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 1-10 of 147 results. Next

A129444 Numbers k such that the centered triangular number A005448(k) = 3*k*(k-1)/2 + 1 is a perfect square.

Original entry on oeis.org

0, 1, 2, 7, 16, 65, 154, 639, 1520, 6321, 15042, 62567, 148896, 619345, 1473914, 6130879, 14590240, 60689441, 144428482, 600763527, 1429694576, 5946945825, 14152517274, 58868694719, 140095478160, 582740001361, 1386802264322
Offset: 1

Views

Author

Alexander Adamchuk, Apr 15 2007

Keywords

Comments

Corresponding numbers m > 0 such that m^2 is a centered triangular number are listed in A129445 = {1, 2, 8, 19, 79, 188, 782, 1861, 7741, 18422, 76628, 182359, ...}.

Examples

			G.f. = x^2 + 2*x^3 + 7*x^4 + 16*x^5 + 65*x^6 + 154*x^7 + 639*x^8 + 1520*x^9 + ...

Links

Crossrefs

Cf. A005448 (centered triangular numbers).
Cf. A129445 (numbers k > 0 such that k^2 is a centered triangular number).
Cf. A000290, A004189, A249483.

Programs

  • Magma
    I:=[0,1,2,7,16,65]; [n le 6 select I[n] else 11*Self(n-2) -11*Self(n-4) +Self(n-6): n in [1..40]]; // G. C. Greubel, Feb 07 2024
  • Mathematica
    Do[ f = 3n(n-1)/2 + 1; If[ IntegerQ[ Sqrt[f] ], Print[ n ] ], {n,1,150000} ]
a[1]=0;a[2]=1;a[3]=2;a[4]=7;a[5]=16;a[6]=65;a[n_]:=a[n]=11(a[n-2]-a[n-4])+a[n-6];Table[a[n], {n, 100}] (* Zak Seidov, Apr 17 2007 *)
LinearRecurrence[{1,10,-10,-1,1},{0,1,2,7,16},30] (* Harvey P. Dale, Dec 06 2012 *)
  • PARI
    {a(n) = my(m); m = if( n<1, 2-n, n-1); (n<1) + (-1)^(n<1) * polcoeff( (x + x^2 - 5*x^3 - x^4) / ((1 - x) * (1 - 10*x^2 + x^4)) + x * O(x^m), m)}; /* Michael Somos, Apr 05 2008 */
  • SageMath
    def A129444_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2*(1+x-5*x^2-x^3)/((1-x)*(1-10*x^2+x^4)) ).list()
a=A129444_list(40); a[1:] # G. C. Greubel, Feb 07 2024

Formula

a(n) = 1/2 + sqrt(1/4 + (2/3)*(A129445(n)^2 - 1)).
a(n) = 11*(a(n-2) - a(n-4)) + a(n-6); a(1)=0; a(2)=1; a(3)=2; a(4)=7; a(5)=16; a(6)=65. - Zak Seidov, Apr 17 2007
a(n) = 1 - a(-n+3) for all n in Z. - Michael Somos, Apr 05 2008
G.f.: x^2*(1 + x - 5*x^2 - x^3) / ((1 - x) * (1 - 10*x^2 + x^4)). - Michael Somos, Apr 05 2008
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - a(n-4) + a(n-5); a(1)=0, a(2)=1, a(3)=2, a(4)=7, a(5)=16. - Harvey P. Dale, Dec 06 2012
a(n) = (1/2)*(2*[n=0] + 1 + ((1+(-1)^n)/2)*(31*b(n/2) - 3*b(n/2 + 1)) + ((1-(-1)^n)/2)*(13*b((n-1)/2) - b((n+1)/2))), where b(n)=A004189(n). - G. C. Greubel, Feb 07 2024

Extensions

More terms from Zak Seidov, Apr 17 2007

A253673 Indices of centered triangular numbers (A005448) that are also centered octagonal numbers (A016754).

