login
Heptagonal square numbers.
5

%I #47 Jul 08 2024 21:42:50

%S 1,81,5929,2307361,168662169,12328771225,4797839017609,

%T 350709705290025,25635978392186449,9976444135331412025,

%U 729252434211108535809,53306479301521270428241,20744638830126197732344369,1516379800105728357531817761,110843467413344235941816109721

%N Heptagonal square numbers.

%C From _Ant King_, Nov 11 2011: (Start)

%C This sequence is also the union of the three sequences defined by:

%C a(3n-2) = ((10 - sqrt(10)) * (3 + sqrt(10))^(4*n-3) - (10 + sqrt(10)) * (-3 + sqrt(10))^(4*n-3))^2 / 1600.

%C a(3n-1) = 9/160 * ((3 + sqrt(10))^(4*n-2) - (-3 + sqrt(10))^(4*n-2))^2.

%C a(3n) = ((20 - 7*sqrt(10)) * (3 + sqrt(10))^(4*n) + (20 + 7*sqrt(10)) * (-3 + sqrt(10))^(4*n))^2 / 1600.

%C Equivalent short forms for these subsequences are:

%C a(3n-2) = floor((10 - sqrt(10))^2 * (3 + sqrt(10))^(8*n - 6) / 1600).

%C a(3n-1) = floor( 9/160 * (3 + sqrt(10))^(8*n - 4)).

%C a(3n) = floor((20 - 7*sqrt(10))^ 2 * (3 + sqrt(10))^(8*n) / 1600).

%C (End)

%C Also heptagonal numbers (A000566) which are also centered octagonal numbers (A016754). - _Colin Barker_, Jan 19 2015

%H Colin Barker, <a href="/A036354/b036354.txt">Table of n, a(n) for n = 1..475</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeptagonalSquareNumber.html">Heptagonal Square Number.</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2079362,-2079362,0,-1,1).

%F 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) ).

%F From _Richard Choulet_, May 08 2009: (Start)

%F With the first values, for n>=0, a(n+9) = 2079363*(a(n+6) - a(n+3)) + a(n).

%F On every bisection modulo 2: a(n+1) = 1039681*a(n) + 116964 + 164388*sqrt(40*a(n)^2 + 9*a(n)).

%F On every bisection modulo 2: a(n+2) = 2079362*a(n+1) - a(n) + 233928. (End)

%F From _Ant King_, Nov 11 2011: (Start)

%F a(n) = a(n-1) + 2079362*a(n-3) - 2079362*a(n-4) - a(n-6) + a(n-7).

%F a(n) = 2079362*a(n-3) - a(n-6) + 233928.

%F (End)

%F From _Jonathan Pappas_, Jan 16 2022: (Start)

%F Define the three sequences

%F b(n) = 1442*b(n-1) - b(n-2) for n >= 2, with b(0) = -77, b(1) = 1;

%F c(n) = 1442*c(n-1) - c(n-2) for n >= 2, with c(0) = -9, c(1) = 9; and

%F d(n) = 1442*d(n-1) - d(n-2) for n >= 2, with d(0) = -1, d(1) = 77.

%F Then, for n >= 1,

%F a(3n - 2) = b(n)^2,

%F a(3n - 1) = c(n)^2, and

%F a(3n) = d(n)^2.

%F (End)

%p A036354 := proc(n)

%p if n <= 7 then

%p op(n,[1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609]);

%p else

%p procname(n-1) +2079362 *(procname(n-3)-procname(n-4)) -procname(n-6) +procname(n-7) ;

%p end if;

%p end proc:

%p seq(A036354(n),n=1..12) ;

%t LinearRecurrence[{ 1, 0, 2079362, -2079362, 0, -1, 1 }, {1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609 }, 13] (* _Ant King_, Nov 11 2011 *)

%o (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

%Y Cf. A046195, A046196, A253920.

%K nonn,easy

%O 1,2

%A Jean-Francois Chariot (jeanfrancois.chariot(AT)afoc.alcatel.fr)

%E More terms from _Eric W. Weisstein_

%E One more term from _Richard Choulet_, May 08 2009