A108741 -id:A108741 - cp's OEIS frontend

A132596 X-values of solutions to the equation X(X + 1) - 6Y^2 = 0.

0, 2, 24, 242, 2400, 23762, 235224, 2328482, 23049600, 228167522, 2258625624, 22358088722, 221322261600, 2190864527282, 21687323011224, 214682365584962, 2125136332838400, 21036680962799042, 208241673295152024

Offset: 0

Mohamed Bouhamida, Nov 14 2007

Or, 3*A000217(X) is a square, (3*A004189(n))^2. - Zak Seidov, Apr 08 2009

"You can find an infinite number of [different] triangular numbers such that when multiplied together form a square number. For example, for every triangular number, T_n, there are an infinite number of other triangular numbers, T_m, such that T_n*T_m is a square. For example, T_2 * T_24 = 30^2." [Pickover] - Robert G. Wilson v, Apr 01 2010

Clifford A. Pickover, The Loom of God, Tapestries of Mathematics and Mysticism, Sterling, NY, 2009, page 33.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

Cf. A007654, A001079, A000217, A098297, A108741 (Y^2), A004189 (Y).

LinearRecurrence[{11, -11, 1}, {0, 2, 24}, 19] (* Jean-François Alcover, Feb 26 2019 *)

a(n) = 10*a(n-1) - a(n-2) + 4, a(0)=0, a(1)=2.
a(n) = (A001079(n) - 1)/2. - Max Alekseyev, Nov 13 2009
From R. J. Mathar, Apr 20 2010: (Start)
a(n) = 11*a(n-1) - 11*a(n-2) + a(n-3) = 2*A098297(n).
G.f.: -2*x*(1+x) / ( (x-1)*(x^2-10*x+1) ). (End)
a(n) = 2*A098297(n) = (1/2)*(T(2*n,sqrt(3)) - 1), where T(n,x) is the n-th Chebyshev polynomial of the first kind. - Peter Bala, Dec 31 2012

A173130 a(n) = Cosh[(2 n - 1) ArcCosh[n]].

Original entry on oeis.org

0, 1, 26, 3363, 937444, 456335045, 343904160606, 371198523608647, 543466014742175624, 1036834190110356583689, 2499384905955651114739810, 7429238104512325157021090411, 26695718139185294187938997247212

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128.

Table[Round[Cosh[(2 n - 1) ArcCosh[n]]], {n, 0, 20}] (* Artur Jasinski *)

a(n) ~ 2^(2*n-2) * n^(2*n-1). - Vaclav Kotesovec, Apr 05 2016

A173131 a(n) = (Cosh[(2n-1)ArcSinh[n]])^2.

Original entry on oeis.org

1, 2, 1445, 19740250, 1361599599377, 298514762397852026, 160545187370375075046277, 179656719395983409634002348450, 373368546362937441101158606899394625

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130.

Table[Round[Cosh[(2 n - 1) ArcSinh[n]]^2], {n, 0, 10}] (* Artur Jasinski *)

a(n) ~ 2^(4*n-4) * n^(4*n-2). - Vaclav Kotesovec, Apr 05 2016

A173133 a(n) = Sinh[(2n-1) ArcSinh[n]].

Original entry on oeis.org

0, 1, 38, 4443, 1166876, 546365045, 400680904674, 423859315570607, 611038907405197432, 1151555487914640463209, 2748476184146759127540190, 8102732939160371170806346243, 28915133156938367486730067779348

Offset: 0

Artur Jasinski, Feb 10 2010

nonn

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131.

Table[Round[Sinh[(2 n - 1) ArcSinh[n]]], {n, 0, 20}] (* Artur Jasinski *)
Round[Table[1/2 (n - Sqrt[1 + n^2])^(2 n - 1) + 1/2 (n + Sqrt[1 + n^2])^(2 n - 1), {n, 0, 10}]] (* Artur Jasinski, Feb 14 2010 *)

a(n) = 1/2 (n - sqrt(1 + n^2))^(2 n - 1) + 1/2 (n + sqrt(1 + n^2))^(2 n - 1). - Artur Jasinski, Feb 14 2010

Minor edits by Vaclav Kotesovec, Apr 05 2016

A173134 a(n) = Sinh[(2n-1)ArcCosh[n]]^2.

Original entry on oeis.org

-1, 0, 675, 11309768, 878801253135, 208241673295152024, 118270071682117442287235, 137788343929239264227213170608, 295355309179742652677310128859789375

Offset: 0

Artur Jasinski, Feb 10 2010

sign

Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133.

