A203408 Numbers which are both heptagonal and decagonal.

1, 540, 2887450, 1123674201, 6004054625647, 2336525434757970, 12484603034492528512, 4858482201068079159687, 25960009135002449017962445, 10102543266574986692211140472, 53980256514964477791853933850326, 21006844571867038996088473395797925

Offset

1,2

Comments

As n increases, the ratios of consecutive terms settle into an approximate 2-cycle with a(n)/a(n-1) bounded above and below by 1/81*(216401+68432*sqrt(10)) and 1/81*(15761+4984*sqrt(10)) respectively.

Links

Table of n, a(n) for n=1..12.

Index entries for linear recurrences with constant coefficients, signature (1,2079362,-2079362,-1,1).

Formula

G.f.: x(1+539*x+807548*x^2+10633*x^3+27*x^4) / ((1-x)*(1-1442*x+x^2)*(1+1442*x+x^2)).

a(n) = 2079362*a(n-2)-a(n-4)+818748.

a(n) = a(n-1)+2079362*a(n-2)-2079362*a(n-3)-a(n-4)+a(n-5).

a(n) = 1/320*((11-2*sqrt(10)*(-1)^n)*(1+sqrt(10))* (3+sqrt(10))^(4*n-3)+(11+2*sqrt(10)*(-1)^n)*(1-sqrt(10))*(3-sqrt(10))^(4*n-3)-126).

a(n) = floor(1/320*(11-2*sqrt(10)*(-1)^n)*(1+sqrt(10))* (3+sqrt(10))^(4*n-3)).

Example

The second number that is both heptagonal and decagonal is 540. Hence a(2)=540.

Mathematica

LinearRecurrence[{1, 2079362, -2079362, -1, 1}, {1, 540, 2887450, 1123674201, 6004054625647}, 15]

Crossrefs

Cf. A203409, A203410, A001107, A000566.

Sequence in context: A056934 A238933 A003746 * A259165 A293393 A146356

Adjacent sequences: A203405 A203406 A203407 * A203409 A203410 A203411

Keyword

nonn,easy

Author

Ant King, Jan 02 2012

Status

approved