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A133324
7-gonal numbers which are sum of 2 consecutive 7-gonal numbers.
5
1, 144841, 927821665, 222590743768705, 1425873367156486249, 342076743178546829707489, 2191277630703059899650524953, 525702444955366082679116505052393, 3367548455158599463971494297793284977, 807897836210987628258457093971387133310617, 5175237696467776056839029025306896109635889521
OFFSET
1,2
COMMENTS
We write (5*p^2 - 3*p)/2 = (5*r^2 - 3*r)/2 + (5*(r+1)^2 - 3*(r+1))/2; X = 10*p - 3 and Y = 10*r + 2 satisfy the Diophantine equation X^2 = 2*Y^2 + 41.
Both bisections of the sequence satisfy the recurrence relation b(n+2) = 1536796802*b(n+1) - b(n) - 441829080.
LINKS
FORMULA
a(n) = a(n-1) + 1536796802*a(n-2) - 1536796802*a(n-3) - a(n-4) + a(n-5). - Colin Barker, Dec 07 2014
G.f.: -x*(697*x^4 + 167145360*x^3 - 609119978*x^2 + 144840*x + 1) / ((x-1)*(x^2 - 39202*x + 1)*(x^2 + 39202*x + 1)). - Colin Barker, Dec 05 2014
EXAMPLE
a(2) = (5/2)*241^2 - (3/2)*241 = 144841 = 5*r^2 + 4*r + 1 with r = 170.
MAPLE
F:= gfun[rectoproc]({a(n) = a(n-1)+1536796802*a(n-2)-1536796802*a(n-3)-a(n-4)+a(n-5),
a(1)=1, a(2)=144841, a(3)=927821665, a(4)=222590743768705, a(5) = 1425873367156486249}, a(n), remember):
seq(F(n), n=1..20); # Robert Israel, Dec 07 2014
MATHEMATICA
LinearRecurrence[{1, 1536796802, -1536796802, -1, 1}, {1, 144841, 927821665, 222590743768705, 1425873367156486249}, 20] (* Harvey P. Dale, Dec 21 2016 *)
PROG
(PARI) Vec(-x*(697*x^4+167145360*x^3-609119978*x^2+144840*x+1) / ((x-1)*(x^2-39202*x+1)*(x^2+39202*x+1)) + O(x^100)) \\ Colin Barker, Dec 05 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Richard Choulet, Oct 18 2007
EXTENSIONS
More terms from Colin Barker, Dec 05 2014
Edited by Michel Marcus and Colin Barker, Dec 07 2014
STATUS
approved