Table[Round[Sinh[(2 n - 1) ArcCosh[n]]^2], {n, 0, 20}]

a(n) ~ 2^(4*n-4) * n^(4*n-2). - Vaclav Kotesovec, Apr 05 2016

A097726 Pell equation solutions (5a(n))^2 - 26b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

Original entry on oeis.org

1, 103, 10505, 1071407, 109273009, 11144775511, 1136657829113, 115927953794015, 11823514629160417, 1205882564220568519, 122988198035868828521, 12543590317094399940623, 1279323224145592925115025, 130478425272533383961791927, 13307520054574259571177661529

Offset: 0

Wolfdieter Lang, Aug 31 2004

a(-1) = -1. - Artur Jasinski, Feb 10 2010

5*a(n) gives the x-values in the solution to the Pell equation x^2 - 26*y^2 = -1. - Colin Barker, Aug 24 2013

			(x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 = -1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (102,-1).

Cf. A097725 for S(n, 102).
Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121. - Artur Jasinski, Feb 10 2010
Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

Table[(1/5) Round[N[Sinh[(2 n - 1) ArcSinh[5]], 100]], {n, 1, 50}] (* Artur Jasinski, Feb 10 2010 *)
CoefficientList[Series[(1 + x)/(1 - 102 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 13 2014 *)
LinearRecurrence[{102,-1},{1,103},20] (* Harvey P. Dale, Aug 20 2017 *)

x='x+O('x^99); Vec((1+x)/(1-102*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 102*x + x^2).
a(n) = S(n, 2*51) + S(n-1, 2*51) = S(2*n, 2*sqrt(26)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 5*i)/(5*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 102*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=103. - Philippe Deléham, Nov 18 2008
a(n) = (1/5)*sinh((2*n-1)*arcsinh(5)), n >= 1. - Artur Jasinski, Feb 10 2010

More terms from Harvey P. Dale, Aug 20 2017

A278620 Expansion of x/(1 - 99x + 99x^2 - x^3).

Original entry on oeis.org

0, 1, 99, 9702, 950698, 93158703, 9128602197, 894509856604, 87652837344996, 8589083549953005, 841642535058049495, 82472379352138897506, 8081451533974553906094, 791899777950154143899707, 77598096787581131548265193, 7603821585405000737586089208, 745096917272902491151888477192

Offset: 0

Bruno Berselli, Nov 24 2016

Colin Barker, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

First differences: A173205.

Cf. A045502, A108741, A123479, A200993, A200994.

P:=proc(q) local a,b,c,n; a:=0; b:=1; print(a); print(b);for n from 1 to q do
c:=98*b-a+1; a:=b; b:=c; print(b); od; end: P(100); # Paolo P. Lava, Nov 30 2016

CoefficientList[x/(1 - 99 x + 99 x^2 - x^3) + O[x]^20, x]
LinearRecurrence[{99,-99,1},{0,1,99},20] (* Harvey P. Dale, Aug 22 2020 *)

makelist(coeff(taylor(x/((1-x)*(1-98*x+x^2)), x, 0, n), x, n), n, 0, 20);

concat(0, Vec(1/(1-99*x+99*x^2-x^3) + O(x^20)))

gf = x/((1-x)*(1-98*x+x^2)); print(taylor(gf, x, 0, 20).list())

O.g.f.: x/((1 - x)*(1 - 98*x + x^2)).
E.g.f.: ((5-2*sqrt(6))*exp((5-2*sqrt(6))^2*x) + (5+2*sqrt(6))*exp((5+2*sqrt(6))^2*x) - 10*exp(x))/960.
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n>2.
a(n) = 98*a(n-1) - a(n-2) + 1 for n>1.
a(n) = a(-n-1) = ((5+2*sqrt(6))^(2*n+1) + (5-2*sqrt(6))^(2*n+1))/960 - 1/96.
a(n) = floor((5+2*sqrt(6))^(2*n+1)/960).
a(n)*a(n-2) = a(n-1)*(a(n-1)-1) for n>1.
Lim_{i -> infinity} a(i)/a(i-1) = (5 + 2*sqrt(6))^2.
From the closed form: a(n) + a(-n) = A108741(n).
a(n) = A200993(n)/10 = A200994(n)/15.
a(n) = A123479(n)/20 for n>0.
a(n) = A045502(n)/40.

A220186 Numbers n >= 0 such that n^2 + n*(n+1)/2 is a square.

Original entry on oeis.org

0, 8, 800, 78408, 7683200, 752875208, 73774087200, 7229107670408, 708378777612800, 69413891098384008, 6801852948864020000, 666512175097575576008, 65311391306613542428800, 6399849835873029582446408, 627119972524250285537319200

Offset: 1

Alex Ratushnyak, Apr 12 2013

Equivalently, numbers n such that triangular(2*n) - triangular(n) is a square.

Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A005449 (n^2 + n(n+1)/2).
Cf. A011916 (numbers n such that n^2 + n(n+1)/2 is a triangular number).
Cf. A014105 (n^2 + n(n+1)).
Cf. A084703 (numbers n such that n^2 + n(n+1) is a square).
Cf. A220185 (numbers n such that n^2 + n(n+1) is an oblong number).