Original entry on oeis.org

1, 16, 65, 1520, 6321, 148896, 619345, 14590240, 60689441, 1429694576, 5946945825, 140095478160, 582740001361, 13727927165056, 57102573187505, 1345196766697280, 5595469432374081, 131815555209168336, 548298901799472385, 12916579213731799600
Offset: 1

Views

Author

Colin Barker, Jan 08 2015

Keywords

Comments

Also positive integers x in the solutions to 3*x^2 - 8*y^2 - 3*x + 8*y = 0, the corresponding values of y being A253674.
Also indices of centered square numbers (A001844) that are also octagonal numbers (A000567). - Colin Barker, Feb 10 2015

Examples

			16 is in the sequence because the 16th centered triangular number is 361, which is also the 10th centered octagonal number.

Links

Crossrefs

Cf. A005448, A016754, A253674, A253675.
Cf. A000567, A001844, A254895, A254896.

Programs

  • Mathematica
    LinearRecurrence[{1,98,-98,-1,1},{1,16,65,1520,6321},20] (* Harvey P. Dale, Aug 07 2023 *)
  • PARI
    Vec(x*(3*x-1)*(5*x^2+18*x+1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(3*x-1)*(5*x^2+18*x+1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)).

A249483 Squares (A000290) which are also centered triangular numbers (A005448).

Original entry on oeis.org

1, 4, 64, 361, 6241, 35344, 611524, 3463321, 59923081, 339370084, 5871850384, 33254804881, 575381414521, 3258631508224, 56381506772644, 319312633001041, 5524812282304561, 31289379402593764, 541375222159074304, 3066039868821187801, 53049246959306977201
Offset: 1

Views

Author

Colin Barker, Jan 13 2015

Keywords

Comments

Apart from the first term the same as A165516. - R. J. Mathar, Jan 20 2015

Examples

			64 is in the sequence because the 8th square is 64, which is also the 7th centered triangular number.

Links

Crossrefs

Cf. A000290, A005448, A129444, A129445.

Programs

  • Magma
    I:=[1,4,64,361,6241]; [n le 5 select I[n] else Self(n-1)+98*Self(n-2)-98*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Jan 20 2015
  • Mathematica
    CoefficientList[Series[(x^2 - 5 x + 1) (x^2 + 8 x + 1) / ((1 - x) (x^2 - 10 x + 1) (x^2 + 10 x + 1)), {x, 0, 70}], x] (* Vincenzo Librandi, Jan 20 2015 *)
  • PARI
    Vec(-x*(x^2-5*x+1)*(x^2+8*x+1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^2-5*x+1)*(x^2+8*x+1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)).

A253476 Indices of centered triangular numbers (A005448) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 15, 70, 1596, 7645, 175491, 840826, 19302360, 92483161, 2123084055, 10172306830, 233519943636, 1118861268085, 25685070715851, 123064567182466, 2825124258799920, 13535983528803121, 310737983397275295, 1488835123601160790, 34178353049441482476
Offset: 1

Views

Author

Colin Barker, Jan 02 2015

Keywords

Comments

Also positive integers x in the solutions to 3*x^2 - 7*y^2 - 3*x + 7*y = 0, the corresponding values of y being A253477.

Examples

			15 is in the sequence because the 15th centered triangular number is 316, which is also the 10th centered heptagonal number.

Links

Crossrefs

Cf. A005448, A069099, A253477, A253689.

Programs

  • Mathematica
    LinearRecurrence[{1,110,-110,-1,1},{1,15,70,1596,7645},30] (* Harvey P. Dale, Jun 14 2016 *)
  • PARI
    Vec(x*(14*x^3+55*x^2-14*x-1)/((x-1)*(x^4-110*x^2+1)) + O(x^100))

Formula

a(n) = a(n-1)+110*a(n-2)-110*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(14*x^3+55*x^2-14*x-1) / ((x-1)*(x^4-110*x^2+1)).

