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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A253878 Indices of triangular numbers (A000217) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 22, 358, 5713, 91057, 1451206, 23128246, 368600737, 5874483553, 93623136118, 1492095694342, 23779907973361, 378986431879441, 6040003002097702, 96261061601683798, 1534136982624843073, 24449930660395805377, 389664753583708042966, 6210186126678932882086
Offset: 1

Views

Author

Colin Barker, Jan 17 2015

Keywords

Comments

Also positive integers x in the solutions to x^2 - 7*y^2 + x + 7*y - 2 = 0, the corresponding values of y being A253879.

Examples

			22 is in the sequence because the 22nd triangular number is 253, which is also the 9th centered heptagonal number.

Links

Crossrefs

Cf. A000217, A069099, A253879, A253880.

Programs

  • Mathematica
    LinearRecurrence[{17,-17,1},{1,22,358},20] (* Harvey P. Dale, Sep 10 2022 *)
  • PARI
    Vec(-x*(x^2+5*x+1)/((x-1)*(x^2-16*x+1)) + O(x^100))

Formula

a(n) = 17*a(n-1)-17*a(n-2)+a(n-3).
G.f.: -x*(x^2+5*x+1) / ((x-1)*(x^2-16*x+1)).
a(n) = (-2+(8-3*sqrt(7))^n*(3+sqrt(7))-(-3+sqrt(7))*(8+3*sqrt(7))^n)/4. - Colin Barker, Mar 04 2016

A253879 Indices of centered heptagonal numbers (A069099) which are also triangular numbers (A000217).

Original entry on oeis.org

1, 9, 136, 2160, 34417, 548505, 8741656, 139317984, 2220346081, 35386219305, 563959162792, 8987960385360, 143243407002961, 2282906551662009, 36383261419589176, 579849276161764800, 9241205157168647617, 147279433238536597065, 2347229726659416905416
Offset: 1

Views

Author

Colin Barker, Jan 17 2015

Keywords

Comments

Also positive integers y in the solutions to x^2 - 7*y^2 + x + 7*y - 2 = 0, the corresponding values of x being A253878.

Examples

			9 is in the sequence because the 9th centered heptagonal number is 253, which is also the 22nd triangular number.

Links

Crossrefs

Cf. A000217, A069099, A253878, A253880.

Programs

  • PARI
    Vec(x*(8*x-1)/((x-1)*(x^2-16*x+1)) + O(x^100))

Formula

a(n) = 17*a(n-1)-17*a(n-2)+a(n-3).
G.f.: x*(8*x-1) / ((x-1)*(x^2-16*x+1)).
a(n) = (14-(8-3*sqrt(7))^n*(7+3*sqrt(7))+(-7+3*sqrt(7))*(8+3*sqrt(7))^n)/28. - Colin Barker, Mar 04 2016

A254652 Indices of pentagonal numbers (A000326) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 4, 88, 421, 9661, 46288, 1062604, 5091241, 116876761, 559990204, 12855381088, 61593831181, 1413975042901, 6774761439688, 155524399338004, 745162164534481, 17106269952137521, 81961063337353204, 1881534170335789288, 9014971804944317941
Offset: 1

Views

Author

Colin Barker, Feb 04 2015

Keywords

Comments

Also positive integers x in the solutions to 3*x^2 - 7*y^2 - x + 7*y - 2 = 0, the corresponding values of y being A254653.

Examples

			4 is in the sequence because the 4th pentagonal number is 22, which is also the 3rd centered heptagonal number.

Links

Crossrefs

Cf. A000326, A069099, A254653, A254654.

Programs

  • Mathematica
    LinearRecurrence[{1,110,-110,-1,1},{1,4,88,421,9661},30] (* Harvey P. Dale, Dec 09 2018 *)
  • PARI
    Vec(-x*(x^2-4*x+1)*(x^2+7*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^2-4*x+1)*(x^2+7*x+1) / ((x-1)*(x^4-110*x^2+1)).

A254653 Indices of centered heptagonal numbers (A069099) which are also pentagonal numbers (A000326).