#include 
#include 
int main() {
  unsigned long long a, i, t;
  for (i=0; i < (1L<<32); ++i) {
      a = (i*i) + ((i+1)*i/2);
      t = sqrt(a);
      if (a == t*t)  printf("%llu\n", i);
  }
  return 0;
}

a[n_]:=Floor[(1/12)*(49 + 20*Sqrt[6])^n]; Table[a[n],{n,0,10}] (* Giovanni Resta, Apr 12 2013 *)
LinearRecurrence[{99,-99,1},{0,8,800},20] (* Harvey P. Dale, Nov 01 2022 *)

lista(nn) = for(n=0, nn, if(issquare(n^2 + n*(n+1)/2), print1(n, ", "))); \\ Altug Alkan, Mar 05 2016

a(n) = A098308(2*n-2).
a(1) = 0, a(2) = 8, a(3) = 800 and a(n) = 99*a(n-1)-99*a(n-2)+a(n-3) for n>3. - Giovanni Resta, Apr 12 2013
G.f.: -8*x^2*(x+1) / ((x-1)*(x^2-98*x+1)). - Colin Barker, May 31 2013
a(n) = (49+20*sqrt(6))^(-n)*(49+20*sqrt(6)-2*(49+20*sqrt(6))^n+(49-20*sqrt(6))*(49+20*sqrt(6))^(2*n))/12. - Colin Barker, Mar 05 2016
a(n) = 8*A108741(n). - R. J. Mathar, Feb 19 2017

A226500 Triangular numbers representable as 3 * x^2.

Original entry on oeis.org

0, 3, 300, 29403, 2881200, 282328203, 27665282700, 2710915376403, 265642041604800, 26030209161894003, 2550694855824007500, 249942065661590841003, 24491771739980078410800, 2399943688452386093417403, 235169989696593857076494700, 23044259046577745607403063203

Offset: 1

Alex Ratushnyak, Jun 09 2013

Colin Barker, Table of n, a(n) for n = 1..503
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A029549 (triangular numbers representable as x^2 + x).

Cf. A000217, A001219, A165892, A069017.

#include 
#include 
typedef unsigned long long U64;
U64 isTriangular(U64 a) {   // input must be < 1ULL<<63
    U64 r = sqrt(a*2);
    return (r*(r+1) == a*2);
}
int main() {
  for (U64 j, i = 0; (j=i*i*3) < (1ULL<<63); i++)
      if (isTriangular(j)) printf("%llu, ", j);
  return 0;
}

a[1]=0; a[2]=3; a[3]=300; a[n_] := a[n] = 99*(a[n-1] - a[n-2]) + a[n-3]; Array[a, 10] (* Giovanni Resta, Jun 09 2013 *)
Rest@ CoefficientList[Series[3 x^2 (1 + x)/((1 - x) (1 - 98 x + x^2)), {x, 0, 16}], x] (* or *)
3 LinearRecurrence[{99, -99, 1}, {0, 1, 100}, 16] (* Michael De Vlieger, Mar 03 2016, latter after Vincenzo Librandi at A108741 *)

a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3), for n > 3. a(n) = floor((49 + 20*sqrt(6))^(n-1)/32). - Giovanni Resta, Jun 09 2013
G.f.: 3*x^2*(1+x)/((1-x)*(1-98*x+x^2)); a(n)=3*A108741(n-1). - Joerg Arndt, Jun 10 2013
a(n) = (49+20*sqrt(6))^(-n)*(49+20*sqrt(6)-2*(49+20*sqrt(6))^n+(49-20*sqrt(6))*(49+20*sqrt(6))^(2*n))/32. - Colin Barker, Mar 03 2016

a(12)-a(15) from Giovanni Resta, Jun 09 2013

cp's OEIS Frontend

A132596 X-values of solutions to the equation X*(X + 1) - 6*Y^2 = 0.

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A173130 a(n) = Cosh[(2 n - 1) ArcCosh[n]].

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A173131 a(n) = (Cosh[(2n-1)ArcSinh[n]])^2.

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A173133 a(n) = Sinh[(2n-1) ArcSinh[n]].

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A173134 a(n) = Sinh[(2n-1)ArcCosh[n]]^2.

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A097726 Pell equation solutions (5*a(n))^2 - 26*b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

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A278620 Expansion of x/(1 - 99*x + 99*x^2 - x^3).

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A220186 Numbers n >= 0 such that n^2 + n*(n+1)/2 is a square.

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C

A132596 X-values of solutions to the equation X(X + 1) - 6Y^2 = 0.

A097726 Pell equation solutions (5a(n))^2 - 26b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

A278620 Expansion of x/(1 - 99x + 99x^2 - x^3).