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.

Previous Showing 21-24 of 24 results.

A017032 a(n) = (7*n + 4)^4.

Original entry on oeis.org

256, 14641, 104976, 390625, 1048576, 2313441, 4477456, 7890481, 12960000, 20151121, 29986576, 43046721, 59969536, 81450625, 108243216, 141158161, 181063936, 228886641, 285610000, 352275361, 429981696, 519885601, 623201296, 741200625, 875213056, 1026625681
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000583 (n^4), A017029 (7*n+4).

Programs

  • Magma
    [(7*n+4)^4: n in [0..40] ]; // Vincenzo Librandi, Jul 16 2011
  • Mathematica
    (7*Range[0, 20] + 4)^4 (* Harvey P. Dale, Mar 03 2012 *)
  • PARI
    Vec(-(81*x^4+9595*x^3+34331*x^2+13361*x+256)/(x-1)^5 + O(x^50)) \\ Colin Barker, Oct 21 2015

Formula

G.f.: -(81*x^4 + 9595*x^3 + 34331*x^2 + 13361*x + 256) / (x-1)^5. - Colin Barker, Oct 21 2015
a(n) = A000583(A017029(n)). - Michel Marcus, Oct 21 2015

A065450 Make an infinite chessboard from the squares in the first quadrant; sequence gives number of squares a knight can reach in n moves starting at the origin.

Original entry on oeis.org

1, 2, 10, 22, 37, 54, 76, 100, 129, 160, 196, 234, 277, 322, 372, 424, 481, 540, 604, 670, 741, 814, 892, 972, 1057, 1144, 1236, 1330, 1429, 1530, 1636, 1744, 1857, 1972, 2092, 2214, 2341, 2470, 2604, 2740, 2881, 3024, 3172, 3322, 3477, 3634, 3796, 3960
Offset: 0

Views

Author

Bodo Zinser, Nov 18 2001

Keywords

Comments

The first conjecture is true: Partial sums of A047356 = b(n) = (14*(n*(n+1)) + 2*n + 5 + 3*(-1)^n )/8, since A047356(n)=(14*n+1+3*(-1)^n)/4. And b(n) has g.f. (4*x^2 + 2*x + 1)/(-x^4 + 2*x^3 - 2*x + 1). The difference a(n) - b(n) = 0,2,2,0,0,0,0,0,0..., which has g.f. 2*x^2 + 2*x. Then (4*x^2 + 2*x + 1)/(-x^4 + 2*x^3 - 2*x + 1) + 2*x^2 + 2*x = (-2*x^6 + 2*x^5 + 4*x^4 - 4*x^3 + 2*x^2 + 4*x + 1)/(-x^4 + 2*x^3 - 2*x + 1). - Vim Wenders, Apr 16 2008

Crossrefs

Cf. A098498.

Formula

Conjectures: G.f.: [1+6x^2+4x^3-4x^4-2x^5+2x^6]/[(1+x)*(1-x)^3]. For n>3, partial sums of A047356. - Ralf Stephan, Mar 06 2004
The second conjecture "For n>3, partial sums of A047356" is also true. From the last possible move, we can either move back to the second last possible move or to b(n)=A047883(n) new squares. So a(n) = a(n-2)+b(n). For n>6, b(n)=7(n-1)+4=A017029(n-1). But a number of the form 7n+4 is naturally the sum of two consecutive terms in A047356 (4=1+3,11=3+8,18=8+10, ...). The conjecture follows. - Vim Wenders, Apr 12 2008

Extensions

More terms from Don Reble, Nov 28 2001

A137186 Lucky numbers (A000959) which are congruent to 4 mod 7.

Original entry on oeis.org

25, 67, 151, 193, 235, 319, 361, 487, 529, 613, 655, 739, 781, 823, 991, 1117, 1201, 1285, 1369, 1495, 1579, 1663, 1705, 1831, 1915, 2083, 2125, 2209, 2251, 2335, 2419, 2461, 2545, 2587, 2671, 2755, 2797, 2923, 3007, 3049, 3091, 3133, 3175, 3259, 3301, 3427
Offset: 1

Views

Author

N. J. A. Sloane, Mar 07 2008

Keywords

Links

Crossrefs

Intersection of A000959 and A017029.

A253671 a(n) = floor(A000111(n)/A000111(n-1)).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40
Offset: 1

Views

Author

Paul Curtz, Jan 08 2015, with the help of Jean-François Alcover

Keywords

Comments

1, 2, 3, 4, ... first appear at n = 1, 3, 5, 7, 8, 10, 11, 13, ... . a(500) = 318.
Numbers appearing only once: interleave 4+7*n, 6+7*n, 9+7*n = 4, 6, 9, 11, 13, 16, ... .
This is a nondecreasing sequence.
The ratio a(n)/n asymptotically tends to 7/11 = 0.6363... - Jean-François Alcover, Jul 21 2015

Examples

			Floor of 1/1, 1/1, 2/1, 5/2, 16/5, 61/16, ... .
1=1*1+0, 1=1*1+0, 2=2*1+0, 5=2*2+1, 16=3*5+1, 61=3*16+13, 272=4*61+28, ... .

Links

Crossrefs

Cf. A000111, A010727, A017005, A017029, A017053.

Programs

  • Mathematica
    max = 500; ee = Table[2^n*EulerE[n, 1] + EulerE[n] - 1, {n, 0, max}]; A000111 = Table[Differences[ee, n] // First // Abs, {n, 0, max}]; Table[Quotient[A000111[[n + 1]], A000111[[n]]], {n, 1, max}] (* Jean-François Alcover, Jan 08 2015 *)
  • PARI
    Vec(x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1)/((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)) + O(x^100)) \\ Colin Barker, Jan 22 2015
  • Python
    # requires python 3.2 or higher
from itertools import accumulate
A253671_list, blist, l1, l2 = [1], [1], 1, 1
for n in range(10**2):
    blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))
    l2, l1 = l1, sum(blist)
    A253671_list.append(l1//l2) # Chai Wah Wu, Jan 29 2015

Formula

a(n+2) = a(n+1) + (0, 1, 0, followed by a sequence of period 11: repeat 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1).
a(n+12) = a(n+1) + (6, 7, 6, followed by 7's = A010727).
a(n) = a(n-1) + a(n-11) - a(n-12) for n>15. - Colin Barker, Jan 22 2015
G.f.: x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1) / ((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)). - Colin Barker, Jan 22 2015
Previous Showing 21-24 of 24 results.

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

Content is available under The OEIS End-User License Agreement.