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A133301 a(n) is the n-th pentagonal number which is the sum of two consecutive pentagonal numbers. 9
1, 1926, 850137, 2564464982, 1132138928657, 3415133918621062, 1507685261236261801, 4547981651299964079126, 2007805569980855008712097, 6056610836775865229750164742, 2673822786819976661810784866297, 8065673443881586606920210924732502 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

We solve the equation P(p)=P(r)+P(r+1) with unknowns p and r, equivalent to (6*p-1)^2=2*(6*r+2)^2+17. The Diophantine equation X^2=2*Y^2+17 whose solutions give p and r are obtained by (x(n), y(n)) such that:

x(1)=5, x(2)=215, x(3)=4517, x(4)=248087 and the same recurrence relation on the odd and even indices x(n+2)=1154*x(n+1)-x(n)

y(1)=2, y(2)=152, y(3)=3194, y(4)=175424 and the same recurrence relation on the odd and even indices y(n+2)=1154*y(n+1)-y(n)

The solutions (p,r) are given by the (u(n),v(n)) such that

u(1)=1, u(2)=36, u(3)=753, u(4)=41348 and the same recurrence relation on the odd and even indices u(n+2)=1154*u(n+1)-u(n) -192 or u(n+1)=577*u(n)-96+68*(72*u(n)^2-24*u(n)-32)^0.5

v(1)=0, v(2)=25, v(3)=532, v(4)=29237 and the same recurrence relation on the odd and even indices v(n+2)=1154*v(n+1)-v(n) +384 or v(n+1)=577*v(n)+192+68*(72*u(n)^2+48*u(n)+15)^0.5

LINKS

Colin Barker, Table of n, a(n) for n = 1..300

Index entries for linear recurrences with constant coefficients, signature (1,1331714,-1331714,-1,1).

FORMULA

For odd and even indices respectively : a(n+2)=1331714*a(n+1)-a(n)-416160. on the odd and the even indices respectively we have also : a(n+1)=665857*a(n)-208080+19618*(1152*a(n)^2-720*a(n)-32)^0.5. the g.f function h such that h(z)=a(1)*z+a(2)*z^2+... is given by h(z)=((z+1925*z^2-483503*z^3+65395*z^4+22*z^5)/((1-z)*(1-1331714*z^2+z^4)).

EXAMPLE

with P(m)=m*(3*m-1)/2, a(1)=1 because a(1)=P(1)=P(0)+P(1); a(2)=1926 because P(36)=1926=P(25)+P(26)=925+1001 ; a(3)=850137 because P(753)=850137=P(532)+P(533)=424270+425867 ...

MAPLE

a:=proc(m) if type (sqrt(18*m^2-6*m-8)/6-1/3), integer=true then m*(3*m-1)/2 else fi end : seq(a(m), m=1..100000)od; # Emeric Deutsch

MATHEMATICA

# (3 # - 1)/2 &@ Select[Range[10^6], IntegerQ[Sqrt[18 #^2 - 6 # - 8]/6 - 1/3] &] (* or *)

Rest@ CoefficientList[Series[-x (22 x^4 + 65395 x^3 - 483503 x^2 + 1925 x + 1)/((x - 1) (x^2 - 1154 x + 1) (x^2 + 1154 x + 1)), {x, 0, 12}], x] (* Michael De Vlieger, Jul 14 2016 *)

PROG

(PARI) Vec(-x*(22*x^4+65395*x^3-483503*x^2+1925*x+1)/((x-1)*(x^2-1154*x+1)*(x^2+1154*x+1)) + O(x^20)) \\ Colin Barker, Oct 20 2014

CROSSREFS

Sequence in context: A255867 A202051 A283949 * A258841 A099482 A253337

Adjacent sequences:  A133298 A133299 A133300 * A133302 A133303 A133304

KEYWORD

nonn,easy

AUTHOR

Richard Choulet, Dec 20 2007

EXTENSIONS

Fixed typo in g.f. in formula, and more terms from Colin Barker, Oct 20 2014

STATUS

approved

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Last modified February 16 02:39 EST 2019. Contains 320140 sequences. (Running on oeis4.)