A157879 -id:A157879 - cp's OEIS frontend

A157014 Expansion of x(1-x)/(1 - 22x + x^2).

1, 21, 461, 10121, 222201, 4878301, 107100421, 2351330961, 51622180721, 1133336644901, 24881784007101, 546265911511321, 11992968269241961, 263299036011811821, 5780585823990618101, 126909589091781786401, 2786230374195208682721, 61170158643202809233461

Offset: 1

Paul Weisenhorn, Feb 21 2009

This sequence is part of a solution of a general problem involving 2 equations, three sequences a(n), b(n), c(n) and a constant A:
    A    * c(n)+1 = a(n)^2,
   (A+1) * c(n)+1 = b(n)^2, where solutions are given by the recurrences:
a(1) = 1, a(2) = 4*A+1, a(n) = (4*A+2)*a(n-1)-a(n-2) for n>2, resulting in a(n) terms 1, 4*A+1, 16*A^2+12*A+1, 64*A^3+80*A^2+24*A+1, ...;
b(1) = 1, b(2) = 4*A+3, b(n) = (4*A+2)*b(n-1)-b(n-2) for n>2, resulting in b(n) terms 1, 4*A+3, 16*A^2+20*A+5, 64*A^3+112*A^2+56*A+7, ...;
c(1) = 0, c(2) = 16*A+8, c(3) = (16*A^2+16*A+3)*c(2), c(n) = (16*A^2+16*A+3) * (c(n-1)-c(n-2)) + c(n-3) for n>3, resulting in c(n) terms 0, 16*A+8, 256*A^3+384*A^2+176*A+24, 4096*A^5 + 10240*A^4 + 9472*A^3 + 3968*A^2 + 736*A + 48, ... .
A157014 is the a(n) sequence for A=5.
For other A values the a(n), b(n) and c(n) sequences are in the OEIS:
  A   a-sequence    b-sequence     c-sequence
  1     A001653       A002315(n-1)   A078522
  2     A072256       A054320(n-1)   A045502(n-1)
  3     A001570       A028230        A059989(n-1)
  4     A007805       A049629(n-1)   A157459
  5  -> A157014 <-    A133283        A157460
  6     A153111       A157461        A157874
  7     A157877       A157878        A157879
  8     A077420(n-1)  A046176        A157880
  9     A097315(n-1)  A097314(n-1)   A157881
Positive values of x (or y) satisfying x^2 - 22xy + y^2 + 20 = 0. - Colin Barker, Feb 19 2014
From Klaus Purath, Apr 22 2025: (Start)
Nonnegative solutions to the Diophantine equation 5*b(n)^2 - 6*a(n)^2 = -1. The corresponding b(n) are A133283(n). Note that (b(n+1)^2 - b(n)*b(n+2))/4 = 6 and (a(n)*a(n+2) - a(n+1)^2)/4 = 5.
(a(n) + b(n))/2 = (b(n+1) - a(n+1))/2 = A077421(n-1) = Lucas U(22,1). Also b(n)*a(n+1) - b(n+1)*a(n) = -2.
a(n)=(t(i+2*n-1) + t(i))/(t(i+n) + t(i+n-1)) as long as t(i+n) + t(i+n-1) != 0 for any integer i and n >= 1 where (t) is a sequence satisfying t(i+3) = 21*t(i+2) - 21*t(i+1) + t(i) or t(i+2) = 22*t(i+1) - t(i) without regard to initial values and including this sequence itself. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (22,-1).

Cf. similar sequences listed in A238379.

a:=[1,21];; for n in [3..20] do a[n]:=22*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 14 2020

I:=[1,21]; [n le 2 select I[n] else 22*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 21 2014

seq( simplify(ChebyshevU(n-1,11) - ChebyshevU(n-2,11)), n=1..20); # G. C. Greubel, Jan 14 2020

CoefficientList[Series[(1-x)/(1-22x+x^2), {x,0,20}], x] (* Vincenzo Librandi, Feb 21 2014 *)
a[c_, n_] := Module[{},
   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
   d := Denominator[Convergents[Sqrt[c], n p]];
   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
   Return[t];
] (* Complement of A041049 *)
a[30, 20] (* Gerry Martens, Jun 07 2015 *)
Table[ChebyshevU[n-1, 11] - ChebyshevU[n-2, 11], {n,20}] (* G. C. Greubel, Jan 14 2020 *)

Vec((1-x)/(1-22*x+x^2)+O(x^20)) \\ Charles R Greathouse IV, Sep 23 2012

[chebyshev_U(n-1,11) - chebyshev_U(n-2,11) for n in (1..20)] # G. C. Greubel, Jan 14 2020

G.f.: x*(1-x)/(1-22*x+x^2).
a(1) = 1, a(2) = 21, a(n) = 22*a(n-1) - a(n-2) for n>2.
5*A157460(n)+1 = a(n)^2 for n>=1.
6*A157460(n)+1 = A133283(n)^2 for n>=1.
a(n) = (6+sqrt(30)-(-6+sqrt(30))*(11+2*sqrt(30))^(2*n))/(12*(11+2*sqrt(30))^n). - Gerry Martens, Jun 07 2015
a(n) = ChebyshevU(n-1, 11) - ChebyshevU(n-2, 11). - G. C. Greubel, Jan 14 2020

Edited by Alois P. Heinz, Sep 09 2011

A245031 Numbers m such that 3m+1 and 8m+1 are both squares.

