematica
    LinearRecurrence[{32,-128,64},{32,768,20672},20] (* Harvey P. Dale, Feb 17 2018 *)

Formula

a(n) = 4^n + (14+6*sqrt(5))^n + (14-6*sqrt(5))^n.
G.f.: -32*x*(6*x^2-8*x+1) / ((4*x-1)*(16*x^2-28*x+1)). - Colin Barker, Dec 01 2012

A245561 a(n) = 5^n - ( (sqrt(5)*phi)^n + (sqrt(5)/phi)^n ) + 1, where phi = golden ratio A001622.

Original entry on oeis.org

0, 1, 11, 76, 451, 2501, 13376, 70001, 361251, 1846876, 9381251, 47437501, 239109376, 1202500001, 6037656251, 30279296876, 151725781251, 759820312501, 3803412109376, 19032656250001, 95219707031251, 476302685546876, 2382252050781251, 11913932617187501
Offset: 0

Views

Author

N. J. A. Sloane, Aug 08 2014

Keywords

References

  • Roger L. Bagula, email message, Aug 08 2014.

Links

Crossrefs

Cf. A020876, A001622.

Programs

  • Magma
    [5^n + 1 - Floor(((5+Sqrt(5))/2)^n+((5-Sqrt(5))/2)^n): n in [0..30]]; // Vincenzo Librandi, Aug 08 2014
  • Maple
    g:=n->simplify(rationalize(simplify(expand( (sqrt(5)*p)^n + (sqrt(5)*q)^n ))); # A020876
h:=n->5^n-g(n)+1;
[seq(h(n),n=0..40)];
  • Mathematica
    CoefficientList[Series[-x (5 x^2 - 1)/((1 - 5 x + 5 x^2) (x - 1)(5 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 08 2014 *)
LinearRecurrence[{11,-40,55,-25},{0,1,11,76},30] (* Harvey P. Dale, Nov 05 2017 *)

Formula

a(n) = 5^n - A020876(n) + 1.
G.f.: -x*(5*x^2-1)/((1-5*x+5*x^2)*(x-1)*(5*x-1)). - Vincenzo Librandi, Aug 08 2014

A303930 Number of no-leaf subgraphs of the 2 X n grid up to horizontal and vertical reflection.

Original entry on oeis.org

1, 2, 4, 10, 26, 76, 232, 750, 2493, 8514, 29524, 103708, 367225, 1308542, 4682276, 16807286, 60462082, 217855460, 785863048, 2837177434, 10249053629, 37039804078, 133902392980, 484178868612, 1751030978481, 6333341963706, 22909148647012, 82872738727330
Offset: 1

Views

Author

Peter Kagey, May 02 2018

Keywords

Comments

The limit lim_{n -> infinity} A020876(n - 1)/a(n) = 4.

Examples

			For n = 4 the a(4) = 10 subgraphs of the 2 X 4 grid are:
+   +   +   +  +---+   +   +  +   +---+   +
               |   |              |   |
+   +   +   +, +---+   +   +, +   +---+   +,
+---+   +---+  +---+---+   +  +---+---+---+
|   |   |   |  |       |      |       |   |
+---+   +---+, +---+---+   +, +---+---+---+,
+---+---+---+  +---+---+---+  +---+---+---+
|           |  |   |   |   |  |   |   |   |
+---+---+---+, +---+---+---+, +---+   +---+, and
+---+---+   +
|   |   |
+---+---+   +.

Links

Crossrefs

Cf. A020876, A301976.
A093129 is analogous for 2 X (n+1) grids where reflections are considered distinct.

Formula

Conjectures from Colin Barker, May 03 2018: (Start)
G.f.: x*(1 - 6*x + 4*x^2 + 30*x^3 - 45*x^4 - 22*x^5 + 60*x^6 - 20*x^7) / ((1 - 3*x + x^2)*(1 - 5*x + 5*x^2)*(1 - 5*x^2 + 5*x^4)).
a(n) = 8*a(n-1) - 16*a(n-2) - 20*a(n-3) + 95*a(n-4) - 60*a(n-5) - 80*a(n-6) + 100*a(n-7) - 25*a(n-8) for n>8.
(End)

A376484 Array read by ascending antidiagonals: A(n,k)=4^k*Sum_{j=1..n} sin(2*j*Pi/(2*n+1))^(2*k).

