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.

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A253823 Octagonal numbers (A000567) which are also centered triangular numbers (A005448).

Original entry on oeis.org

1, 97921, 1039585, 130402572385, 1384429704481, 173658931280825761, 1843664419471976641, 231264030011583194717761, 2455233718711319470593985, 307977546462671843639087352385, 3269669116478082433043125969441, 410138010309759307549971991199125921
Offset: 1

Views

Author

Colin Barker, Jan 14 2015

Keywords

Examples

			97921 is in the sequence because it is the 181st octagonal number and the 256th centered triangular number.

Links

Crossrefs

Cf. A000567, A005448, A253821, A253822.

Programs

  • Mathematica
    LinearRecurrence[{1,1331714,-1331714,-1,1},{1,97921,1039585,130402572385,1384429704481},20] (* Harvey P. Dale, Jan 22 2025 *)
  • PARI
    Vec(-x*(x^4+97920*x^3-390050*x^2+97920*x+1)/((x-1)*(x^2-1154*x+1)*(x^2+1154*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+1331714*a(n-2)-1331714*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+97920*x^3-390050*x^2+97920*x+1) / ((x-1)*(x^2-1154*x+1)*(x^2+1154*x+1)).

A254283 Indices of hexagonal numbers (A000384) which are also centered triangular numbers (A005448).

Original entry on oeis.org

1, 31, 115, 5965, 22261, 1157131, 4318471, 224477401, 837761065, 43547458615, 162521328091, 8447982493861, 31528299888541, 1638865056350371, 6116327657048815, 317931372949478065, 1186536037167581521, 61677047487142394191, 230181874882853766211
Offset: 1

Views

Author

Colin Barker, Jan 28 2015

Keywords

Comments

Also positive integers x in the solutions to 4*x^2 - 3*y^2 - 2*x + 3*y - 2 = 0, the corresponding values of y being A254284.

Examples

			31 is in the sequence because the 31st hexagonal number is 1891, which is also the 36th centered triangular number.

Links

Crossrefs

Cf. A000384, A005448, A254284, A254285.

Programs

  • PARI
    Vec(-x*(x^4+30*x^3-110*x^2+30*x+1)/((x-1)*(x^2-14*x+1)*(x^2+14*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+194*a(n-2)-194*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+30*x^3-110*x^2+30*x+1) / ((x-1)*(x^2-14*x+1)*(x^2+14*x+1)).

A254284 Indices of centered triangular numbers (A005448) which are also hexagonal numbers (A000384).

Original entry on oeis.org

1, 36, 133, 6888, 25705, 1336140, 4986541, 259204176, 967363153, 50284273908, 187663465045, 9754889933880, 36405744855481, 1892398362898716, 7062526838498173, 367115527512416928, 1370093800923789985, 71218519939045985220, 265791134852376758821
Offset: 1

Views

Author

Colin Barker, Jan 28 2015

Keywords

Comments

Also positive integers y in the solutions to 4*x^2 - 3*y^2 - 2*x + 3*y - 2 = 0, the corresponding values of x being A254283.

Examples

			36 is in the sequence because the 36th centered triangular number is 1891, which is also the 31st hexagonal number.

Links

Crossrefs

Cf. A000384, A005448, A254283, A254285.

Programs

  • Mathematica
    LinearRecurrence[{1,194,-194,-1,1},{1,36,133,6888,25705},20] (* Harvey P. Dale, Nov 11 2020 *)
  • PARI
    Vec(x*(35*x^3+97*x^2-35*x-1)/((x-1)*(x^2-14*x+1)*(x^2+14*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+194*a(n-2)-194*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(35*x^3+97*x^2-35*x-1) / ((x-1)*(x^2-14*x+1)*(x^2+14*x+1)).

A254285 Hexagonal numbers (A000384) which are also centered triangular numbers (A005448).

Original entry on oeis.org

1, 1891, 26335, 71156485, 991081981, 2677903145191, 37298379237211, 100780206894952201, 1403687203222107385, 3792762303606727977835, 52826364168762410080471, 142736816433155393822880781, 1988067387723517337746328821, 5371757345852607787523567324911
Offset: 1

Views

Author

Colin Barker, Jan 28 2015

Keywords

Examples

			1891 is in the sequence because it is the 31st hexagonal number and the 36th centered triangular number.

Links

Crossrefs

Cf. A000384, A005448, A254283, A254284.

