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 1-6 of 6 results.

A262242 Triangular numbers representable as 2^x + 2^y.

Original entry on oeis.org

3, 6, 10, 36, 66, 136, 528, 2080, 8256, 32896, 131328, 524800, 2098176, 8390656, 33558528, 134225920, 536887296, 2147516416, 8590000128, 34359869440, 137439215616, 549756338176, 2199024304128, 8796095119360, 35184376283136, 140737496743936, 562949970198528
Offset: 1

Views

Author

Alex Ratushnyak, Sep 15 2015

Keywords

Links

Crossrefs

Cf. A000217, A225390, A226499, A227027, A259745, A259746.

Programs

  • Python
    def isTriangular(a):
    sr = 1 << (int.bit_length(a) >> 1)
    a += a
    while a < sr*(sr+1):  sr>>=1
    b = sr>>1
    while b:
      s = sr+b
      if a >= s*(s+1):  sr = s
      b>>=1
    return (a==sr*(sr+1))
for a in range(1,200):
    for b in range(a):
        c = (1<

Formula

Conjectures from Colin Barker, Sep 16 2015: (Start)
a(n) = 2^(n-5)*(2^n+4) for n>5.
a(n) = 6*a(n-1)-8*a(n-2) for n>7.
G.f.: x*(240*x^6+28*x^5-70*x^4+24*x^3-2*x^2-12*x+3) / ((2*x-1)*(4*x-1)).
(End)

A225440 Triangular numbers that are the product of three distinct triangular numbers greater than 1.

Original entry on oeis.org

378, 630, 990, 3240, 4095, 4950, 5460, 9180, 15400, 19110, 25200, 31878, 37128, 37950, 39060, 52650, 61425, 79800, 97020, 103740, 105570, 122265, 145530, 157080, 161028, 176715, 192510, 221445, 265356, 288420, 304590, 306936, 346528, 437580, 500500, 545490, 583740
Offset: 1

Views

Author

Alex Ratushnyak, May 08 2013

Keywords

Comments

Triangular numbers of the form triangular(x) * triangular(y) * triangular(z), x > y > z > 1.

Examples

			378 = 3 * 6 * 21.
630 = 3 * 10 * 21.
990 = 3 * 6 * 55.

Links

Crossrefs

Cf. A000217, A085780, A140089, A188630, A225390.

Programs

  • C
    #include 
typedef unsigned long long U64;
U64 isTriangular(U64 a) {  // ! Must be a < (1<<63)
    U64 s = sqrt(a*2);
    if (a>=(1ULL<<63)) exit(1);
    return (s*(s+1)/2 == a);
}
int compare64(const void *p1, const void *p2) {
  if (*(U64*)p1 == *(U64*)p2) return 0;
  if (*(U64*)p1 < *(U64*)p2) return -1;
  return 1;
}
#define TOP (1<<21)
U64 d[TOP];
int main() {
  U64 c, x, tx, y, ty, z, tz, p = 0;
    for (x = tx = 3; tx <= TOP; tx+=x, ++x) {
    for (y = ty = 3; ty < tx;   ty+=y, ++y) {
    for (z = tz = 3; tz < ty;   tz+=z, ++z) {
    c = tx*ty*tz;
    if (c <= TOP*18 && isTriangular(c))  d[p++] = c;
  }}}
  qsort(d, p, 8, compare64);
  for (x=c=0; cx) printf("%llu, ", y), x=y;
  return 0;
}

A226389 Triangular numbers representable as triangular(x)*triangular(y)+1.

Original entry on oeis.org

1, 10, 91, 136, 946, 1711, 1891, 5671, 8911, 10585, 11026, 11935, 13861, 19306, 21736, 26335, 32131, 36856, 44551, 49141, 65341, 107416, 138601, 239086, 305371, 351541, 366796, 459361, 849556, 873181, 933661, 1100386, 1413721, 1516411, 1904176, 2297296, 2467531, 3837835
Offset: 1

Views

Author

Alex Ratushnyak, Jun 06 2013

Keywords

Comments

Cases with x=y are included.
The associated indices in A000217 are 1, 4, 13, 16, 43, 58, 61, 106, 133, 145, 148, 154,...

Examples

			10 = 3 * 3 + 1,
91 = 6 * 15 + 1,
3837835 = 378 * 10153 + 1.

Links

Crossrefs

Cf. A000217, A188630, A225390.

