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 1-3 of 3 results.

A167963 a(n) = n*(n^5 + 1)/2.

Original entry on oeis.org

0, 1, 33, 366, 2050, 7815, 23331, 58828, 131076, 265725, 500005, 885786, 1492998, 2413411, 3764775, 5695320, 8388616, 12068793, 17006121, 23522950, 32000010, 42883071, 56689963, 74017956, 95551500, 122070325, 154457901, 193710258, 240945166, 297411675
Offset: 0

Views

Author

N. J. A. Sloane, Dec 11 2009

Keywords

Links

Crossrefs

Sequences of the form n*(n^m + 1)/2: A001477 (m=0), A000217 (m=1), A006003 (m=2), A027441 (m=3), A021003 (m=4), this sequence (m=5), A168029 (m=6), A168067 (m=7), A168116 (m=8), A168118 (m=9), A168119 (m=10).

Programs

  • Magma
    [n*(n^5+1)/2: n in [0..40]]; // Vincenzo Librandi, Dec 10 2014
  • Maple
    A167963:=n->n*(n^5+1)/2; seq(A167963(n), n=0..100); # Wesley Ivan Hurt, Nov 23 2013
  • Mathematica
    Table[n(n^5+1)/2, {n,0,100}] (* Wesley Ivan Hurt, Nov 23 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,33,366,2050,7815,23331},30] (* Harvey P. Dale, Dec 09 2014 *)
CoefficientList[Series[x (1 + 26 x + 156 x^2 + 146 x^3 + 31 x^4) / (1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2014 *)
  • SageMath
    [n*(n^5+1)/2 for n in range(41)] # G. C. Greubel, Jan 17 2023

Formula

G.f.: x*(1 + 26*x + 156*x^2 + 146*x^3 + 31*x^4)/(1-x)^7. - Vincenzo Librandi, Dec 10 2014
E.g.f.: (1/2)*x*(2 + 31*x + 90*x^2 + 65*x^3 + 15*x^4 + x^5)*exp(x). - G. C. Greubel, Jan 17 2023

A168526 a(n) = n^6*(n + 1)/2.

Original entry on oeis.org

0, 1, 96, 1458, 10240, 46875, 163296, 470596, 1179648, 2657205, 5500000, 10629366, 19408896, 33787663, 56471520, 91125000, 142606336, 217238121, 323116128, 470458810, 672000000, 943427331, 1303868896, 1776430668, 2388787200, 3173828125, 4170362976
Offset: 0

Views

Author

N. J. A. Sloane, Dec 11 2009

Keywords

Links

Crossrefs

Cf. A168029.

Programs

Formula

From R. J. Mathar, Dec 16 2009: (Start)
a(n) = 8*a(n-1) -28*a(n-2) +56*a(n-3) -70*a(n-4) +56*a(n-5) -28*a(n-6) +8*a(n-7) -a(n-8).
G.f.: x*(1 + 88*x + 718*x^2 + 1208*x^3 + 473*x^4 + 32*x^5)/(1-x)^8. (End)
E.g.f.: (1/2)*x*(2 + 94*x + 391*x^2 + 415*x^3 + 155*x^4 + 22*x^5 + x^6)*exp(x). - G. C. Greubel, Mar 20 2025

A361263 Numbers of the form k*(k^5 +- 1)/2.

Original entry on oeis.org

0, 1, 31, 33, 363, 366, 2046, 2050, 7810, 7815, 23325, 23331, 58821, 58828, 131068, 131076, 265716, 265725, 499995, 500005, 885775, 885786, 1492986, 1492998, 2413398, 2413411, 3764761, 3764775, 5695305, 5695320, 8388600, 8388616, 12068776, 12068793, 17006103, 17006121, 23522931, 23522950
Offset: 1

Views

Author

Thomas Scheuerle, Mar 06 2023

Keywords

Comments

Integer solutions of x + y = (x - y)^6. If x = a(n) then y = a(n - (-1)^n).

Links

Crossrefs

Cf. A006003, A021003, A027441, A057587, A057590, A135503, A168029.
Cf. A167963 (bisection).

Programs

  • Maple
    map(k -> (k*(k^5-1)/2, k*(k^5+1)/2), [$1..100]);
  • PARI
    concat(0, Vec(x^2*(1+30*x-4*x^2+150*x^3+6*x^4+150*x^5-4*x^6+30*x^7+x^8)/((1-x)^7*(1+x)^6) + O(x^100)))
  • Python
    def A361263(n): return (k:=n+1>>1)*(k**5+1-((n&1)<<1))>>1 # Chai Wah Wu, Mar 22 2023

Formula

G.f.: x^2*(1+30*x-4*x^2+150*x^3+6*x^4+150*x^5-4*x^6+30*x^7+x^8) / ((1-x)^7*(1+x)^6).
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13).
Showing 1-3 of 3 results.

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

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