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 1-5 of 5 results.

A292309 Numbers equal to the sum of three triangular numbers in arithmetic progression.

Original entry on oeis.org

9, 63, 84, 108, 234, 315, 459, 513, 570, 630, 759, 975, 1053, 1134, 1395, 1584, 1998, 2109, 2709, 2838, 2970, 3105, 3384, 3528, 3825, 4134, 4455, 4620, 4788, 4959, 5133, 5310, 5673, 5859, 6834, 7038, 7668, 7884, 8325, 8778, 9009, 9243, 9480, 10209, 10710, 11223
Offset: 1

Views

Author

Antonio Roldán, Sep 14 2017

Keywords

Comments

Subsequence of A045943, because a(n) = 3*k*(k+1)/2 = 3*A000217(k) for some k.

Examples

			9 = A000217(0) + A000217(2) + A000217(3) = 0 + 3 + 6, with 6 - 3 = 3 - 0 = 3.
513 = A000217(11) + A000217(18) + A000217(23) = 66 + 171 + 276, with 171 - 66 = 276 - 171 = 105.

Crossrefs

Cf. A000217, A045943, A127329, A292310, A292313, A292314, A292316.

Programs

  • Mathematica
    Module[{t = 3, k = 2, i, e, v, m}, Reap[While[t <= 5000, i = k; e = 0; v = t+i; While[i > 0 && e == 0, If[IntegerQ @ Sqrt[8v+1], m = 3t; e = 1; Sow[m]]; i--; v += i]; k++; t += k]][[2, 1]]] (* Jean-François Alcover, Jun 25 2023, after PARI code *)
  • PARI
    t=3;k=2;while(t<=5000,i=k;e=0;v=t+i;while(i>1&&e==0,if(issquare(8*v+1),m=3*t;e=1;print1(m,", "));i+=-1;v+=i);k+=1;t+=k)

Formula

a(n) = 3*A292310(n).

A292310 Triangular numbers that are equidistant from two other triangular numbers.

Original entry on oeis.org

3, 21, 28, 36, 78, 105, 153, 171, 190, 210, 253, 325, 351, 378, 465, 528, 666, 703, 903, 946, 990, 1035, 1128, 1176, 1275, 1378, 1485, 1540, 1596, 1653, 1711, 1770, 1891, 1953, 2278, 2346, 2556, 2628, 2775, 2926, 3003, 3081, 3160, 3403, 3570, 3741, 3828, 4095, 4186, 4278, 4371, 4656
Offset: 1

Views

Author

Antonio Roldán, Sep 14 2017

Keywords

Comments

Triangular numbers which are the arithmetic mean of two other triangular numbers. - R. J. Mathar, Oct 01 2017

Examples

			3 is in the sequence because 0 = A000217(0), 6 = A000217(3), and the distances from 3 to 0 and 3 to 6 are the same.
153 is in the sequence because 153 = A000217(17), 6 = A000217(2), 300 = A000217(24), and the two distances 300-153 = 153-6 = 147 are the same.

Crossrefs

Cf. A000217, A292309, A292313, A292314, A292316.

Programs

  • Maple
    isA292310 := proc(n)
    local ilow ;
    if isA000217(n) then
        for ilow from 0 do
            tilow := A000217(ilow) ;
            if tilow >= n then
                return false ;
            elif isA000217(2*n-tilow) then
                return true ;
            end if;
        end do:
    else
        false;
    end if;
end proc:
for n from 1 to 5000 do
    if isA292310(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 01 2017
  • Mathematica
    Module[{t = 3, k = 2, i, e, v}, Reap[While[t <= 6000, i = k; e = 0; v = t + i; While[i > 0 && e == 0, If[IntegerQ@Sqrt[8v + 1], e = 1; Sow[t]]; i--; v += i]; k++; t += k]][[2, 1]]] (* Jean-François Alcover, Jun 25 2023, after first PARI code *)
  • PARI
    t=3; k=2; while(t<=6000, i=k; e=0; v=t+i; while(i>0&&e==0, if(issquare(8*v+1), e=1; print1(t,", ")); i--; v+=i); k++; t+=k)
  • PARI
    upto(n) = {my(t = 0, i = 0, triangulars = List([0]), res = List); while(t <= n, i++; t+=i; listput(triangulars, t)); for(i=2,#triangulars, tr = triangulars[i]<<1; for(j = 1, i-1, if(issquare(8 * (tr - triangulars[j]) + 1), listput(res, triangulars[i]); next(2)))); res} \\ David A. Corneth, Oct 04 2017

Formula

a(n) = A292309(n)/3.

Extensions

Term 105 added by David A. Corneth, Oct 04 2017

A292314 Numbers equal to the sum of three oblong numbers in arithmetic progression.

