A004253 -id:A004253 - cp's OEIS frontend

A237255 Values of x in the solutions to x^2 - 5xy + y^2 + 17 = 0, where 0 < x < y.

2, 3, 7, 13, 33, 62, 158, 297, 757, 1423, 3627, 6818, 17378, 32667, 83263, 156517, 398937, 749918, 1911422, 3593073, 9158173, 17215447, 43879443, 82484162, 210239042, 395205363, 1007315767, 1893542653, 4826339793, 9072507902, 23124383198, 43468996857

Offset: 1

Colin Barker, Feb 05 2014

The corresponding values of y are given by a(n+2).

			3 is in the sequence because (x, y) = (3, 13) is a solution to x^2 - 5xy + y^2 + 17 = 0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-1).

Cf. A004253, A237254.

Vec(-x*(x-1)*(x+2)*(2*x+1)/(x^4-5*x^2+1) + O(x^100))

a(n) = 5*a(n-2)-a(n-4).

G.f.: -x*(x-1)*(x+2)*(2*x+1) / (x^4-5*x^2+1).

A334673 a(n) = 23*a(n-1) - a(n-2) + 1 for n > 1, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 24, 552, 12673, 290928, 6678672, 153318529, 3519647496, 80798573880, 1854847551745, 42580695116256, 977501140122144, 22439945527693057, 515141245996818168, 11825808712399124808, 271478459139183052417, 6232178751488811080784, 143068632825103471805616

Offset: 0

Francesca Arici, Sep 11 2020

Michael De Vlieger, Table of n, a(n) for n = 0..735
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A004253, A004254, A030221, A097778 (first differences).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

CoefficientList[Series[x/((1 - x) (x^2 - 23 x + 1)), {x, 0, 18}], x] (* Michael De Vlieger, Apr 07 2021 *)

a(n) = A004254(n)*A004254(n+1)/5 = A160695(n+1)/5.
G.f.: x/((1-x)*(x^2-23*x+1)). - Alois P. Heinz, Sep 11 2020
From Klaus Purath, Jun 18 2025: (Start)
a(n) = (A004253(n+1)^2 - 1) / 15.
a(n) = (A030221(n)^2 - 1) / 35.
a(n) + a(n+1) = A004253(n+1)^2. (End)

a(13)-a(14) corrected and more terms added by Alois P. Heinz, Sep 11 2020

A085348 Ratio-determined insertion sequence I(0.264) (see the link below).

Original entry on oeis.org

1, 4, 19, 72, 341, 1292, 6119, 23184, 109801, 416020, 1970299, 7465176, 35355581, 133957148, 634430159, 2403763488, 11384387281, 43133785636, 204284540899, 774004377960, 3665737348901, 13888945017644, 65778987739319

Offset: 0

John W. Layman, Jun 24 2003

nonn

This is one of the "twin" ratio-determined insertion sequences (RDIS) that are "children" in the next generation below the "parent" sequences I(0.25024) (A004253) and I(0.26816) (A001353) in the recurrence tree of RDIS sequences. The RDIS twin of this sequence is A085349. See the link for an explanation of RDIS twin. See A082630 or A082981 for other recent examples of RDIS sequences.

Assuming that a(n) = 18a(n-2) - a(n-4) is true: For n >= 2, a(n) = (t(i+2n+2) - t(i))/(t(i+n+2) + t(i+n)*(-1)^(n-1)), where (t) is any recurrence of the form (4,1) without regard to initial values. With an additional initional 0 is this sequence the union of A060645 for even n and A049629 for odd n. - Klaus Purath, Sep 22 2024

John W. Layman, Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types, June 2003 [Broken link]
John W. Layman, Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types, June 2003 [local copy, corrected]
John W. Layman, Sequences Generated by Age-Determined Insertion Trees, Jan 2006
John W. Layman, Sequences Generated by Age-Determined Insertion Trees, Jan 2006 [Local copy]

Cf. A001353, A004253, A082630, A082981, A085349.

It appears that a(n)=18a(n-2)-a(n-4).
Apparently a(n)a(n+3) = -4 + a(n+1)a(n+2). - Ralf Stephan, May 29 2004
From Klaus Purath, Sep 22 2024: (Start)
Assuming that a(n) = 18a(n-2) - a(n-4) is true:
a(2n) = 5a(2n-1) - a(2n-2), n >= 1.
a(2n+1) = 4a(2n) - a(2n-1), n >= 1. (End)

A085349 Ratio-determined insertion sequence I(0.26688) (see the link below).

