A255184 25-gonal numbers: a(n) = n*(23*n-21)/2.

0, 1, 25, 72, 142, 235, 351, 490, 652, 837, 1045, 1276, 1530, 1807, 2107, 2430, 2776, 3145, 3537, 3952, 4390, 4851, 5335, 5842, 6372, 6925, 7501, 8100, 8722, 9367, 10035, 10726, 11440, 12177, 12937, 13720, 14526, 15355, 16207, 17082, 17980

Offset: 0

Author

Luciano Ancora, Apr 03 2015

Keywords

• nonn
• easy

Comments

If b(n,k) = n*((k-2)*n-(k-4))/2 is n-th k-gonal number, then b(n,k) = A000217(n) + (k-3)* A000217(n-1) (see Deza in References section, page 21, where the formula is attributed to Bachet de Méziriac).

Also, b(n,k) = b(n,k-1) + A000217(n-1) (see Deza and Picutti in References section, page 20 and 137 respectively, where the formula is attributed to Nicomachus). Some examples:

for k=4, A000290(n) = A000217(n) + A000217(n-1);

for k=5, A000326(n) = A000290(n) + A000217(n-1);

for k=6, A000384(n) = A000326(n) + A000217(n-1), etc.

This is the case k=25.

References

• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (23rd row of the table).
• E. Picutti, Sul numero e la sua storia, Feltrinelli Economica (1977), pages 131-147.

Links

• Luciano Ancora, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. k-gonal numbers: A000217 (k=3), A000290 (k=4), A000326 (k=5), A000384 (k=6), A000566 (k=7), A000567 (k=8), A001106 (k=9), A001107 (k=10), A051682 (k=11), A051624 (k=12), A051865 (k=13), A051866 (k=14), A051867 (k=15), A051868 (k=16), A051869 (k=17), A051870 (k=18), A051871 (k=19), A051872 (k=20), A051873 (k=21), A051874 (k=22), A051875 (k=23), A051876 (k=24), this sequence (k=25), A255185 (k=26), A255186 (k=27), A161935 (k=28), A255187 (k=29), A254474 (k=30).

Programs

Magma

k:=25; [n*((k-2)*n-(k-4))/2: n in [0..40]]; // Bruno Berselli, Apr 10 2015

Mathematica

Table[n (23 n - 21)/2, {n, 40}]

PARI

a(n)=n*(23*n-21)/2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: x*(-1 - 22*x)/(-1 + x)^3.

a(n) = A000217(n) + 22*A000217(n-1) = A051876(n) + A000217(n-1), see comments.

Product_{n>=2} (1 - 1/a(n)) = 23/25. - Amiram Eldar, Jan 22 2021

E.g.f.: exp(x)*(x + 23*x^2/2). - Nikolaos Pantelidis, Feb 05 2023