Original entry on oeis.org

18, 126, 168, 216, 468, 918, 1026, 1140, 1260, 1518, 1950, 2106, 2268, 2790, 3168, 3996, 4218, 5418, 5676, 5940, 6210, 6768, 7056, 7650, 8268, 8910, 9240, 9576, 9918, 10266, 10620, 11346, 11718, 13668, 14076, 15336, 15768, 16650, 17556, 18018, 18486, 18960, 20418, 21420, 22446
Offset: 1

Views

Author

Antonio Roldán, Sep 14 2017

Keywords

Comments

Subsequence of A028896.

Examples

			126 = 3*4 + 6*7 + 8*9 = 12 + 42 + 72, with 72 - 42 = 42 - 12 = 30;
468 = 8*9 + 12*13 + 15*16 = 72 + 156 + 240, with 240 - 156 = 156 - 72 = 84.

Crossrefs

Cf. A292309, A292310, A292313, A292316.

Programs

  • Mathematica
    o[n_] := n(n+1); s[x_] := Reduce[ x+k == o[y] && x-k == o[z] && k>0 && z>0, {z, y, k}, Integers]; 3 Select[o@ Range@ 93, s[#] =!= False &] (* Giovanni Resta, Sep 18 2017 *)
  • PARI
    t=2; k=2; while(t<=10^4, i=k; e=0; v=t+i; while(i>2&&e==0, if(issquare(4*v+1), m=3*t; e=1; print1(m,", ")); i+=-2; v+=i); k+=2; t+=k)

Formula

a(n) = 3*A292316(n).

A292316 Oblong numbers equidistant from two other oblong numbers.

Original entry on oeis.org

6, 42, 56, 72, 156, 306, 342, 380, 420, 506, 650, 702, 756, 930, 1056, 1332, 1406, 1806, 1892, 1980, 2070, 2256, 2352, 2550, 2756, 2970, 3080, 3192, 3306, 3422, 3540, 3782, 3906, 4556, 4692, 5112, 5256, 5550, 5852, 6006, 6162, 6320, 6806, 7140, 7482, 7656, 8190, 8372, 8556, 8742
Offset: 1

Views

Author

Antonio Roldán, Sep 14 2017

Keywords

Comments

Oblong numbers (A002378) that are the arithmetic mean of two other oblong numbers. - R. J. Mathar, Oct 05 2017

Examples

			6 = 2*3 is an oblong number and equidistant from 12 = 3*4 and 0 = 0*1.
342 = 18*19 is oblong number and equidistant from 132 = 11*12 and 552 = 23*24 (552-342 = 210; 342-132 = 210).

Crossrefs

Cf. A002378, A292309, A292310, A292313, A292314.

Programs

  • Mathematica
    o[n_] := n(n+1); s[x_] := Reduce[ x+k == o[y] && x-k == o[z] && k>0 && z>0, {z, y, k}, Integers]; Select[o@ Range@ 93, s[#] =!= False &] (* Giovanni Resta, Sep 18 2017 *)
  • PARI
    t=2;k=2;while(t<=10^4,i=k;e=0;v=t+i;while(i>2&&e==0,if(issquare(4*v+1),e=1;print1(t,", "));i+=-2;v+=i);k+=2;t+=k)

Formula

a(n) = A292314(n)/3.

A306212 Numbers that are the sum of squares of three distinct positive integers in arithmetic progression.

Original entry on oeis.org

14, 29, 35, 50, 56, 66, 77, 83, 93, 107, 110, 116, 126, 140, 149, 155, 158, 165, 179, 194, 197, 200, 210, 219, 224, 242, 245, 251, 261, 264, 275, 290, 293, 302, 308, 315, 318, 332, 341, 350, 365, 371, 372, 381, 395, 398, 413, 428, 434, 435, 440, 450, 461, 462, 464, 482
Offset: 1

Views

Author

Antonio Roldán, Jan 29 2019

Keywords

Examples

			35 = 1^2 + 3^2 + 5^2, with 3 - 1 = 5 - 3 = 2;
371 = 1^2 + 9^2 + 17^2, with 9 - 1 = 17 - 9 = 8. Also 371 = 9^2 + 11^2 + 13^2, with 11 - 9 = 13 - 11 = 2.

Links

Crossrefs

Cf. A000290, A000378, A000408, A085317, A120328, A292313, A306213, A306214.

Programs

  • Maple
    N:= 1000: # for terms <= N
S:= {seq(seq(3*a^2+2*b^2, b=1..min(a-1, floor(sqrt((N-3*a^2)/2)))),a=1..floor(sqrt(N/3)))}:
sort(convert(S,list)); # Robert Israel, Jun 08 2020
  • PARI
    for(n=3, 600, k=sqrt(n/3); a=2; v=0; while(a<=k&&v==0, b=(n-3*a^2)/2; if(b==truncate(b)&&issquare(b), d=sqrt(b); if(d>=1&&d<=a-1, v=1; print1(n,", "))); a+=1))
  • PARI
    w=List(); for(n=3, 600, k=sqrt(n/3); for(a=2, k, for(c=1, a-1, v=(a-c)^2+a^2+(a+c)^2; if(v==n, listput(w,n))))); print(vecsort(Vec(w),,8))
Showing 1-5 of 5 results.

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