Original entry on oeis.org

1, 4, 15, 71, 269, 1274, 4827, 22861, 86617, 410224, 1554279, 7361171, 27890405, 132090854, 500473011, 2370274201, 8980623793, 42532844764, 161150755263, 763220931551, 2891732970941, 13695443923154, 51890042721675

Offset: 1

John W. Layman, Jun 24 2003

nonn

This is one of the "twin" ratio-determined insertion sequences (RDIS) that are "children" in the next generation below the "parent" sequences I(0.25024) (A004253) and I(0.26816) (A001353) in the recurrence tree of RDIS sequences. The RDIS twin of this sequence is A085348. See the link for an explanation of RDIS twin. See A082630 or A082981 for other recent examples of RDIS sequences.

John W. Layman, Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types, June 2003 [Broken link]
John W. Layman, Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types, June 2003 [local copy, corrected]
John W. Layman, Sequences Generated by Age-Determined Insertion Trees, Jan 2006
John W. Layman, Sequences Generated by Age-Determined Insertion Trees, Jan 2006 [Local copy]
Index entries for linear recurrences with constant coefficients, signature (0, 18, 0, -1).

Cf. A001353, A004253, A082630, A082981, A085348.

LinearRecurrence[{0,18,0,-1},{1,4,15,71},30] (* Harvey P. Dale, Mar 04 2013 *)

It appears that a(n)=18a(n-2)-a(n-4).

Apparently a(n)a(n+3) = 11 + a(n+1)a(n+2). - Ralf Stephan, May 29 2004

A099867 a(n) = 5*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=9.

Original entry on oeis.org

1, 9, 44, 211, 1011, 4844, 23209, 111201, 532796, 2552779, 12231099, 58602716, 280782481, 1345309689, 6445765964, 30883520131, 147971834691, 708975653324, 3396906431929, 16275556506321, 77980876099676, 373628823992059, 1790163243860619, 8577187395311036

Offset: 0

Creighton Dement, Oct 28 2004

From Klaus Purath, Mar 07 2023: (Start)
For any two terms (a(n), a(n+1)) = (x, y), x^2 - 5*x*y + y^2 = 37 = A082111(4). This is valid in general for all recursive sequences (t) with constant coefficients (5,-1) and t(0) =  1: x^2 - 5*x*y + y^2 = A082111(t(1)-5). This includes and interprets the Feb 04 2014 comment in A004253 by Colin Barker.
By analogy to all this, for three consecutive terms (x, y, z) of any sequence (t) of the form (5,-1) with t(0) = 1: y^2 - x*z = A082111(t(1)-5). (End)

Colin Barker, Table of n, a(n) for n = 0..1000
A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (5,-1).

Cf. A099868, A003501, A004254.

I:=[1,9]; [n le 2 select I[n] else 5*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jul 30 2015

a[0] = 1; a[1] = 9; a[n_] := a[n] = 5 a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 21}] (* Robert G. Wilson v, Dec 14 2004 *)
LinearRecurrence[{5, -1}, {1, 9}, 30] (* or *) CoefficientList[Series[(1 + 4 x)/(1 - 5 x + x^2), {x, 0, 30}], x] (* Harvey P. Dale, Jun 26 2011 *)

Vec((1+4*x) / (1-5*x+x^2) + O(x^30)) \\ Colin Barker, Mar 31 2017

|2*a(n) + A099868(n) - A003501(n+1)| = 20*A004254(n).
From R. J. Mathar, Sep 11 2008: (Start)
G.f.: (1+4*x) / (1-5*x+x^2).
a(n) = A004254(n+1) + 4*A004254(n).
(End)
a(n) = 2^(-1-n)*((5-sqrt(21))^n*(-13+sqrt(21)) + (5+sqrt(21))^n*(13+sqrt(21))) / sqrt(21). - Colin Barker, Mar 31 2017

A123971 Triangle T(n,k), read by rows, defined by T(n,k)=3*T(n-1,k)-T(n-1,k-1)-T(n-2,k), T(0,0)=1, T(1,0)=2, T(1,1)=-1, T(n,k)=0 if k<0 or if k>n.

