A027669 -id:A027669 - cp's OEIS frontend

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

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

Kelvin Voskuijl, Jun 14 2024

nonn

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

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

Pontus von Brömssen, Table of n, a(n) for n = 1..46
Brady Haran and Matt Parker, 90,525,801,730 Cannon Balls, Numberphile video (2019).
Brady Haran and Matt Parker, Cannon Ball Numbers, Numberphile (2019).
Index to sequences related to polygonal numbers.

Subsequences: A027568, A344376, A344410.

Cf. A027669, A027696, A057145, A080851, A379973, A379974, A379975.

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

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

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

Seiichi Manyama, May 17 2021

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Numberphile, Cannon Ball Numbers.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

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

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 *)

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

p(k, n) = n*((k-2)*n-k+4)/2;
a(n) = p(3*n+2, 3*n^3-3*n+1);

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)

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

A379973 Least k >= 3 such that A373711(n) is both k-gonal and k-gonal pyramidal.

Original entry on oeis.org

3, 3, 3, 3, 10, 14, 6, 8, 3, 4, 8, 3, 30, 11, 88, 14, 43, 50, 276, 17, 322, 20, 23, 26, 41, 29, 145, 32, 823, 35, 2378, 38, 41, 44, 47, 50, 53, 56, 59, 374, 62, 65, 2386, 68, 71, 74

Offset: 1

Pontus von Brömssen, Jan 08 2025

For n <= 46, there is a unique k >= 3 such that A373711(n) is both k-gonal and k-gonal pyramidal. If this were true for all n, A027669 would be the sorted distinct terms of this sequence.

Cf. A027669, A057145, A080851, A373711, A379974, A379975.

A057145(a(n),A379974(n)) = A080851(a(n)-2,A379975(n)-1) = A373711(n).

A027696 Numbers k >= 2 such that for some m >= 2, the sum of the first m k-gonal numbers is again a k-gonal number, excluding the parametric solution m = (k^2-4*k-2)/3 when k==2 (mod 3).

Original entry on oeis.org

3, 4, 6, 8, 10, 11, 14, 17, 30, 41, 43, 50, 60, 88, 145, 276, 322, 374, 823, 1152

Offset: 1

Masanobu Kaneko (mkaneko(AT)math.kyushu-u.ac.jp)

The parametric solution: if k==2 (mod 3) and k >= 5, the sum of the first (k^2-4*k-2)/3 k-gonal numbers is the ((k^3-6*k^2+3*k+19)/9)-th k-gonal number A057145(k,(k^3-6*k^2+3*k+19)/9) = A344410((k-2)/3).
2378, 2386, and 31265 are also terms. See link "Cannon Ball Numbers". - Pontus von Brömssen, Jan 08 2025
Number k is a term iff the elliptic curve (3*k-6)*y^2 - (3*k-12)*y = (k-2)*x^3 + 3*x^2 - (k-5)*x has an integral point with x >= 2 different from (k^2-4*k-2)/3. The listed values may be incomplete. For example, I was not able to verify that k = 273 is not a term. - Max Alekseyev, Feb 27 2025

Brady Haran and Matt Parker, Cannon Ball Numbers, Numberphile (2019).

Cf. A027669, A057145, A344410, A373711.

More terms from Masanobu Kaneko (mkaneko(AT)math.kyushu-u.ac.jp), Jan 05 1998

Name clarified by Max Alekseyev, Feb 27 2025

A308488 a(n) is the smallest n-gonal pyramidal number greater than 1 which is also n-gonal; a(n) = 0 when one does not exist.

Original entry on oeis.org

10, 4900, 0, 946, 0, 1045, 0, 175, 23725, 0, 0, 441, 0, 0, 975061, 0, 0, 3578401, 0, 0, 10680265, 0, 0, 27453385, 0, 0, 63016921, 23001, 0, 132361021, 0, 0, 258815701, 0, 0, 477132085, 0, 0, 55202400, 0, 245905, 1408778281, 0, 0, 2286380881, 0, 0, 314755, 0, 0

Offset: 3

Davis Smith, Aug 22 2019

nonn

a(n) is the smallest n-gonal number, N, such that, for some m > 1, N is the sum of the first m n-gonal numbers, 0 when one does not exist.

For n > 5, if n == 2 (mod 3), then a(n) > 0 and a(n) <= A080851(n - 2,((n-2)^2)/3 - 3), but there are cases where a(n) > 0 and n !== 2 (mod 3), e.g., a(10).

James Grime and Brady Haran, The Best Way to Pack Spheres, Numberphile video (2018).
M. Kaneko and K. Tachibana, When is a Polygonal Pyramid Number Again Polygonal?, Rocky Mountain Journal of Mathematics, 32 (2002).
Matt Parker and Brady Haran, 90,525,801,730 Cannon Balls, Numberphile video (2019).

Cf. A027669, A057145, A080851.

A308488_vec(lim,J=10^6)={my(
    pyramid(s,n)=(3*n^2 + n^3*(s-2)-n*(s-5))/6,
    check(s)=j=if(lift(Mod(s,3))==2,((s-2)^2)/3-2,J);m=3;while(m<=j,if(ispolygonal(pyramid(s,m),s),return(pyramid(s,m)),m++));0);
vector(lim,s,check(s+2))}

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

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

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A379973 Least k >= 3 such that A373711(n) is both k-gonal and k-gonal pyramidal.

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A027696 Numbers k >= 2 such that for some m >= 2, the sum of the first m k-gonal numbers is again a k-gonal number, excluding the parametric solution m = (k^2-4*k-2)/3 when k==2 (mod 3).

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A308488 a(n) is the smallest n-gonal pyramidal number greater than 1 which is also n-gonal; a(n) = 0 when one does not exist.

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