A256645 25-gonal pyramidal numbers: a(n) = n*(n+1)*(23*n-20)/6.

0, 1, 26, 98, 240, 475, 826, 1316, 1968, 2805, 3850, 5126, 6656, 8463, 10570, 13000, 15776, 18921, 22458, 26410, 30800, 35651, 40986, 46828, 53200, 60125, 67626, 75726, 84448, 93815, 103850, 114576, 126016, 138193, 151130, 164850, 179376, 194731, 210938, 228020

Offset: 0

Author

Luciano Ancora, Apr 07 2015

Keywords

• nonn
• easy

Comments

If b(n,k) = n*(n+1)*((k-2)*n-(k-5))/6 is n-th k-gonal pyramidal number, then b(n,k) = A000292(n) + (k-3)*A000292(n-1) (see Deza in References section, p. 96).

Also, b(n,k) = b(n,k-1) + A000292(n-1) (see Deza in References section, p. 95). Some examples:

for k=4, A000330(n) = A000292(n) + A000292(n-1);

for k=5, A002411(n) = A000330(n) + A000292(n-1);

for k=6, A002412(n) = A002411(n) + A000292(n-1), etc.

This is the case k=25.

References

• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (23rd row of the table).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index to sequences related to polygonal numbers.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Partial sums of A255184.

Cf. similar sequences listed in A237616.

Cf. A000292, A000330, A002411, A002412, A256716.

Programs

Magma

k:=25; [n*(n+1)*((k-2)*n-(k-5))/6: n in [0..40]]; // Vincenzo Librandi, Apr 08 2015

Mathematica

Table[n (n + 1) (23 n - 20)/6, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 26, 98}, 40] (* Vincenzo Librandi, Apr 08 2015 *)

PARI

concat(0, Vec(x*(1 + 22*x)/(1 - x)^4 + O(x^100))) \\ Colin Barker, Apr 07 2015

Formula

G.f.: x*(1 + 22*x)/(1 - x)^4.

a(n) = A000292(n) + 22*A000292(n-1) = A256716(n) + A000292(n-1), see comments.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, Apr 07 2015

E.g.f.: exp(x)*x*(6 + 72*x + 23*x^2)/6. - Elmo R. Oliveira, Aug 04 2025