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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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)).

A253514 Centered heptagonal numbers (A069099) which are also centered octagonal numbers (A016754).

Original entry on oeis.org

1, 841, 755161, 678133681, 608963290321, 546848356574521, 491069215240629481, 440979608437728699361, 395999197307865131396641, 355606838202854450265484201, 319334544706965988473273415801, 286762065540017254794549261905041
Offset: 1

Views

Author

Colin Barker, Jan 03 2015

Keywords

Examples

			841 is in the sequence because it is the 16th centered heptagonal number and the 15th centered octagonal number.

Links

Crossrefs

Cf. A001110, A016754, A069099, A157877, A157879, A253446, A253447.

Programs

  • PARI
    Vec(-x*(x^2-58*x+1)/((x-1)*(x^2-898*x+1)) + O(x^100))

Formula

a(n) = 899*a(n-1)-899*a(n-2)+a(n-3).
G.f.: -x*(x^2-58*x+1) / ((x-1)*(x^2-898*x+1)).
From Peter Bala, Apr 15 2025; (Start)
a(n) = (1/64)*(-4 + sqrt(14))^2*(15 + 4*sqrt(14) + (449 + 120*sqrt(14))^n)^2 *(449 + 120*sqrt(14))^(-n).
a(-n) = a(n+1).
a(n) = (1/16) * (1 - T(2*n+1, -15)), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. Cf. A001110.
a(n) = A157877(n)^2 = 1 + 7*A157879(n).
a(2) divides a(3*n+2); a(3) divides a(5*n+3); a(4) divides a(7*n+4); a(5) divides a(9*n+5). In general, a(k) divides a((2*k-1)*n + k). (End)

A253546 Centered hexagonal numbers (A003215) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 547, 368551, 248402701, 167423051797, 112842888508351, 76055939431576651, 51261590333994154297, 34550235829172628419401, 23286807687272017560521851, 15695273830985510663163308047, 10578591275276546914954509101701, 7129954824262561635168675971238301
Offset: 1

Views

Author

Colin Barker, Jan 03 2015

Keywords

Examples

			547 is in the sequence because it is the 14th centered hexagonal number and the 13th centered heptagonal number.

Links

Crossrefs

Cf. A003215, A069099, A253457, A253458.

Programs

  • Mathematica
    LinearRecurrence[{675,-675,1},{1,547,368551},20] (* Harvey P. Dale, Aug 03 2021 *)
  • PARI
    Vec(-x*(x^2-128*x+1) / ((x-1)*(x^2-674*x+1)) + O(x^100))

Formula

a(n) = 675*a(n-1)-675*a(n-2)+a(n-3).
G.f.: -x*(x^2-128*x+1) / ((x-1)*(x^2-674*x+1)).

A253599 Centered square numbers (A001844) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 841, 43513, 54236113, 2807177521, 3498988573081, 181102250526121, 225733748749491361, 11683630587634972513, 14562987063325697070313, 753757743549580366157401, 939516547177660272044661361, 48627927055673997154643575441, 60611970510056587727363585953081
Offset: 1

Views

Author

Colin Barker, Jan 05 2015

Keywords

Examples

			841 is in the sequence because it is the 21st centered square number and the 16th centered heptagonal number.

Links

Crossrefs

Cf. A001844, A069099, A253459, A253460.

Programs

  • Mathematica
    LinearRecurrence[{1,64514,-64514,-1,1},{1,841,43513,54236113,2807177521},20] (* Harvey P. Dale, Mar 26 2023 *)
  • PARI
    Vec(-x*(x^4+840*x^3-21842*x^2+840*x+1)/((x-1)*(x^2-254*x+1)*(x^2+254*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+64514*a(n-2)-64514*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+840*x^3-21842*x^2+840*x+1) / ((x-1)*(x^2-254*x+1)*(x^2+254*x+1)).

A253621 Indices of centered heptagonal numbers (A069099) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 6, 66, 781, 9301, 110826, 1320606, 15736441, 187516681, 2234463726, 26626048026, 317278112581, 3780711302941, 45051257522706, 536834378969526, 6396961290111601, 76226701102369681, 908323451938324566, 10823654722157525106, 128975533213951976701
Offset: 1

Views

Author

Colin Barker, Jan 06 2015

Keywords

Comments

Also positive integers y in the solutions to 5*x^2 - 7*y^2 - 5*x + 7*y = 0, the corresponding values of x being A133272.

Examples

			6 is in the sequence because the 6th centered heptagonal number is 106, which is also the 7th centered pentagonal number.

Links

Crossrefs

Cf. A005891, A069099, A133272, A253622.

