A203629 -id:A203629 - cp's OEIS frontend

A203627 Numbers which are both 9-gonal (nonagonal) and 10-gonal (decagonal).

1, 1212751, 977965238701, 788633124418157851, 635955328796073362530201, 512835649051022518566661395751, 413551693065406705688396809494274501, 333488912390817262631483541451235285166451, 268926125929366270527488184087670639619302551601

Offset: 1

Ant King, Jan 06 2012

As n increases, this sequence is approximately geometric with common ratio r = lim(n->Infinity, a(n)/a(n-1)) = (2*sqrt(2)+sqrt(7))^8 = 403201+107760*sqrt(14).

			The second number that is both nonagonal and decagonal is A001106(589) = A001107(551) = 1212751. Hence a(2) = 1212751.

Index entries for linear recurrences with constant coefficients, signature (806403, -806403, 1).

Cf. A203628, A203629, A001107, A001106.

LinearRecurrence[{806403, -806403, 1}, {1, 1212751, 977965238701}, 9]

G.f.: x*(1+406348*x+451*x^2) / ((1-x)*(1-806402*x+x^2)).
a(n) = 806402*a(n-1)-a(n-2)+406800.
a(n) = 806403*a(n-1)-806403*a(n-2)+a(n-3).
a(n) = 1/448*((15+2*sqrt(14))*(2*sqrt(2)+sqrt(7))^(8*n-6)+(15-2*sqrt(14))*(2*sqrt(2)-sqrt(7))^(8*n-6)-226).
a(n) = floor(1/448*(15+2*sqrt(14))*(2*sqrt(2)+sqrt(7))^(8*n-6)).

A203628 Indices of 9-gonal (nonagonal) numbers which are also 10-gonal (decagonal).

Original entry on oeis.org

1, 589, 528601, 474682789, 426264615601, 382785150126589, 343740638549061001, 308678710631906651989, 277193138406813624424801, 248919129610608002826818989, 223529101197187579724859027001, 200728883955944835984920579427589

Offset: 1

Ant King, Jan 06 2012

As n increases, this sequence is approximately geometric with common ratio r = lim(n->Infinity, a(n)/a(n-1)) = (2*sqrt(2)+sqrt(7))^4 = 449+120*sqrt(14).

			The second number that is both 9-gonal (nonagonal) and 10-gonal (decagonal) is A001106(589) = 1212751. Hence a(2) = 589.

Index entries for linear recurrences with constant coefficients, signature (899, -899, 1).

Cf. A203627, A203629, A001107, A001106.

LinearRecurrence[{899, -899, 1}, {1, 589, 528601}, 12]

G.f.: x*(1-310*x-11*x^2) / ((1-x)*(1-898*x+x^2)).
a(n) = 898*a(n-1)-a(n-2)-320.
a(n) = 899*a(n-1)-899*a(n-2)+a(n-3).
a(n) = 1/56*((sqrt(2)+2*sqrt(7))*(2*sqrt(2)+sqrt(7))^(4*n-3)+(sqrt(2)-2*sqrt(7))*(2*sqrt(2)-sqrt(7))^(4*n-3)+20).
a(n) = ceiling(1/56*(sqrt(2)+2*sqrt(7))*(2*sqrt(2)+sqrt(7))^(4*n-3)).

cp's OEIS Frontend

A203627 Numbers which are both 9-gonal (nonagonal) and 10-gonal (decagonal).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula