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 1-2 of 2 results.

A194713 13 times hexagonal numbers: a(n) = 13*n*(2*n-1).

Original entry on oeis.org

0, 13, 78, 195, 364, 585, 858, 1183, 1560, 1989, 2470, 3003, 3588, 4225, 4914, 5655, 6448, 7293, 8190, 9139, 10140, 11193, 12298, 13455, 14664, 15925, 17238, 18603, 20020, 21489, 23010, 24583, 26208, 27885, 29614, 31395, 33228, 35113, 37050, 39039, 41080, 43173
Offset: 0

Views

Author

Omar E. Pol, Oct 02 2011

Keywords

Comments

Sequence found by reading the line from 0, in the direction 0, 13, ..., in the square spiral whose vertices are the generalized 15-gonal numbers.

Links

Crossrefs

Cf. A000384, A002939, A094159, A143698, A152745, A154617, A195320.

Programs

Formula

a(n) = 26*n^2 - 13*n = 13*A000384(n).
a(n) = a(n-1) + 52*n - 39, a(0)=0. - Vincenzo Librandi, Oct 03 2011
From Elmo R. Oliveira, Dec 15 2024: (Start)
G.f.: 13*x*(1 + 3*x)/(1 - x)^3.
E.g.f.: 13*x*(1 + 2*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

A267140 Least m>0 for which m + n^2 is a square and m + triangular(n) is a triangular number (A000217).

Original entry on oeis.org

1, 35, 12, 72, 180, 336, 45, 792, 1092, 208, 1836, 2280, 112, 315, 3900, 4536, 644, 5952, 6732, 7560, 225, 715, 10332, 627, 12420, 13536, 924, 1575, 17172, 840, 396, 21240, 22692, 3267, 2565, 27336, 28980, 3392, 32412, 34200, 1881, 3795, 637, 1400, 1785, 45936, 2240
Offset: 0

Views

Author

Alex Ratushnyak, Jan 10 2016

Keywords

Comments

For n>1, a(n) <= b(n), where b(n) = 24*n^2 - 60*n + 36 = A143698(n-1), because b(n) + n^2 = (5*n-6)^2, and b(n) + n*(n+1)/2 = (7*n-9)*(7*n-8)/2 = triangular(7*n-9).

Examples

			12 + 2^2 = 16 is a square, and 12 + 2*3/2 = 15 is a triangular number, and 12 is the least such integer, so a(2)=12.

Links

Crossrefs

Cf. A000217, A000290, A143698, A267077.

Programs

  • Mathematica
    lmst[n_]:=Module[{m=1,n2=n^2,nt=(n(n+1))/2},While[ !IntegerQ[Sqrt[ m+n2]] || !OddQ[Sqrt[1+8(m+nt)]],m++];m]; Join[{1},Array[lmst,50]] (* Harvey P. Dale, Aug 15 2021 *)
  • PARI
    a(n) = {m = 1; while (! (issquare(m+n^2) && ispolygonal(m+n*(n+1)/2, 3)), m++); m;} \\ Michel Marcus, Jan 11 2016
  • Python
    from math import sqrt
def A267140(n):
    u,r,k,m = 2*n+1,4*n*(n+1)+1,0,2*n+1
    while True:
        if int(sqrt(8*m+r))**2 == 8*m+r:
            return m
        k += 2
        m += u + k # Chai Wah Wu, Jan 13 2016
Showing 1-2 of 2 results.

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

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