Programs

  • Magma
    I:=[1,6]; [n le 2 select I[n] else 12*Self(n-1)-Self(n-2)-5: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016
  • Mathematica
    RecurrenceTable[{a[1] == 1, a[2] == 6, a[n] == 12 a[n-1] - a[n-2] - 5}, a, {n, 20}] (* Vincenzo Librandi, Mar 05 2016 *)
  • PARI
    Vec(-x*(x^2-7*x+1)/((x-1)*(x^2-12*x+1)) + O(x^100))

Formula

a(n) = 13*a(n-1)-13*a(n-2)+a(n-3).
G.f.: -x*(x^2-7*x+1) / ((x-1)*(x^2-12*x+1)).
a(n) = (14-(-7+sqrt(35))*(6+sqrt(35))^n+(6-sqrt(35))^n*(7+sqrt(35)))/28. - Colin Barker, Mar 05 2016
a(n) = 12*a(n-1) - a(n-2) - 5. - Vincenzo Librandi, Mar 05 2016
a(n) = (5*a(n-1) + a(n-1)^2) / a(n-2), n >= 3. - Seiichi Manyama, Aug 11 2016

A253622 Centered heptagonal numbers (A069099) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 106, 15016, 2132131, 302747551, 42988020076, 6103996103206, 866724458635141, 123068769130086781, 17474898492013687726, 2481312517096813570276, 352328902529255513291431, 50028222846637186073812891, 7103655315319951166968139056
Offset: 1

Views

Author

Colin Barker, Jan 06 2015

Keywords

Examples

			106 is in the sequence because it is the 6th centered heptagonal number and the 7th centered pentagonal number.

Links

Crossrefs

Cf. A005891, A069099, A133272, A253621.

Programs

  • Mathematica
    LinearRecurrence[{143,-143,1},{1,106,15016},20] (* Harvey P. Dale, Feb 25 2016 *)
  • PARI
    Vec(-x*(x^2-37*x+1)/((x-1)*(x^2-142*x+1)) + O(x^100))

Formula

a(n) = 143*a(n-1)-143*a(n-2)+a(n-3).
G.f.: -x*(x^2-37*x+1) / ((x-1)*(x^2-142*x+1)).
a(n) = (4+(6+sqrt(35))*(71+12*sqrt(35))^(-n)-(-6+sqrt(35))*(71+12*sqrt(35))^n)/16. - Colin Barker, Mar 07 2016

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)).

A253714 Indices of hexagonal numbers (A000384) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 2857, 45529, 184300369, 2937241777, 11889953986681, 189493215939721, 767068491312421537, 12224965330197902689, 49486656636639609035209, 788681413122894278122297, 3192582165489099245985035761, 50880992673985436128583949841
Offset: 1

Views

Author

Colin Barker, Jan 10 2015

Keywords

Comments

Also positive integers x in the solutions to 4*x^2-7*y^2-2*x+7*y-2 = 0, the corresponding values of y being A253715.

Examples

			2857 is in the sequence because the 2857th hexagonal number is 16322041, which is also the 2160th centered heptagonal number.

Links

Crossrefs

Cf. A000384, A069099, A253715, A253716.

Programs

  • PARI
    Vec(-x*(x^4+2856*x^3-21842*x^2+2856*x+1)/((x-1)*(x^2-254*x+1)*(x^2+254*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+64514*a(n-2)-64514*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+2856*x^3-21842*x^2+2856*x+1) / ((x-1)*(x^2-254*x+1)*(x^2+254*x+1)).

A253715 Indices of centered heptagonal numbers (A069099) which are also hexagonal numbers (A000384).

Original entry on oeis.org

1, 2160, 34417, 139317984, 2220346081, 8987960385360, 143243407002961, 579849276161764800, 9241205157168647617, 37408396193312133889584, 596187109366334725327921, 2413365271435489729590825120, 38462415164418513312636815521, 155695847083980788221510357869840
Offset: 1

Views

Author

Colin Barker, Jan 10 2015

Keywords

Comments

Also positive integers y in the solutions to 4*x^2-7*y^2-2*x+7*y-2 = 0, the corresponding values of x being A253714.

Examples

			2160 is in the sequence because the 2160th centered heptagonal number is 16322041, which is also the 2857th hexagonal number.

Links

Crossrefs

Cf. A000384, A069099, A253714, A253716.

Programs

  • Mathematica
    LinearRecurrence[{1,64514,-64514,-1,1},{1,2160,34417,139317984,2220346081},20] (* Harvey P. Dale, Apr 10 2019 *)
  • PARI
    Vec(x*(2159*x^3+32257*x^2-2159*x-1)/((x-1)*(x^2-254*x+1)*(x^2+254*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+64514*a(n-2)-64514*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(2159*x^3+32257*x^2-2159*x-1) / ((x-1)*(x^2-254*x+1)*(x^2+254*x+1)).

A253716 Hexagonal numbers (A000384) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 16322041, 4145734153, 67933251842771953, 17254778510170993681, 282742011610770921096804841, 71815357774355276244995175961, 1176788140728629029198108610250463201, 298899554649081431834808455098428958753, 4897858370145334123819452782766901335994312153
Offset: 1

Views

Author

Colin Barker, Jan 10 2015

Keywords

Examples

			16322041 is in the sequence because it is the 2857th hexagonal number and the 2160th centered heptagonal number.

Links

Crossrefs

Cf. A000384, A069099, A253714, A253715.

Programs

  • Mathematica
    LinearRecurrence[{0,4162056194,0,-1},{1,16322041,4145734153,67933251842771953},20] (* Harvey P. Dale, Jan 05 2017 *)
  • PARI
    Vec(-x*(x-1)*(x^2+16322042*x+1)/((x^2-64514*x+1)*(x^2+64514*x+1)) + O(x^100))

Formula

a(n) = 4162056194*a(n-2)-a(n-4).
G.f.: -x*(x-1)*(x^2+16322042*x+1) / ((x^2-64514*x+1)*(x^2+64514*x+1)).
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