Programs

  • Maple
    A000217inv:=proc(n) local t1; t1:=floor(sqrt(2*n)); if n = t1*(t1+1)/2 then return t1 ; else return -1; end if; end proc:
A000217 := proc(n)
    n*(n+1)/2 ;
end proc:
isA226389 := proc(n)
    local Tx, Ty;
    if n = 1 then
        return true;
    elif A000217inv(n) >= 0 then
        for x from 0 do
            Tx := A000217(x) ;
            if Tx+1 > n then
                return false;
            end if;
            for y from 0 to x do
                Ty := A000217(y) ;
                if Tx*Ty+1 >n then
                    break;
                elif Tx*Ty+1 = n then
                    return true;
                end if;
            end do:
        end do:
    else
        false;
    end if;
end proc:
for n from 0 do
    Tn := A000217(n) ;
    if isA226389(Tn) then
        printf("%d,\n",Tn) ;
    end if;
end do: # R. J. Mathar, Jun 06 2013

A262251 Triangular numbers representable as 2^x + 3^y.

Original entry on oeis.org

3, 10, 28, 91
Offset: 1

Views

Author

Alex Ratushnyak, Sep 16 2015

Keywords

Comments

No other terms such that 0 <= x,y < 2000.
No other terms such that 0 <= x,y < 5250. - Michael S. Branicky, Mar 10 2021

Examples

			a(1) = 3 = 2^1 + 3^0.
a(4) = 91 = 2^6 + 3^3.

Crossrefs

Cf. A225390, A226499, A227027, A259745, A259746, A262242.
Intersection of A000217 and A004050.

Programs

  • PARI
    isok(t) = {for (k=0, logint(t, 2), my(tt = t - 2^k); if (tt, p = valuation(tt, 3); if (tt == 3^p, return(1))););}
lista(nn) = for (n=1, nn, if (isok(t=n*(n+1)/2), print1(t, ", "))); \\ Michel Marcus, Sep 20 2015
  • PARI
    select(x->ispolygonal(x, 3), setbinop(f, [0..20], [0..20])) \\ Michel Marcus, Mar 10 2021
  • Python
    from sympy import integer_nthroot
def auptoexponent(maxexp):
  sums = set(2**x + 3**y for x in range(maxexp) for y in range(maxexp))
  iroots = set(integer_nthroot(2*s, 2)[0] for s in sums)
  return sorted(set(r*(r+1)//2 for r in iroots if r*(r+1)//2 in sums))
print(auptoexponent(500)) # Michael S. Branicky, Mar 10 2021

A379609 Triples (x,y,z) such that T(x)*T(y) = T(z) with 2 <= x <= y, where T(n) is the n-th triangular number, sorted first by values of z, then by values of x.

Original entry on oeis.org

3, 3, 8, 2, 5, 9, 4, 6, 20, 2, 20, 35, 3, 14, 35, 4, 12, 39, 5, 11, 44, 7, 10, 55, 5, 19, 75, 3, 34, 84, 9, 13, 90, 6, 21, 98, 2, 76, 132, 6, 33, 153, 4, 55, 175, 14, 18, 189, 13, 20, 195, 12, 23, 207, 15, 20, 224, 14, 22, 230, 11, 28, 231, 12, 29, 260, 7, 51, 272, 10, 38, 285, 11, 36, 296, 5, 90, 350, 3, 143, 351, 10, 50, 374, 7, 75, 399, 9, 68, 459, 19, 34, 475
Offset: 1

Views

Author

Kelvin Voskuijl, Dec 27 2024

Keywords

Examples

			Triples begin:
  3,  3,  8;
  2,  5,  9;
  4,  6, 20;
  2, 20, 35;
  3, 14, 35;
  4, 12, 39;
  5, 11, 44;
  7, 10, 55;
  ...
The triple 2,5,9 is in this sequence because the second triangular number (3) times the fifth (15) is the ninth (45).

Crossrefs

Cf. A000217, A188630, A198453 (additive), A225390.

A262724 Triangular numbers representable as 3^x + y^3.

Original entry on oeis.org

1, 3, 10, 28, 36, 91, 1081, 2278, 2926, 8001, 46665, 5639761, 10911456, 166066200, 341532180, 3137785371, 1647882316985625, 875366737297292691171, 465198187808352499674075441
Offset: 1

Views

Author

Alex Ratushnyak, Sep 28 2015

Keywords

Comments

Any further terms are greater than 10^30. - Charles R Greathouse IV, Oct 02 2015

Crossrefs

Cf. A000217, A225390, A226499, A227027, A259745, A259746, A262242, A262251.

Programs

  • PARI
    list(lim)=my(v=List(),X,t); for(x=0,logint(lim\=1,3), X=3^x; for(y=0, sqrtnint(lim-X,3), if(ispolygonal(t=X+y^3,3), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Sep 28 2015

Extensions

a(17)-a(19) from Charles R Greathouse IV, Sep 28 2015
Showing 1-6 of 6 results.

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