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 1-7 of 7 results.

A334012 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero octagonal numbers in exactly n ways.

Original entry on oeis.org

1, 1045, 5985
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 12 2020

Keywords

Examples

			From _Seiichi Manyama_, May 16 2021: (Start)
Let S(k, m) denote the sum of m octagonal numbers starting from k*(3*k-2). We have
a(1) = S(1, 1);
a(2) = S(19, 1) = S(1, 10);
a(3) = S(45, 1) = S(11, 9) = S(1, 18). (End)

Links

Crossrefs

Cf. A000567, A054859, A068314, A186337, A298467, A322637, A334007, A334008, A334010, A334011, A344376.

A373711 Numbers that are simultaneously k-gonal and k-gonal pyramidal for some k >= 3.

Original entry on oeis.org

0, 1, 10, 120, 175, 441, 946, 1045, 1540, 4900, 5985, 7140, 23001, 23725, 48280, 195661, 245905, 314755, 801801, 975061, 1169686, 3578401, 10680265, 27453385, 55202400, 63016921, 101337426, 132361021, 197427385, 258815701, 432684460, 477132085, 837244045
Offset: 1

Views

Author

Kelvin Voskuijl, Jun 14 2024

Keywords

Comments

Matt Parker calls these numbers cannonball numbers, after the cannonball problem involving finding a number both square and square pyramidal.
If m==2 (mod 3), the m-gonal number A057145(m,(m^3-6*m^2+3*m+19)/9) = (m^2-4*m-2)*(m^2-4*m+1)*(m^3-6*m^2+3*m+19)/162 = A344410((m-2)/3) is a term. See comment in A027696. - Pontus von Brömssen, Dec 09 2024

Examples

			4900 is a term because it is both the 70th square and the 24th square pyramidal number.

Links

Crossrefs

Subsequences: A027568, A344376, A344410.
Cf. A027669, A027696, A057145, A080851, A379973, A379974, A379975.

Formula

a(n) = A057145(A379973(n),A379974(n)) = A080851(A379973(n)-2,A379975(n)-1). - Pontus von Brömssen, Jan 09 2025

Extensions

a(13)-a(33) from Pontus von Brömssen, Dec 08 2024

A344410 a(n) = (3*n^2 - 1) * (3*n^2 - 2) * (3*n^3 - 3*n + 1)/2.

Original entry on oeis.org

1, 1, 1045, 23725, 195661, 975061, 3578401, 10680265, 27453385, 63016921, 132361021, 258815701, 477132085, 837244045, 1408778281, 2286380881, 3595928401, 5501691505, 8214519205, 12001111741, 17194450141, 24205450501, 33535911025, 45792819865, 61704091801
Offset: 0

Views

Author

Seiichi Manyama, May 17 2021

Keywords

Comments

a(n) is both (3*n+2)-gonal number and (3*n+2)-gonal pyramidal number.

Links

Crossrefs

Cf. A016789, A027669, A057145, A344376, A344377.

Programs

  • Mathematica
    Table[PolygonalNumber[3*n + 2, 3*n^3 - 3*n + 1], {n, 0, 24}] (* Amiram Eldar, May 17 2021 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,1,1045,23725,195661,975061,3578401,10680265},30] (* Harvey P. Dale, Aug 10 2021 *)
  • PARI
    a(n) = (3*n^2-1)*(3*n^2-2)*(3*n^3-3*n+1)/2;
  • PARI
    p(k, n) = n*((k-2)*n-k+4)/2;
a(n) = p(3*n+2, 3*n^3-3*n+1);
  • PARI
    my(N=40, x='x+O('x^N)); Vec((1-7*x+1065*x^2+15337*x^3+35135*x^4+15567*x^5+943*x^6-x^7)/(1-x)^8)

Formula

Let p(k,m) = A057145(k,m) denote m-th k-gonal number. Then
a(n) = p(3*n+2, 3*n^3-3*n+1);
a(n) = Sum_{j=1..3*n^2-2} p(3*n+2, j) for n > 0.
G.f.: (1-7*x+1065*x^2+15337*x^3+35135*x^4+15567*x^5+943*x^6-x^7)/(1-x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Wesley Ivan Hurt, Sep 05 2022

A344377 Numbers that are both 11-gonal numbers (A051682) and 11-gonal pyramidal numbers (A007586).

Original entry on oeis.org

0, 1, 23725, 1519937678700, 7248070597636
Offset: 1

Views

Author

Seiichi Manyama, May 17 2021

Keywords

Comments

Intersection of A051682 and A007586.

Links

Crossrefs

Cf. A007586, A027568, A051682, A344280, A344376.

Programs

  • PARI
    for(k=0, 1e5, if(ispolygonal(m=k*(k+1)*(3*k-2)/2, 11), print1(m", ")))

A378361 Octagonal indices of numbers that are both octagonal and octagonal pyramidal.

Original entry on oeis.org

0, 1, 19, 45, 6413415, 16068720
Offset: 1

Views

Author

Kelvin Voskuijl, Nov 23 2024

Keywords

Examples

			19 is a term because the 19th octagonal number (1045) is also the 10th octagonal pyramidal number.

Crossrefs

Cf. A000567 (octagonal numbers), A002414 (octagonal pyramidal numbers).
Cf. A344376.

A344280 Numbers that are both 10-gonal numbers (A001107) and 10-gonal pyramidal numbers (A007585).

Original entry on oeis.org

0, 1, 175, 368050005576
Offset: 1

Views

Author

Seiichi Manyama, May 17 2021

Keywords

Comments

Intersection of A001107 and A007585.

Links

Crossrefs

Cf. A001107, A007585, A027568, A344376, A344377.

Programs

  • PARI
    for(k=0, 1e4, if(ispolygonal(m=k*(k+1)*(8*k-5)/6, 10), print1(m", ")))

A378918 Indices of octagonal pyramidal numbers that are both octagonal and octagonal pyramidal.

Original entry on oeis.org

0, 1, 10, 18, 49785, 91839
Offset: 1

Views

Author

Kelvin Voskuijl, Dec 10 2024

Keywords

Examples

			10 is a term because the 10th octagonal pyramidal number (1045) is also the 19th octagonal number.

Crossrefs

Cf. A000567 (octagonal numbers), A002414 (octagonal pyramidal numbers).
Cf. A344376.
Cf. A378361 (octagonal indices).

Formula

A344376(n) = A002414(a(n)).
Showing 1-7 of 7 results.

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