Original entry on oeis.org

1, 2, -1, 5, -5, 1, 13, -19, 8, -1, 34, -65, 42, -11, 1, 89, -210, 183, -74, 14, -1, 233, -654, 717, -394, 115, -17, 1, 610, -1985, 2622, -1825, 725, -165, 20, -1, 1597, -5911, 9134, -7703, 3885, -1203, 224, -23, 1, 4181, -17345, 30691, -30418, 18633, -7329

Offset: 0

Gary W. Adamson and Roger L. Bagula, Oct 30 2006

This entry is the result of merging two sequences, this one and a later submission by Philippe Deléham, Nov 29 2013 (with edits from Ralf Stephan, Dec 12 2013). Most of the present version is the work of Philippe Deléham, the only things remaining from the original entry are the sequence data and the Mathematica program. - N. J. A. Sloane, May 31 2014
Subtriangle of the triangle given by (0, 2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Apart from signs, equals A126124.
Row sums = 1.
Sum_{k=0..n} T(n,k)*(-x)^k = A001519(n+1), A079935(n+1), A004253(n+1), A001653(n+1), A049685(n), A070997(n), A070998(n), A072256(n+1), A078922(n+1), A077417(n), A085260(n+1), A001570(n+1) for x=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.

			Triangle begins:
  1
  2, -1
  5, -5, 1
  13, -19, 8, -1
  34, -65, 42, -11, 1
  89, -210, 183, -74, 14, -1
  233, -654, 717, -394, 115, -17, 1
Triangle (0, 2, 1/2, 1/2, 0, 0, ...) DELTA (1, -2, 0, 0, ...) begins:
  1
  0, 1
  0, 2, -1
  0, 5, -5, 1
  0, 13, -19, 8, -1
  0, 34, -65, 42, -11, 1
  0, 89, -210, 183, -74, 14, -1
  0, 233, -654, 717, -394, 115, -17, 1

Cf. A094954, A098495, A123971, A126124, A152063, A001519, A079935, A004253, A001653, A049685, A070997, A070998, A072256, A078922, A077417, A085260, A001570, A001870, A126124.

Mathematica ( general k th center) Clear[M, T, d, a, x, k] k = 3 T[n_, m_, d_] := If[ n == m && n < d && m < d, k, If[n == m - 1 || n == m + 1, -1, If[n == m == d, k - 1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[{M[1]}, Table[CoefficientList[ Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a] Table[NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x], {d, 1, 10}] Table[x /. NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x][[d]], {d, 1, 10}]

T(n,k)=polcoeff(polcoeff(Ser((1-x)/(1+(y-3)*x+x^2)),n,x),n-k,y) \\ Ralf Stephan, Dec 12 2013

@CachedFunction
def A123971(n,k): # With T(0,0) = 1!
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*A123971(n-1,k) if n==1 else 3*A123971(n-1,k)
    return A123971(n-1,k-1) - A123971(n-2,k) - h
for n in (0..9): [A123971(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

T(n,k) = (-1)^n*A126124(n+1,k+1).
T(n,k) = (-1)^k*Sum_{m=k..n} binomial(m,k)*binomial(m+n,2*m). - Wadim Zudilin, Jan 11 2012
G.f.: (1-x)/(1+(y-3)*x+x^2).
T(n,0) = A001519(n+1) = A000045(2*n+1).
T(n+1,1) = -A001870(n).

Edited by N. J. A. Sloane, May 31 2014

A340476 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4sin(aPi/(2n+1))^2 + 4cos(bPi/(2k+1))^2).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 19, 11, 1, 1, 91, 176, 29, 1, 1, 436, 2911, 1471, 76, 1, 1, 2089, 48301, 79808, 11989, 199, 1, 1, 10009, 801701, 4375897, 2091817, 97021, 521, 1, 1, 47956, 13307111, 240378643, 372713728, 53924597, 783511, 1364, 1

Offset: 0

Seiichi Manyama, Jan 09 2021

			Square array begins:
  1,  1,     1,       1,         1, ...
  1,  4,    19,      91,       436, ...
  1, 11,   176,    2911,     48301, ...
  1, 29,  1471,   79808,   4375897, ...
  1, 76, 11989, 2091817, 372713728, ...

Wikipedia, Chebyshev polynomials
Wikipedia, Resultant

Column k=0..1 give A000012, A002878.
Row n=0..7 give A000012, A004253(n+1), A003729, A028478, A028480, A028482, A028484, A028486.
Main diagonal gives A127606.
Cf. A187617, A340475.

default(realprecision, 120);
{T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*sin(a*Pi/(2*n+1))^2+4*cos(b*Pi/(2*k+1))^2)))}

{T(n, k) = sqrtint(4^k*polresultant(polchebyshev(2*n+1, 1, I*x/2), polchebyshev(2*k, 2, x/2)))}

T(n,k) = 2^k * sqrt(Resultant(T_{2*n+1}(i*x/2), U_{2*k}(x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

A101463 Expansion of g.f. (x^3+x^2+2x+1)/(x^4+5x^2+1).