A253477 Indices of centered heptagonal numbers (A069099) which are also centered triangular numbers (A005448).

Original entry on oeis.org

1, 10, 46, 1045, 5005, 114886, 550450, 12636361, 60544441, 1389884770, 6659338006, 152874688285, 732466636165, 16814825826526, 80564670640090, 1849477966229521, 8861381303773681, 203425761459420730, 974671378744464766, 22374984282570050725
Offset: 1

Views

Author

Colin Barker, Jan 02 2015

Keywords

Comments

Also positive integers y in the solutions to 3*x^2 - 7*y^2 - 3*x + 7*y = 0, the corresponding values of x being A253476.

Examples

			10 is in the sequence because the 10th centered heptagonal number is 316, which is also the 15th centered triangular number.

Links

Crossrefs

Cf. A005448, A069099, A253476, A253689.

Programs

  • Mathematica
    LinearRecurrence[{1,110,-110,-1,1},{1,10,46,1045,5005},30] (* Harvey P. Dale, Aug 13 2018 *)
  • PARI
    Vec(-x*(x^4+9*x^3-74*x^2+9*x+1)/((x-1)*(x^4-110*x^2+1)) + O(x^100))

Formula

a(n) = a(n-1)+110*a(n-2)-110*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+9*x^3-74*x^2+9*x+1) / ((x-1)*(x^4-110*x^2+1)).

A253674 Indices of centered octagonal numbers (A016754) which are also centered triangular numbers (A005448).

Original entry on oeis.org

1, 10, 40, 931, 3871, 91180, 379270, 8934661, 37164541, 875505550, 3641745700, 85790609191, 356853914011, 8406604195120, 34968041827330, 823761420512521, 3426511245164281, 80720212606031890, 335763133984272160, 7909757073970612651, 32901360619213507351
Offset: 1

Views

Author

Colin Barker, Jan 08 2015

Keywords

Comments

Also positive integers y in the solutions to 3*x^2 - 8*y^2 - 3*x + 8*y = 0, the corresponding values of x being A253673.

Examples

			10 is in the sequence because the 10th centered octagonal number is 361, which is also the 16th centered triangular number.

Links

Crossrefs

Cf. A005448, A016754, A253673, A253675.

Programs

  • Mathematica
    LinearRecurrence[{1,98,-98,-1,1},{1,10,40,931,3871},30] (* Harvey P. Dale, Oct 01 2015 *)
  • PARI
    Vec(-x*(x^2-5*x+1)*(x^2+14*x+1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^2-5*x+1)*(x^2+14*x+1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)).

A253675 Centered triangular numbers (A005448) which are also centered octagonal numbers (A016754).

Original entry on oeis.org

1, 361, 6241, 3463321, 59923081, 33254804881, 575381414521, 319312633001041, 5524812282304561, 3066039868821187801, 53049246959306977201, 29440114501108412261161, 509378863778453312776441, 282683976373603105710477121, 4891055796951461749972406281
Offset: 1

Views

Author

Colin Barker, Jan 08 2015

Keywords

Examples

			361 is in the sequence because it is the 16th centered triangular number and the 10th centered octagonal number.

Links

Crossrefs

Cf. A005448, A016754, A253673, A253674.

Programs

  • Mathematica
    LinearRecurrence[{1,9602,-9602,-1,1},{1,361,6241,3463321,59923081},20] (* Harvey P. Dale, Dec 09 2017 *)
  • PARI
    Vec(-x*(x^4+360*x^3-3722*x^2+360*x+1)/((x-1)*(x^2-98*x+1)*(x^2+98*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+9602*a(n-2)-9602*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+360*x^3-3722*x^2+360*x+1) / ((x-1)*(x^2-98*x+1)*(x^2+98*x+1)).