Programs

  • PARI
    Vec(-x*(x^4+1890*x^3-13190*x^2+1890*x+1)/((x-1)*(x^2-194*x+1)*(x^2+194*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+37634*a(n-2)-37634*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+1890*x^3-13190*x^2+1890*x+1) / ((x-1)*(x^2-194*x+1)*(x^2+194*x+1)).

A254674 Indices of heptagonal numbers (A000566) which are also centered triangular numbers (A005448).

Original entry on oeis.org

1, 10, 34, 601, 2089, 37234, 129466, 2307889, 8024785, 143051866, 497407186, 8866907785, 30831220729, 549605230786, 1911038277994, 34066657400929, 118453542014881, 2111583153626794, 7342208566644610, 130884088867460281, 455098477589950921
Offset: 1

Views

Author

Colin Barker, Feb 05 2015

Keywords

Comments

Also positive integers x in the solutions to 5*x^2 - 3*y^2 - 3*x + 3*y - 2 = 0, the corresponding values of y being A254675.

Examples

			10 is in the sequence because the 10th heptagonal number is 235, which is also the 13th centered triangular number.

Links

Crossrefs

Cf. A000566, A005448, A254675, A254676.

Programs

  • PARI
    Vec(-x*(x^4+9*x^3-38*x^2+9*x+1)/((x-1)*(x^2-8*x+1)*(x^2+8*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+62*a(n-2)-62*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+9*x^3-38*x^2+9*x+1) / ((x-1)*(x^2-8*x+1)*(x^2+8*x+1)).

A254675 Indices of centered triangular numbers (A005448) which are also heptagonal numbers (A000566).

Original entry on oeis.org

1, 13, 44, 776, 2697, 48069, 167140, 2979472, 10359953, 184679165, 642149916, 11447128728, 39802934809, 709537301941, 2467139808212, 43979865591584, 152922865174305, 2726042129376237, 9478750500998668, 168970632155735080, 587529608196743081
Offset: 1

Views

Author

Colin Barker, Feb 05 2015

Keywords

Comments

Also positive integers y in the solutions to 5*x^2 - 3*y^2 - 3*x + 3*y - 2 = 0, the corresponding values of x being A254674.

Examples

			13 is in the sequence because the 13th centered triangular number is 235, which is also the 10th heptagonal number.

Links

Crossrefs

Cf. A000566, A005448, A254674, A254676.

Programs

  • PARI
    Vec(x*(12*x^3+31*x^2-12*x-1)/((x-1)*(x^2-8*x+1)*(x^2+8*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+62*a(n-2)-62*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(12*x^3+31*x^2-12*x-1) / ((x-1)*(x^2-8*x+1)*(x^2+8*x+1)).

A254676 Heptagonal numbers (A000566) which are also centered triangular numbers (A005448).

Original entry on oeis.org

1, 235, 2839, 902101, 10906669, 3465871039, 41903418691, 13315875628969, 160992923703385, 51159590700627091, 618534770964985711, 196555134155933653885, 2376410429054551397509, 755164774267506397598311, 9130168249892815504243099, 2901342866180625423639056209
Offset: 1

Views

Author

Colin Barker, Feb 05 2015

Keywords

Examples

			235 is in the sequence because it is the 10th heptagonal number and the 13th centered triangular number.

Links

Crossrefs

Cf. A000566, A005448, A254674, A254675.

Programs

  • Mathematica
    LinearRecurrence[{1,3842,-3842,-1,1},{1,235,2839,902101,10906669},20] (* Harvey P. Dale, Oct 12 2024 *)
  • PARI
    Vec(-x*(x^4+234*x^3-1238*x^2+234*x+1)/((x-1)*(x^2-62*x+1)*(x^2+62*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+3842*a(n-2)-3842*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+234*x^3-1238*x^2+234*x+1) / ((x-1)*(x^2-62*x+1)*(x^2+62*x+1)).

A140701 Partial products of A005448.

Original entry on oeis.org

1, 1, 4, 40, 760, 23560, 1083760, 69360640, 5895654400, 642626329600, 87397180825600, 14507932017049600, 2887078471392870400, 678463440777324544000, 185898982772986925056000, 58744078556263868317696000, 21206612358811256462688256000
Offset: 0

Views

Author

Jonathan Vos Post, May 24 2008

Keywords

Examples

			a(10) = 87397180825600 = 1 * 4 * 10 * 19 * 31 * 46 * 64 * 85 * 109 * 136.