Original entry on oeis.org

1, 2, -4, -9, 19, 43, -91, -206, 436, 987, -2089, -4729, 10009, 22658, -47956, -108561, 229771, 520147, -1100899, -2492174, 5274724, 11940723, -25272721, -57211441, 121088881, 274116482, -580171684, -1313370969, 2779769539, 6292738363, -13318676011

Offset: 0

Creighton Dement, Jan 20 2005

A floretion-generated sequence relating to Pythagoras' theorem generalized.

Floretion Algebra Multiplication Program. FAMP code: em[J* ]sigcycseq[ + .75'i + .5'k + .25i' + .5j' + .5k' - .25'ii' + .25'jj' - .25'kk' - .75'jk' + .5'ki' - .25'kj' + .25e]

F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971.

James A. Sellers, Domino Tilings and Products of Fibonacci and Pell Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2.
Index entries for linear recurrences with constant coefficients, signature (0,-5,0,-1)

Elements of even index in the sequence gives A004253. Elements of odd index in the sequence gives A002310.

Cf. A004253, A002310.

CoefficientList[Series[(x^3+x^2+2x+1)/(x^4+5x^2+1),{x,0,30}],x] (* or *) LinearRecurrence[{0,-5,0,-1},{1,2,-4,-9},31] (* Harvey P. Dale, Apr 15 2012 *)

Let b(1)=1, b(2)=2, b(3)=4 and b(n)=(b(n-1)*b(n-2)+(3+(-1)^n)/2)/b(n-3) then b(n)=abs(a(n)) - Benoit Cloitre, Mar 03 2007
a(n) = -5*a(n-2)-a(n-4), n>3. [Harvey P. Dale, Apr 15 2012]
G.f.: ( 1+2*x+x^2+x^3 ) / ( 1+5*x^2+x^4 ). - R. J. Mathar, Jun 18 2014
a(n) = -3a(n-1)+2a(n-2) if n even. a(n) = (5*a(n-1)+a(n-2))/2 if n odd. - R. J. Mathar, Jun 18 2014

A261522 Positive integers k such that x^2 - 23xy + y^2 + k = 0 has integer solutions.

Original entry on oeis.org

21, 41, 59, 75, 84, 89, 101, 111, 119, 125, 129, 131, 164, 189, 201, 236, 251, 269, 300, 311, 329, 336, 356, 369, 381, 404, 419, 425, 444, 461, 476, 479, 489, 500, 509, 516, 521, 524, 525, 531, 579, 581, 629, 656, 675, 719, 731, 756, 761, 801, 804, 831, 839

Offset: 1

Colin Barker, Aug 23 2015

nonn

			41 is in the sequence because x^2 - 23xy + y^2 + 41 = 0 has integer solutions; for example, (x, y) = (2, 45).

Cf. A004253, A237254.

Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236330, A236331, A238240, A238245.

cp's OEIS Frontend

A237255 Values of x in the solutions to x^2 - 5xy + y^2 + 17 = 0, where 0 < x < y.

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A334673 a(n) = 23*a(n-1) - a(n-2) + 1 for n > 1, a(0)=0, a(1)=1.

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A085348 Ratio-determined insertion sequence I(0.264) (see the link below).

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A085349 Ratio-determined insertion sequence I(0.26688) (see the link below).

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A099867 a(n) = 5*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=9.

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A123971 Triangle T(n,k), read by rows, defined by T(n,k)=3*T(n-1,k)-T(n-1,k-1)-T(n-2,k), T(0,0)=1, T(1,0)=2, T(1,1)=-1, T(n,k)=0 if k<0 or if k>n.

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A340476 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4*sin(a*Pi/(2*n+1))^2 + 4*cos(b*Pi/(2*k+1))^2).

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A101463 Expansion of g.f. (x^3+x^2+2*x+1)/(x^4+5*x^2+1).

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A340476 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4sin(aPi/(2n+1))^2 + 4cos(bPi/(2k+1))^2).

A101463 Expansion of g.f. (x^3+x^2+2x+1)/(x^4+5x^2+1).