Original entry on oeis.org

0, 1, 0, 2, 3, 0, 3, 5, 9, 0, 4, 7, 15, 27, 0, 5, 9, 21, 50, 81, 0, 6, 11, 27, 70, 175, 243, 0, 7, 13, 33, 90, 245, 625, 729, 0, 8, 15, 39, 110, 315, 882, 2250, 2187, 0, 9, 17, 45, 130, 385, 1134, 3234, 8125, 6561, 0, 10, 19, 51, 150, 455, 1386, 4158, 12005, 29375, 19683, 0
Offset: 0

Views

Author

Cheng-Jun Li, Sep 24 2024

Keywords

Comments

It is only a conjecture that A(n,k) is always an integer.
It appears that A(n,k) is divisible by 2*n+1 when n, k are positive integers.

Examples

			For n = 0 to 10 and k = 0 to 10, A(n, k) shows as below :
  0  0  0   0   0    0    0     0      0      0       0
  1  3  9  27  81  243  729  2187   6561  19683   59049
  2  5 15  50 175  625 2250  8125  29375 106250  384375
  3  7 21  70 245  882 3234 12005  44933 169099  638666
  4  9 27  90 315 1134 4158 15444  57915 218781  831222
  5 11 33 110 385 1386 5082 18876  70785 267410 1016158
  6 13 39 130 455 1638 6006 22308  83655 316030 1200914
  7 15 45 150 525 1890 6930 25740  96525 364650 1385670
  8 17 51 170 595 2142 7854 29172 109395 413270 1570426
  9 19 57 190 665 2394 8778 32604 122265 461890 1755182
 10 21 63 210 735 2646 9702 36036 135135 510510 1939938

Crossrefs

Conjectures: This array is related to existing sequences as follows: (Start)
Rows: A000004, A000244, A020876, A322459 (with alternate signs).
Columns: A001477, A005408, A016945.
Main Diagonals: A033876.
A(0,k) = A000004, A(1,k) = A000244, A(2,k) = A020876, A(3,k) = (-1)^k * A322459 (First 4 rows of the array).
A(n,0) = A001477(n > 0), A(n,1) = A005408(n > 0), A(n,2) = A016945(n > 0) (First 3 columns of the array).
A(n,n) = A033876(n > 0) (Main diagonal from top left corner). (End)

Programs

  • PARI
    A(n,k) = 4^k*sum(j=1,n,(sin(2*j*Pi/(2*n+1)))^(2*k))

A260304 a(n) = 5*a(n-1) - 5*a(n-2) for n>1, a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 5, 10, 25, 75, 250, 875, 3125, 11250, 40625, 146875, 531250, 1921875, 6953125, 25156250, 91015625, 329296875, 1191406250, 4310546875, 15595703125, 56425781250, 204150390625, 738623046875, 2672363281250, 9668701171875, 34981689453125, 126564941406250
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 21 2015

Keywords

Comments

Lim_{n -> infinity} a(n + 1)/a(n) = 2 + phi = 3.6180339887..., where phi is the golden ratio (A001622).

Links

Crossrefs

Cf. A020876, A030191, A098111.
Cf. A093129: initial values 1,2; A081567: initial values 1,3.

Programs

  • Magma
    [n le 2 select n+1 else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 23 2015
  • Mathematica
    Table[((5 + 2 Sqrt[5]) ((5 - Sqrt[5])/2)^n + (5 - 2 Sqrt[5]) ((5 + Sqrt[5])/2)^n)/5, {n, 0, 30}]
RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == 5 a[n - 1] - 5 a[n - 2]}, a, {n, 0, 30}] (* Bruno Berselli, Nov 23 2015 *)
  • PARI
    a(n)=([0,1; -5,5]^n*[2;3])[1,1] \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: (2 - 7*x)/(1 - 5*x + 5*x^2).
a(n) = ((5 + 2*sqrt(5))*((5 - sqrt(5))/2)^n + (5 - 2*sqrt(5))*((5 + sqrt(5))/2)^n)/5.
a(n) = 2*A030191(n) - 7*A030191(n-1). - Bruno Berselli, Nov 23 2015

Extensions

Edited by Bruno Berselli, Nov 23 2015
Previous Showing 11-15 of 15 results.

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