OFFSET
1,2
COMMENTS
From Ant King, Nov 11 2011: (Start)
This sequence is also the union of the three sequences defined by:
a(3n-2) = ((10 - sqrt(10)) * (3 + sqrt(10))^(4*n-3) - (10 + sqrt(10)) * (-3 + sqrt(10))^(4*n-3))^2 / 1600.
a(3n-1) = 9/160 * ((3 + sqrt(10))^(4*n-2) - (-3 + sqrt(10))^(4*n-2))^2.
a(3n) = ((20 - 7*sqrt(10)) * (3 + sqrt(10))^(4*n) + (20 + 7*sqrt(10)) * (-3 + sqrt(10))^(4*n))^2 / 1600.
Equivalent short forms for these subsequences are:
a(3n-2) = floor((10 - sqrt(10))^2 * (3 + sqrt(10))^(8*n - 6) / 1600).
a(3n-1) = floor( 9/160 * (3 + sqrt(10))^(8*n - 4)).
a(3n) = floor((20 - 7*sqrt(10))^ 2 * (3 + sqrt(10))^(8*n) / 1600). (End)
Also heptagonal numbers (A000566) which are also centered octagonal numbers (A016754). - Colin Barker, Jan 19 2015
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 38.
LINKS
Colin Barker, Table of n, a(n) for n = 1..475
Eric Weisstein's World of Mathematics, Heptagonal Square Number.
Index entries for linear recurrences with constant coefficients, signature (1,0,2079362,-2079362,0,-1,1).
FORMULA
O.g.f.: -x*(1 + 80*x + 5848*x^2 + 222070*x^3 + 5848*x^4 + 80*x^5 + x^6) / ( (x-1)*(x^6 - 2079362*x^3 + 1) ).
From Richard Choulet, May 08 2009: (Start)
With the first values, for n>=0, a(n+9) = 2079363*(a(n+6) - a(n+3)) + a(n).
On every bisection modulo 2: a(n+1) = 1039681*a(n) + 116964 + 164388*sqrt(40*a(n)^2 + 9*a(n)).
On every bisection modulo 2: a(n+2) = 2079362*a(n+1) - a(n) + 233928. (End)
From Ant King, Nov 11 2011: (Start)
a(n) = a(n-1) + 2079362*a(n-3) - 2079362*a(n-4) - a(n-6) + a(n-7).
a(n) = 2079362*a(n-3) - a(n-6) + 233928.
(End)
From Jonathan Pappas, Jan 16 2022: (Start)
Define the three sequences
b(n) = 1442*b(n-1) - b(n-2) for n >= 2, with b(0) = -77, b(1) = 1;
c(n) = 1442*c(n-1) - c(n-2) for n >= 2, with c(0) = -9, c(1) = 9; and
d(n) = 1442*d(n-1) - d(n-2) for n >= 2, with d(0) = -1, d(1) = 77.
Then, for n >= 1,
a(3n - 2) = b(n)^2,
a(3n - 1) = c(n)^2, and
a(3n) = d(n)^2. (End)
MAPLE
MATHEMATICA
LinearRecurrence[{ 1, 0, 2079362, -2079362, 0, -1, 1 }, {1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609 }, 13] (* Ant King, Nov 11 2011 *)
PROG
(PARI) Vec(-x*(x^6+80*x^5+5848*x^4+222070*x^3+5848*x^2+80*x+1)/((x-1)*(x^6-2079362*x^3+1)) + O(x^100)) \\ Colin Barker, Jan 19 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jean-Francois Chariot (jeanfrancois.chariot(AT)afoc.alcatel.fr)
EXTENSIONS
More terms from Eric W. Weisstein
One more term from Richard Choulet, May 08 2009
STATUS
approved