Original entry on oeis.org

1, 3, 58, 276, 6325, 30303, 695638, 3333000, 76513801, 366599643, 8415822418, 40322627676, 925663952125, 4435122444663, 101814618911278, 487823146285200, 11198682416288401, 53656110968927283, 1231753251172812778, 5901684383435715876, 135481658946593117125
Offset: 1

Views

Author

Colin Barker, Feb 04 2015

Keywords

Comments

Also positive integers y in the solutions to 3*x^2 - 7*y^2 - x + 7*y - 2 = 0, the corresponding values of x being A254652.

Examples

			3 is in the sequence because the 3rd centered heptagonal number is 22, which is also the 4th pentagonal number.

Links

Crossrefs

Cf. A000326, A069099, A254652, A254654.

Programs

  • PARI
    Vec(x*(2*x^3+55*x^2-2*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*(2*x^3+55*x^2-2*x-1) / ((x-1)*(x^4-110*x^2+1)).

A254654 Pentagonal numbers (A000326) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 22, 11572, 265651, 139997551, 3213845272, 1693690359922, 38881099834501, 20490265834338301, 470383542583947322, 247891234370134405072, 5690700059299494866551, 2998988132919620198222251, 68846088847021746311586172, 36281758184170330787958387022
Offset: 1

Views

Author

Colin Barker, Feb 04 2015

Keywords

Examples

			22 is in the sequence because it is the 4th pentagonal number and the 3rd centered heptagonal number.

Links

Crossrefs

Cf. A000326, A069099, A254652, A254653.

Programs

  • Magma
    I:=[1,22,11572,265651,139997551]; [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 20 2017
  • Mathematica
    CoefficientList[Series[(x^4 + 21*x^3 - 548*x^2 + 21*x + 1)/((1 - x)*(x^2 - 110*x + 1)*(x^2 + 110*x + 1)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jan 19 2017 *)
LinearRecurrence[{1,12098,-12098,-1,1},{1,22,11572,265651,139997551},20] (* Harvey P. Dale, Jan 10 2025 *)
  • PARI
    Vec(-x*(x^4+21*x^3-548*x^2+21*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+21*x^3-548*x^2+21*x+1) / ((x-1)*(x^2-110*x+1)*(x^2+110*x+1)).

A254855 Indices of octagonal numbers (A000567) that are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 2, 16, 43, 407, 1108, 10558, 28757, 274093, 746566, 7115852, 19381951, 184738051, 503184152, 4796073466, 13063405993, 124513172057, 339145371658, 3232546400008, 8804716257107, 83921693228143, 228583477313116, 2178731477531702, 5934365693883901
Offset: 1

Views

Author

Colin Barker, Feb 08 2015

Keywords

Comments

Also positive integers x in the solutions to 6*x^2 - 7*y^2 - 4*x + 7*y - 2 = 0, the corresponding values of y being A254856.

Examples

			16 is in the sequence because the 16th octagonal number is 736, which is also the 15th centered heptagonal number.

Links

Crossrefs

Cf. A000567, A069099, A254856, A254857.

Programs

  • Mathematica
    LinearRecurrence[{1,26,-26,-1,1},{1,2,16,43,407},30] (* Harvey P. Dale, Aug 31 2021 *)
  • PARI
    Vec(-x*(x^4+x^3-12*x^2+x+1)/((x-1)*(x^4-26*x^2+1)) + O(x^100))

Formula

a(n) = a(n-1)+26*a(n-2)-26*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+x^3-12*x^2+x+1) / ((x-1)*(x^4-26*x^2+1)).

A254856 Indices of centered heptagonal numbers (A069099) that are also octagonal numbers (A000567).

Original entry on oeis.org

1, 2, 15, 40, 377, 1026, 9775, 26624, 253761, 691186, 6587999, 17944200, 171034201, 465858002, 4440301215, 12094363840, 115276797377, 313987601826, 2992756430575, 8151583283624, 77696390397561, 211627177772386, 2017113393905999, 5494155038798400
Offset: 1

Views

Author

Colin Barker, Feb 08 2015

Keywords

Comments

Also positive integers y in the solutions to 6*x^2 - 7*y^2 - 4*x + 7*y - 2 = 0, the corresponding values of x being A254855.

