A128922 -id:A128922 - cp's OEIS frontend

A131215 Numbers which are both 11-gonal and centered 11-gonal.

1, 606, 241396, 96075211, 38237692791, 15218505655816, 6056927013322186, 2410641732796574421, 959429352726023297581, 381850471743224475863026, 151975528324450615370186976, 60485878422659601692858553631

Offset: 1

Richard Choulet, Sep 27 2007

nonn

A centered 11-gonal number is defined by (11*r^2 - 11*r + 2)/2 = A069125(r); a 11-gonal number by (9*p^2 - 7*p)/2 = A051682(p).
A number is both these numbers iff exist p and r such that (18*p - 7)^2 = 99*(2*r - 1) + 22.
The Diophantine equation X^2 = 99*Y^2 + 22 is such that : X is given by the sequence 11, 209, 4169, 83171,... in A131216; Y is given by the sequence 1, 21, 419, 8359,... in A083043.
The first equation is such that : p is given by 1, 12, 232, 4621,... which satisfies a(n+2) = 20*a(n+1) - a(n) - 7 and a(n+1) = 10*a(n) - 7/2 + sqrt(396*a(n)^2 - 308*a(n) + 33)/2 with g.f.  (1 -9*x +x^2)/( (1-x) * (1 -20*x + x^2) ); r is given by 1, 11, 210, 4180,... which satisfies a(n+2) = 20*a(n+1) - a(n) - 9 and a(n+1) = 10*a(n) - 9/2 + sqrt(396*a(n)^2 - 396*a(n) + 121)/2 with g.f. (1 - 10*x)/( (1-x)*(1 -20*x +x^2) ).

G. C. Greubel, Table of n, a(n) for n = 1..380
Index entries for linear recurrences with constant coefficients, signature (399,-399,1).

Cf. A128922.

a:=[1,606,241396];; for n in [4..20] do a[n]:=399*a[n-1]-399*a[n-2] +a[n-3]; od; a; # G. C. Greubel, Dec 06 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2)) )); // G. C. Greubel, Dec 06 2019

seq(coeff(series(x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2)), x, n+1), x, n), n = 1..20); # G. C. Greubel, Dec 06 2019

LinearRecurrence[{399,-399,1},{1,606,241396},20] (* Harvey P. Dale, Mar 04 2015 *)

my(x='x+O('x^20)); Vec(x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2))) \\ G. C. Greubel, Dec 06 2019

def A131215_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2)) ).list()
a=A131215_list(20); a[1:] # G. C. Greubel, Dec 06 2019

a(n+2) = 398*a(n+1) - a(n) + 209.
a(n+1) = 199*a(n) + 209/2 + (5/2)*sqrt(6336*a(n)^2 + 6688*a(n) + 1617).
G.f.: z*(1 +207*z +z^2)/((1-z)*(1-398*z+z^2)).
a(1)=1, a(2)=606, a(3)=241396, a(n) = 399*a(n-1) - 399*a(n-2) + a(n-3). - Harvey P. Dale, Mar 04 2015

More terms from Paolo P. Lava, Sep 26 2008

A133273 Indices of centered decagonal numbers which are also decagonal numbers.

Original entry on oeis.org

1, 10, 171, 3060, 54901, 985150, 17677791, 317215080, 5692193641, 102142270450, 1832868674451, 32889493869660, 590178020979421, 10590314883759910, 190035489886698951, 3410048503076821200, 61190837565496082641, 1098025027675852666330, 19703259660599851911291

Offset: 1

Richard Choulet, Oct 16 2007

Numbers k such that 80*k^2 - 80*k + 25 is a square.

Also the indices of centered square numbers which are also centered pentagonal numbers. - Colin Barker, Jan 01 2015

Colin Barker, Table of n, a(n) for n = 1..798
Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Cf. A001107, A001844, A005891, A010532, A062786, A128922.

LinearRecurrence[{19,-19,1},{1,10,171},20] (* Harvey P. Dale, Oct 09 2020 *)

Vec(x*(-1+9*x)/((-1+x)*(1-18*x+x^2)) + O(x^100)) \\ Colin Barker, Jan 01 2015

a(n+2) = 18*a(n+1) - a(n) - 8.
a(n+1) = 9*a(n) - 4 + sqrt(80*a(n)^2 - 80*a(n) + 25).
G.f.: x*(-1+9*x)/(-1+x)/(1 - 18*x + x^2). - R. J. Mathar, Nov 14 2007
a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3). - Colin Barker, Jan 01 2015
Product_{n>=2} (1 - 1/a(n)) = 2/sqrt(5) (= A010532 / 10). - Amiram Eldar, Dec 02 2024

More terms from Paolo P. Lava, Nov 25 2008

A280070 Indices of 10-gonal numbers (A001107) that are also centered 10-gonal numbers (A062786).

Original entry on oeis.org

1, 11, 191, 3421, 61381, 1101431, 19764371, 354657241, 6364065961, 114198530051, 2049209474951, 36771572019061, 659839086868141, 11840331991607471, 212466136762066331, 3812550129725586481, 68413436198298490321, 1227629301439647239291, 22028913989715351816911

Offset: 1

Colin Barker, Dec 25 2016

Also positive integers x in the solutions to 4*x^2 - 5*y^2 - 3*x + 5*y - 1 = 0, the corresponding values of y being A133273.

			11 is in the sequence because the 11th 10-gonal number is 451, which is also the 10th centered 10-gonal number.

Colin Barker, Table of n, a(n) for n = 1..750
Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Cf. A001107, A062786, A128922, A133273.

Vec(x*(1 - 8*x + x^2) / ((1 - x)*(1 - 18*x + x^2)) + O(x^30))

a(n) = (6 + (5+2*sqrt(5))*(9+4*sqrt(5))^(-n) + (5-2*sqrt(5))*(9+4*sqrt(5))^n)/16.
a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 - 8*x + x^2) / ((1 - x)*(1 - 18*x + x^2)).

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A131215 Numbers which are both 11-gonal and centered 11-gonal.

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A133273 Indices of centered decagonal numbers which are also decagonal numbers.

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A280070 Indices of 10-gonal numbers (A001107) that are also centered 10-gonal numbers (A062786).

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