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A036354 Heptagonal square numbers. 3
1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, 350709705290025, 25635978392186449, 9976444135331412025, 729252434211108535809, 53306479301521270428241, 20744638830126197732344369, 1516379800105728357531817761, 110843467413344235941816109721 (list; graph; refs; listen; history; text; internal format)
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

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) ).

With the first values, for n>=0 a(n+9):=2079363*u(a+6)-2079363*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. - Richard Choulet, May 08 2009

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)

MAPLE

A036354 := proc(n)

if n <= 7 then

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

else

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

end if;

end proc:

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

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

Cf. A046195, A046196, A253920.

Sequence in context: A237937 A060349 A212706 * A205887 A205199 A250612

Adjacent sequences:  A036351 A036352 A036353 * A036355 A036356 A036357

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

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Last modified December 7 04:57 EST 2016. Contains 278841 sequences.