Original entry on oeis.org

0, 1, 21, 120, 2080, 11781, 203841, 1154440, 19974360, 113123361, 1957283461, 11084934960, 191793804840, 1086210502741, 18793835590881, 106437544333680, 1841604094101520, 10429793134197921, 180458407386358101, 1022013289607062600

Offset: 1

Bruno Berselli, Jul 15 2014

Naturally, all terms are triangular numbers.
Numbers m such that k*m+1 and 8*m+1 are both squares:
k=1: A006454;
k=3: this sequence;
k=4: A029549;
k=5: 0, 3, 231, 4560, 333336, 6575751, ...
k=6: A200999;
k=7: A157879.
Numbers m such that 3*m+1 and k*m+1 are both squares:
k=1: A045899;
k=2: A045502;
k=4: A059989;
k=5: A159683;
k=6: 8*A029546;
k=7: A160695;
k=8: this sequence.

Bruno Berselli, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (1,98,-98,-1,1).

Cf. A000217.

Cf. A006454, A029546, A029549, A045502, A045899, A059989, A157879, A159683, A160695, A200999.

I:=[0,1,21,120,2080]; [n le 5 select I[n] else Self(n-1)+98*Self(n-2)-98*Self(n-3)-Self(n-4)+Self(n-5): n in [1..20]];

LinearRecurrence[{1, 98, -98, -1, 1}, {0, 1, 21, 120, 2080}, 20] (* or *) CoefficientList[Series[x (1 + 20 x + x^2)/((1 - x) (1 - 10 x + x^2) (1 + 10 x + x^2)), {x, 0, 20}], x]

a[1]:0$ a[2]:1$ a[3]:21$ a[4]:120$ a[5]:2080$ a[n]:=a[n-1]+98*a[n-2]-98*a[n-3]-a[n-4]+a[n-5]$ makelist(a[n], n, 1, 20);

a=vector(20); a[1]=0; a[2]=1; a[3]=21; a[4]=120; a[5]=2080; for(i=6, #a, a[i]=a[i-1]+98*a[i-2]-98*a[i-3]-a[i-4]+a[i-5]); a

G.f.: x^2*(1 + 20*x + x^2)/((1 - x)*(1 - 10*x + x^2)*(1 + 10*x + x^2)).
a(n) = a(n-1) + 98*a(n-2) - 98*a(n-3) - a(n-4) + a(n-5).
G.f. of the quadrisections:
a(4k+1):   40*x*(52 + 3*x)/((1 - x)*(1 - 9602*x + x^2));
a(4k+2): (1 + 2178*x + 21*x^2)/((1 - x)*(1 - 9602*x + x^2));
a(4k+3): (21 + 2178*x + x^2)/((1 - x)*(1 - 9602*x + x^2));
a(4k+4): 40*(3 + 52*x)/((1 - x)*(1 - 9602*x + x^2)).

Changed offset from 0 to 1 and adapted formulas by Bruno Berselli, Mar 03 2016

A157877 Expansion of (1-x)x/(x^2-30x+1).

Original entry on oeis.org

1, 29, 869, 26041, 780361, 23384789, 700763309, 20999514481, 629284671121, 18857540619149, 565096933903349, 16934050476481321, 507456417360536281, 15206758470339607109, 455695297692827676989, 13655652172314490702561, 409213869871741893399841

Offset: 1

Paul Weisenhorn, Mar 08 2009

This sequence is part of a solution of a more general problem involving 2 equations, three sequences a(n), b(n), c(n) and a constant A:
    A    * c(n)+1 = a(n)^2,
   (A+1) * c(n)+1 = b(n)^2, for details see comment in A157014.
A157877 is the a(n) sequence for A=7.
Positive values of x (or y) satisfying x^2 - 30xy + y^2 + 28 = 0. - Colin Barker, Feb 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (30,-1).

Cf. A157879, A157877, A157878.

Cf. similar sequences listed in A238379.

