A256646 26-gonal pyramidal numbers: a(n) = n*(n+1)*(8*n-7)/2.

0, 1, 27, 102, 250, 495, 861, 1372, 2052, 2925, 4015, 5346, 6942, 8827, 11025, 13560, 16456, 19737, 23427, 27550, 32130, 37191, 42757, 48852, 55500, 62725, 70551, 79002, 88102, 97875, 108345, 119536, 131472, 144177, 157675, 171990, 187146, 203167, 220077

Offset: 0

Author

Luciano Ancora, Apr 07 2015

Keywords

• nonn
• easy

Comments

See comments in A256645.

References

• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (24th row of the table).

Links

• Luciano Ancora, Table of n, a(n) for n = 0..1000
• Luciano Ancora, Polygonal and Pyramidal numbers, Section 3.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Partial sums of A255185.

Cf. similar sequences listed in A237616.

Cf. A000292, A051866, A256645.

Programs

Magma

[n*(n+1)*(8*n-7)/2: n in [0..50]]; // Vincenzo Librandi, Apr 08 2015

Mathematica

Table[n (n + 1) (8 n - 7)/2, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 27, 102}, 40] (* Vincenzo Librandi, Apr 08 2015 *)

SageMath

[(8*n-7)*binomial(n+1,2) for n in range(51)] # G. C. Greubel, Jul 12 2024

Formula

G.f.: x*(1 + 23*x)/(1 - x)^4.

a(n) = A000292(n) + 23*A000292(n-1).

a(n) = n*A051866(n) - Sum_{i=0..n-1} A051866(i). - Bruno Berselli, Apr 09 2015

Sum_{n>=1} 1/a(n) = 2*(4*(sqrt(2)+1)*Pi - 4*(sqrt(2)-8)*log(2) + 8*sqrt(2)*log(sqrt(2)+2) - 7)/105. - Amiram Eldar, Jan 10 2022

E.g.f.: (1/2)*x*(2 + 25*x + 8*x^2)*exp(x). - G. C. Greubel, Jul 12 2024