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 1-4 of 4 results.

A108396 Triangle read by rows: T(n,k) = n*(1+n^k)/2, 0<=k<=n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 3, 6, 15, 42, 4, 10, 34, 130, 514, 5, 15, 65, 315, 1565, 7815, 6, 21, 111, 651, 3891, 23331, 139971, 7, 28, 175, 1204, 8407, 58828, 411775, 2882404, 8, 36, 260, 2052, 16388, 131076, 1048580, 8388612, 67108868, 9, 45, 369, 3285, 29529, 265725
Offset: 0

Views

Author

Reinhard Zumkeller, Jun 02 2005

Keywords

Comments

Row sums give A108397;
T(n,0) = A001477(n);
T(n,1) = A000217(n) for n>0;
T(n,2) = A006003(n) for n>1;
T(n,3) = A027441(n) for n>2;
T(n,4) = A021003(n) for n>3;
T(n,n) = A108398(n).

Examples

			.  0:  0
.  1:  1  1
.  2:  2  3   5
.  3:  3  6  15   42
.  4:  4 10  34  130   514
.  5:  5 15  65  315  1565   7815
.  6:  6 21 111  651  3891  23331  139971
.  7:  7 28 175 1204  8407  58828  411775  2882404
.  8:  8 36 260 2052 16388 131076 1048580  8388612  67108868
.  9:  9 45 369 3285 29529 265725 2391489 21523365 193710249 1743392205 .

Links

Crossrefs

Cf. A079901, A000312, A033918, A001477, A000217, A006003, A027441, A021003, A108398, A108397 (row sums), A256512 (central terms).

Programs

  • Haskell
    a108396 n k = a108396_tabl !! n !! k
a108396_row n = a108396_tabl !! n
a108396_tabl = zipWith (\v ws -> map (flip div 2 . (* v) . (+ 1)) ws)
                       [0..] a079901_tabl
-- Reinhard Zumkeller, Mar 31 2015
  • Mathematica
    Join[{0},Flatten[Table[n (1+n^k)/2,{n,10},{k,0,n}]]] (* Harvey P. Dale, Mar 19 2015 *)

Extensions

Offset changed by Reinhard Zumkeller, Mar 31 2015

A168029 a(n) = n*(n^6 + 1)/2.

Original entry on oeis.org

0, 1, 65, 1095, 8194, 39065, 139971, 411775, 1048580, 2391489, 5000005, 9743591, 17915910, 31374265, 52706759, 85429695, 134217736, 205169345, 306110025, 446935879, 640000010, 900544281, 1247178955, 1702412735, 2293235724, 3051757825, 4015905101
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), A167963 (m=5), this sequence (m=6), A168067 (m=7), A168116 (m=8), A168118 (m=9), A168119 (m=10).

Programs

  • Magma
    [n*(n^6+1)/2: n in [0..40]]; // Vincenzo Librandi, Dec 10 2014
  • Mathematica
    CoefficientList[Series[x(1 +57x +603x^2 +1198x^3 +603x^4 +57x^5 +x^6)/ (1-x)^8, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 10 2014 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1}, {0,1,65,1095,8194,39065, 139971,411775}, 41] (* Harvey P. Dale, Jan 24 2019 *)
  • SageMath
    [n*(n^6+1)/2 for n in range(41)] # G. C. Greubel, Jan 12 2023

Formula

G.f.: x*(1+57*x+603*x^2+1198*x^3+603*x^4+57*x^5+x^6)/(1-x)^8. - Vincenzo Librandi, Dec 10 2014
E.g.f.: (x/2)*(2 +63*x +301*x^2 +350*x^3 +140*x^4 +21*x^5 +x^6)*exp(x). - G. C. Greubel, Jan 12 2023

Extensions

More terms from Vincenzo Librandi, Dec 10 2014

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

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 1-4 of 4 results.

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

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