A253689 Centered triangular numbers (A005448) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 316, 7246, 3818431, 87657571, 46195373386, 1060481282176, 558871623400861, 12829702464103141, 6761228853708238456, 155213739350238513106, 81797346113290645435291, 1877775805829483067448711, 989584286517361374767907526, 22717331543711346799755988036
Offset: 1

Views

Author

Colin Barker, Jan 09 2015

Keywords

Examples

			316 is in the sequence because it is the 15th centered triangular number and the 10th centered heptagonal number.

Links

Crossrefs

Cf. A005448, A069099, A253476, A253477.

Programs

  • Magma
    I:=[1,316,7246,3818431,87657571]; [n le 5 select I[n] else  Self(n-1)+12098*Self(n-2)-12098*Self(n-3)-Self(n-4)+Self(n-5): n in [1..20]]; // Vincenzo Librandi, Jan 10 2015
  • Mathematica
    LinearRecurrence[{1, 12098, -12098, -1, 1}, {1, 316, 7246, 3818431, 87657571}, 20] (* or *) CoefficientList[Series[(x^4 + 315 x^3 - 5168 x^2 + 315 x + 1) / ((1 - x) (x^2 - 110 x + 1)(x^2 + 110 x + 1)), {x, 0, 20}], x] (* Vincenzo Librandi, Jan 10 2015 *)
  • PARI
    Vec(-x*(x^4+315*x^3-5168*x^2+315*x+1)/((x-1)*(x^2-110*x+1)*(x^2+110*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+12098*a(n-2)-12098*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+315*x^3-5168*x^2+315*x+1) / ((x-1)*(x^2-110*x+1)*(x^2+110*x+1)).

A253821 Indices of octagonal numbers (A000567) which are also centered triangular numbers (A005448).

Original entry on oeis.org

1, 181, 589, 208489, 679321, 240595741, 783935461, 277647276241, 904660842289, 320404716185989, 1043977828065661, 369746764831354681, 1204749508926930121, 426687446210667115501, 1390279889323849293589, 492396943180345019933089, 1604381787530213157871201
Offset: 1

Views

Author

Colin Barker, Jan 14 2015

Keywords

Comments

Also positive integers x in the solutions to 6*x^2 - 3*y^2 - 4*x + 3*y - 2 = 0, the corresponding values of y being A253822.

Examples

			181 is in the sequence because the 181st octagonal number is 97921, which is also the 256th centered triangular number.

Links

Crossrefs

Cf. A000567, A005448, A253822, A253823.

Programs

  • PARI
    Vec(-x*(x^4+180*x^3-746*x^2+180*x+1)/((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+1154*a(n-2)-1154*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+180*x^3-746*x^2+180*x+1) / ((x-1)*(x^2-34*x+1)*(x^2+34*x+1)).

A253822 Indices of centered triangular numbers (A005448) which are also octagonal numbers (A000567).

Original entry on oeis.org

1, 256, 833, 294848, 960705, 340253760, 1108652161, 392652543616, 1279383632513, 453120695078528, 1476407603267265, 522900889468077120, 1703773094786790721, 603427173325465917376, 1966152674976353224193, 696354435116698200574208, 2268938483149616833927425
Offset: 1

Views

Author

Colin Barker, Jan 14 2015

Keywords

Comments

Also positive integers y in the solutions to 6*x^2 - 3*y^2 - 4*x + 3*y - 2 = 0, the corresponding values of x being A253821.

Examples

			256 is in the sequence because the 256th centered triangular number is 97921, which is also the 181st octagonal number.

Links

Crossrefs

Cf. A000567, A005448, A253821, A253823.

Programs

  • Mathematica
    LinearRecurrence[{1,1154,-1154,-1,1},{1,256,833,294848,960705},20] (* Harvey P. Dale, Jul 19 2019 *)
  • PARI
    Vec(x*(255*x^3+577*x^2-255*x-1)/((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+1154*a(n-2)-1154*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(255*x^3+577*x^2-255*x-1) / ((x-1)*(x^2-34*x+1)*(x^2+34*x+1)).
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