Links

Crossrefs

Cf. A005448.
For analog with centered n-gonal numbers see A140702.
For analog with regular triangular numbers see A006472.
For the analog with a partial sum instead of a partial product see A006003.

Programs

  • Mathematica
    Table[Product[(3*k^2-3*k+2)/2,{k,1,n}],{n,1,20}] (* Vaclav Kotesovec, Jul 11 2015 *)
FoldList[Times,3*Accumulate[Range[0,20]]+1] (* Harvey P. Dale, Aug 05 2018 *)
  • PARI
    a(n) = prod(k=1, n, 3*k*(k-1)/2 + 1); \\ Michel Marcus, Mar 02 2023

Formula

a(n) = Product_{k=1..n} A005448(k).
a(n) ~ cosh(Pi*sqrt(5/3)/2) * 3^n * n^(2*n) / (exp(2*n) * 2^(n-1)). - Vaclav Kotesovec, Jul 11 2015
a(n) = (2/3)^(1 - n) * Pochhammer(1 + (3 - i*sqrt(15))/6, n - 1) * Pochhammer(1 + (3 + i*sqrt(15))/6, n - 1), for n>=1. - Antonio Graciá Llorente, Sep 10 2023

A162703 Palindromes in A005448.

Original entry on oeis.org

1, 4, 15151, 45154, 66466, 92629, 98689, 4976794, 6424246, 648616846, 136287949782631, 479573060375974, 69465717171756496, 4345218593958125434, 42097537753535773579024, 58071646151315164617085, 6220959179720279719590226, 458122911526080625119221854
Offset: 1

Views

Author

Claudio Meller, Jul 11 2009

Keywords

Comments

Essentially the palindromes which are sums of three consecutive triangular numbers T.
Indices of the centered triangular numbers: 1, 2, 101, 174, 211, 249, 257, 1822, 2070, 20795, 9531980, 17880587, 215198695, ..., (A195903). - Robert G. Wilson v
a(18) > 10^25. - Donovan Johnson, Sep 29 2011
a(31) > 10^40. - Patrick De Geest, May 23 2021

Examples

			T(99) + T(100) + T(101) = 15151.
T(172) + T(173) + T(174) = 45154.

Links

Crossrefs

Cf. A000217, A005448, A195903.

Programs

  • Mathematica
    n = 1; lst = {}; While[n < 10^10, ctn = 3 n (n - 1)/2 + 1; id = IntegerDigits@ ctn; If[id == Reverse@id, AppendTo[lst, ctn]; Print[{n, ctn}]]; n++ ]; lst (* Robert G. Wilson v *)

Formula

a(n) = (3*m^2 - 3*m + 2)/2 or a(n) = (3*n^2 + 3*n + 2)/2 with n = m - 1.

Extensions

Edited and extended by R. J. Mathar and Robert G. Wilson v, Jul 13 2009
a(14)-a(17) from Donovan Johnson, Sep 29 2011
a(18)-a(30) from Patrick De Geest, May 23 2021

A253470 Indices of centered triangular numbers (A005448) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 5, 36, 280, 2201, 17325, 136396, 1073840, 8454321, 66560725, 524031476, 4125691080, 32481497161, 255726286205, 2013328792476, 15850904053600, 124793903636321, 982500325036965, 7735208696659396, 60899169248238200, 479458145289246201, 3774765993065731405
Offset: 1

Views

Author

Colin Barker, Jan 01 2015

Keywords

Comments

Also indices of pentagonal numbers (A000326) which are also centered pentagonal numbers (A005891).
Also positive integers x in the solutions to 3*x^2 - 5*y^2 - 3*x + 5*y = 0, the corresponding values of y being A182432.

Examples

			5 is in the sequence because the 5th centered triangular number is 31, which is also the 4th centered pentagonal number.

Links

Crossrefs

Cf. A005448, A005891, A182432, A253654.

Programs

  • PARI
    Vec(x*(4*x-1)/((x-1)*(x^2-8*x+1)) + O(x^100))

Formula

a(n) = 9*a(n-1)-9*a(n-2)+a(n-3).
G.f.: x*(4*x-1) / ((x-1)*(x^2-8*x+1)).
a(n) = (6-(4-sqrt(15))^n*(3+sqrt(15))+(-3+sqrt(15))*(4+sqrt(15))^n)/12. - Colin Barker, Mar 03 2016
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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