Examples

			15 is in the sequence because the 15th centered heptagonal number is 736, which is also the 16th octagonal number.

Links

Crossrefs

Cf. A000567, A069099, A254855, A254857.

Programs

  • Mathematica
    LinearRecurrence[{1,26,-26,-1,1},{1,2,15,40,377},30] (* Harvey P. Dale, Apr 30 2019 *)
  • PARI
    Vec(x*(x^3+13*x^2-x-1)/((x-1)*(x^4-26*x^2+1)) + O(x^100))

Formula

a(n) = a(n-1)+26*a(n-2)-26*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(x^3+13*x^2-x-1) / ((x-1)*(x^4-26*x^2+1)).

A254857 Octagonal numbers (A000567) that are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 8, 736, 5461, 496133, 3680776, 334392976, 2480837633, 225380369761, 1672080883936, 151906034826008, 1126980034935301, 102384442092359701, 759582871465509008, 69006962064215612536, 511957728387718136161, 46510590046839230489633, 345058749350450558263576
Offset: 1

Views

Author

Colin Barker, Feb 08 2015

Keywords

Examples

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

Links

Crossrefs

Cf. A000567, A069099, A254855, A254856.

Programs

  • PARI
    Vec(-x*(x^4+7*x^3+54*x^2+7*x+1)/((x-1)*(x^2-26*x+1)*(x^2+26*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+674*a(n-2)-674*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+7*x^3+54*x^2+7*x+1) / ((x-1)*(x^2-26*x+1)*(x^2+26*x+1)).

A322803 Number of compositions (ordered partitions) of n into centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 45, 55, 67, 82, 101, 125, 155, 192, 239, 297, 368, 455, 562, 694, 857, 1058, 1308, 1619, 2005, 2483, 3074, 3805, 4708, 5822, 7198, 8900, 11007, 13616, 16846, 20845, 25795, 31918, 39489
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 26 2018

Keywords

Links

Crossrefs

Cf. A069099, A280863, A282502, A282504, A322799, A322801, A322802.

Programs

  • Maple
    h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`((7*(t-1)*t+2)/2>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-(7*(i-1)*i+2)/2), i=1..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018
  • Mathematica
    nmax = 54; CoefficientList[Series[1/(1 - Sum[x^(7 k (k + 1)/2 + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=0} x^(7*k*(k+1)/2+1)).

A254375 Digital roots of centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 8, 4, 7, 8, 7, 4, 8, 1, 1, 8, 4, 7, 8, 7, 4, 8, 1, 1, 8, 4, 7, 8, 7, 4, 8, 1, 1, 8, 4, 7, 8, 7, 4, 8, 1, 1, 8, 4, 7, 8, 7, 4, 8, 1, 1, 8, 4, 7, 8, 7, 4, 8, 1, 1, 8, 4, 7, 8, 7, 4, 8, 1, 1, 8, 4, 7, 8, 7, 4, 8, 1, 1, 8, 4, 7, 8, 7, 4, 8, 1, 1, 8, 4, 7, 8
Offset: 1

Views

Author

Colin Barker, Jan 29 2015

Keywords

Comments

The sequence is periodic with period 9.

Examples

			a(3) = 4 because the 3rd centered heptagonal number is 22, the digital root of which is 4.

Links

Crossrefs

Cf. A001844, A069099.

Programs

  • Mathematica
    FixedPoint[Plus @@ IntegerDigits[#] &, #] & /@ FoldList[#1 + #2 &, 1, 7 Range@ 80] (* Michael De Vlieger, Feb 01 2015, after Robert G. Wilson v at A069099 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 8, 4, 7, 8, 7, 4, 8, 1},86] (* Ray Chandler, Aug 26 2015 *)
  • PARI
    m=9; vector(200, n, (m*n*(n-1)/2)%9+1)

Formula

a(n) = A010888(A069099(n)).
a(n) = a(n-9).
G.f.: -x*(x^8+8*x^7+4*x^6+7*x^5+8*x^4+7*x^3+4*x^2+8*x+1) / ((x-1)*(x^2+x+1)*(x^6+x^3+1)).
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