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 11-14 of 14 results.

A281660 The least common multiple of 1+n and 1+n^2.

Original entry on oeis.org

1, 2, 15, 20, 85, 78, 259, 200, 585, 410, 1111, 732, 1885, 1190, 2955, 1808, 4369, 2610, 6175, 3620, 8421, 4862, 11155, 6360, 14425, 8138, 18279, 10220, 22765, 12630, 27931, 15392, 33825, 18530, 40495, 22068, 47989, 26030, 56355, 30440, 65641, 35322
Offset: 0

Views

Author

R. J. Mathar, Jan 26 2017

Keywords

Links

Programs

  • Maple
    A281660 := proc(n)
        ilcm(1+n,1+n^2) ;
end proc:

Formula

a(n) = lcm(1+n,1+n^2) = (1+n)*(1+n^2)/gcd(1+n,1+n^2) = A053698(n)/A000034(n).
G.f.: (5*x^2+1) *(x^4+2*x^3+6*x^2+2*x+1) / ( (x-1)^4 *(1+x)^4 ).
a(2*n+1) = 2*A059722(n+1). - R. J. Mathar, Jan 28 2017
a(n) = ((3 + (-1)^n)*(1+n+n^2+n^3)) / 4. - Colin Barker, Feb 07 2017

A143803 a(n) = 2*A001614(n) - 1 where A001614 lists the Connell numbers.

Original entry on oeis.org

1, 3, 7, 9, 13, 17, 19, 23, 27, 31, 33, 37, 41, 45, 49, 51, 55, 59, 63, 67, 71, 73, 77, 81, 85, 89, 93, 97, 99, 103, 107, 111, 115, 119, 123, 127, 129, 133, 137, 141, 145, 149, 153, 157, 161, 163, 167, 171, 175, 179, 183, 187, 191, 195, 199
Offset: 1

Views

Author

Gary W. Adamson, Sep 01 2008

Keywords

Comments

Row sums = A059722: (1, 10, 39, 100, ...).
Right border of the triangle = A056220: (1, 7, 17, 31, 49, ...).
Left border = A058331: (1, 3, 9, 19, 33, 51, ...).
Connell-like triangle read by rows: odd rows are in the set 4n-3, evens are in 4n-1. Leftmost term in the next row is the next higher term consistent with the modular rule.
Given A056220: (1, 7, 17, 31, 49, 71, ...) as the rightmost diagonal; the triangle is generated starting from the right: (n-th term of A056220, then (n-1) operations of the trajectory (-4), (-4), (-4), ...
Row 3 = (9, 13, 17) since beginning with A056220(3) = 17 as rightmost term, we perform two operations of (-4), -(4)j.

Examples

			First few rows of the triangle =
   1;
   3,  7;
   9, 13, 17;
  19, 23, 27, 31;
  33, 37, 41, 45, 49;
  51, 55, 59, 63, 67, 71;
  ...
Examples: a(5) = 13 = 2*A001614(5) - 1, where 7 = A001614(5).

Crossrefs

Cf. A001614, A059722, A056220, A058331.

Programs

  • Python
    from math import isqrt
def A143803(n): return ((m:=n<<1)-(k:=isqrt(m))-int(m>=k*(k+1)+1)<<1)-1 # Chai Wah Wu, Aug 01 2022

Formula

a(n) = 2*A001614(n) - 1, where A001614 = the Connell numbers.

A335648 Partial sums of A006010.

Original entry on oeis.org

0, 1, 6, 26, 78, 195, 420, 820, 1476, 2501, 4026, 6222, 9282, 13447, 18984, 26216, 35496, 47241, 61902, 80002, 102102, 128843, 160908, 199068, 244140, 297037, 358722, 430262, 512778, 607503, 715728, 838864, 978384, 1135889, 1313046, 1511658, 1733598, 1980883, 2255604
Offset: 0

Views

Author

Stefano Spezia, Jun 15 2020

Keywords

Links

Crossrefs

Cf. A000290, A001844, A005408, A007204, A053755, A058331, A059722.
Cf. A006010 (1st differences), A186424 (3rd differences), A317614 (2nd differences).

Programs

  • Magma
    I:=[0, 1, 6, 26, 78, 195, 420, 820]; [n le 8 select I[n] else 4*Self(n-1)-4*Self(n-2)-4*Self(n-3)+10*Self(n-4)-4*Self(n-5)-4*Self(n-6)+4*Self(n-7)-Self(n-8): n in [1..39]];
  • Mathematica
    Table[(1+n)(5-5(-1)^n+8n+12n^2+8n^3+2n^4)/80,{n,0,38}]
  • PARI
    a(n) = (1 + n)*(5 - 5*(-1)^n + 8*n + 12*n^2 + 8*n^3 + 2*n^4)/80;
  • Sage
    (x*(1+2*x+6*x^2+2*x^3+x^4)/((1-x)^6*(1+x)^2)).series(x, 39).coefficients(x, False)

Formula

a(n) = (1 + n)*(5 - 5*(-1)^n + 8*n + 12*n^2 + 8*n^3 + 2*n^4)/80.
O.g.f.: x*(1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^6*(1 + x)^2).
E.g.f.: (cosh(x) - sinh(x))*(-5 + 5*x + (5 + 65*x + 180*x^2 + 130*x^3 + 30*x^4 + 2*x^5)*(cosh(2*x) + sinh(2*x)))/80.
a(n) = 4*a(n-1) - 4*a(n-2) - 4*a(n-3) + 10*a(n-4) - 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - a(n-8) for n > 7.
a(2*n-1) = n*A053755(n)/5 for n > 0.
a(2*n) = n*A005408(n)*A059722(n-1)/5.
a(2*n+1) - a(2*n-1) = A001844(n)^2 = A007204(n) for n > 0.
a(2*n) - a(2*n-2) = 2*A000290(n)*A058331(n) for n > 0.

A329530 a(n) = n * (7*binomial(n, 2) + 1).

Original entry on oeis.org

0, 1, 16, 66, 172, 355, 636, 1036, 1576, 2277, 3160, 4246, 5556, 7111, 8932, 11040, 13456, 16201, 19296, 22762, 26620, 30891, 35596, 40756, 46392, 52525, 59176, 66366, 74116, 82447, 91380, 100936, 111136, 122001, 133552, 145810, 158796, 172531, 187036, 202332, 218440
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 15 2019

Keywords

Comments

Centered heptagonal prism numbers.

References

  • E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), 144.

Links

Crossrefs

Centered m-gonal prism numbers: A100175 (m = 3), A059722 (m = 4), A006564 (m = 5), A005915 (m = 6), this sequence (m = 7), A139757 (m = 8), A006566 (m = 9).
Cf. A000217, A002413, A006597, A024966, A069099.

Programs

  • Mathematica
    Table[n (7 Binomial[n, 2] + 1), {n, 0, 40}]
nmax = 40; CoefficientList[Series[x (1 + 12 x + 8 x^2)/(1 - x)^4, {x, 0, nmax}], x]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 16, 66}, 41]

Formula

G.f.: x * (1 + 12*x + 8*x^2) / (1 - x)^4.
E.g.f.: exp(x) * x * (2 + 14*x + 7*x^2) / 2.
a(n) = n * (7*n^2 - 7*n + 2) / 2.
a(n) = n * (7*A000217(n-1) + 1).
a(n) = n * A069099(n).
Previous Showing 11-14 of 14 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.