A081496 Start with Pascal's triangle; a(n) is the sum of the numbers on the periphery of the n-th central rhombus containing exactly 4 numbers.

5, 14, 46, 160, 574, 2100, 7788, 29172, 110110, 418132, 1595620, 6113744, 23505356, 90633800, 350351640, 1357278300, 5268292830, 20483876820, 79765662900, 311038321440, 1214362277700, 4746455801880, 18570960418920, 72728638093800

Offset

1,1

Links

Table of n, a(n) for n=1..24.

Qi Wang, Tau-tilting finite simply connected algebras, arXiv:1910.01937 [math.RT], 2019. See Example 5.1 and table page 17.

Formula

a(n) = (9*n-4)*Catalan(n-1) = (9*n-4)*binomial(2*(n-1), (n-1))/n. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004

a(n) = Sum_{k=0..n} A029635(n,k)^2 for n>=1, where A029635 is the Lucas triangle. - Paul D. Hanna, Oct 17 2017

Example

The first three rhombuses are
...1...........2.........6
.1...1.......3...3.....10..10
...2......,....6.....,...20
and the corresponding sums are a(1) =5, a(2) =14 and a(3) =46.

Maple

seq((9*n-4)*binomial(2*(n-1), (n-1))/n, n=1..26); # C. Ronaldo, Dec 20 2004

Prog

(PARI) { A029635(n, k) = if( k<0 || k>n, 0, (n==0) + binomial(n, k) + binomial(n-1, k-1))}; \\ program from Michael Somos in A029635
{a(n) = sum(k=0, n, A029635(n, k)^2)} \\ Paul D. Hanna, Oct 17 2017
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A081494, A081495, A081497, A029635.

Sequence in context: A098730 A244236 A163608 * A152051 A220563 A075827

Adjacent sequences: A081493 A081494 A081495 * A081497 A081498 A081499

Keyword

nonn

Author

Amarnath Murthy, Mar 25 2003

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004

Status

approved