I:=[1,29,869]; [n le 3 select I[n] else 30*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 25 2014

LinearRecurrence[{30,-1},{1,29},30] (* Harvey P. Dale, Dec 14 2011 *)
CoefficientList[Series[(1 - x)/(x^2 - 30 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 25 2014 *)

Vec((1-x)*x/(x^2-30*x+1)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

G.f.: (1-x)*x/(x^2-30*x+1).
a(1)=1, a(2)=29; for n>2, a(n) = 30*a(n-1)-a(n-2).
7*A157879(n)+1 = a(n)^2.
8*A157879(n)+1 = A157878(n)^2.
a(n) = (1/8)*(4-sqrt(14))*(1+(15+4*sqrt(14))^(2*n-1))/(15+4*sqrt(14))^(n-1). - Bruno Berselli, Feb 25 2014
From Andrea Pinos, Oct 05 2022: (Start)
a(n) = ceiling((C^n)/(C+1)), where C = 15 + 4*sqrt(14) = sqrt(225) + sqrt(224).
Limit_{n->oo} a(n+1)/a(n) = C. (End)

Edited by Alois P. Heinz, Sep 09 2011

A157878 Expansion of x(1+x)/(x^2-30x+1).

Original entry on oeis.org

1, 31, 929, 27839, 834241, 24999391, 749147489, 22449425279, 672733610881, 20159558901151, 604114033423649, 18103261443808319, 542493729280825921, 16256708616980969311, 487158764780148253409, 14598506234787466632959, 437468028278843850735361

Offset: 1

Paul Weisenhorn, Mar 08 2009

This sequence is part of a solution of a more general problem involving 2 equations, three sequences a(n), b(n), c(n) and a constant A:
    A    * c(n)+1 = a(n)^2,
   (A+1) * c(n)+1 = b(n)^2, for details see comment in A157014.
this is the b(n) sequence for A=7.

Robert Israel, Table of n, a(n) for n = 1..610
Index entries for linear recurrences with constant coefficients, signature (30,-1).

7*A157879(n)+1 = A157877(n)^2.

8*A157879(n)+1 = A157878(n)^2.

f:= gfun:-rectoproc({a(1) = 1, a(2) = 31, a(n) = 30*a(n-1)-a(n-2)}, a(n), remember):
map(f, [$1..30]); # Robert Israel, Jul 09 2015

CoefficientList[Series[x*(1 + x)/(x^2 - 30 x + 1), {x, 0, 17}], x] (* Michael De Vlieger, Jul 09 2015 *)
LinearRecurrence[{30,-1},{1,31},20] (* Harvey P. Dale, Sep 05 2021 *)

Vec((1+x)/(x^2-30*x+1)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

G.f.: x*(1+x)/(x^2-30*x+1).
a(1) = 1, a(2) = 31, a(n) = 30*a(n-1)-a(n-2) for n>2.
a(n) = ((15-4*sqrt(14))^(n-1)*(7-2*sqrt(14))+(7+2*sqrt(14))*(15+4*sqrt(14))^(n-1))/14. - Gerry Martens, Jul 09 2015

Edited by Alois P. Heinz, Sep 09 2011

A253514 Centered heptagonal numbers (A069099) which are also centered octagonal numbers (A016754).

Original entry on oeis.org

1, 841, 755161, 678133681, 608963290321, 546848356574521, 491069215240629481, 440979608437728699361, 395999197307865131396641, 355606838202854450265484201, 319334544706965988473273415801, 286762065540017254794549261905041

Offset: 1

Colin Barker, Jan 03 2015

			841 is in the sequence because it is the 16th centered heptagonal number and the 15th centered octagonal number.

Colin Barker, Table of n, a(n) for n = 1..339
Index entries for linear recurrences with constant coefficients, signature (899,-899,1).

Cf. A001110, A016754, A069099, A157877, A157879, A253446, A253447.

Vec(-x*(x^2-58*x+1)/((x-1)*(x^2-898*x+1)) + O(x^100))

a(n) = 899*a(n-1)-899*a(n-2)+a(n-3).
G.f.: -x*(x^2-58*x+1) / ((x-1)*(x^2-898*x+1)).
From Peter Bala, Apr 15 2025; (Start)
a(n) = (1/64)*(-4 + sqrt(14))^2*(15 + 4*sqrt(14) + (449 + 120*sqrt(14))^n)^2 *(449 + 120*sqrt(14))^(-n).
a(-n) = a(n+1).
a(n) = (1/16) * (1 - T(2*n+1, -15)), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. Cf. A001110.
a(n) = A157877(n)^2 = 1 + 7*A157879(n).
a(2) divides a(3*n+2); a(3) divides a(5*n+3); a(4) divides a(7*n+4); a(5) divides a(9*n+5). In general, a(k) divides a((2*k-1)*n + k). (End)

cp's OEIS Frontend

A157014 Expansion of x*(1-x)/(1 - 22*x + x^2).

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A245031 Numbers m such that 3*m+1 and 8*m+1 are both squares.

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A157877 Expansion of (1-x)*x/(x^2-30*x+1).

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A157878 Expansion of x*(1+x)/(x^2-30*x+1).

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A253514 Centered heptagonal numbers (A069099) which are also centered octagonal numbers (A016754).

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A157014 Expansion of x(1-x)/(1 - 22x + x^2).

A245031 Numbers m such that 3m+1 and 8m+1 are both squares.

A157877 Expansion of (1-x)x/(x^2-30x+1).

A157878 Expansion of x(1+x)/(x^2-30x+1).