A094954 -id:A094954 - cp's OEIS frontend

A078988 Chebyshev sequence with Diophantine property.

1, 65, 4289, 283009, 18674305, 1232221121, 81307919681, 5365090477825, 354014663616769, 23359602708228929, 1541379764079492545, 101707704826538279041, 6711167138787446924161, 442835323455144958715585, 29220420180900779828304449, 1928104896615996323709378049

Offset: 0

Wolfdieter Lang, Jan 10 2003

Bisection (even part) of A041025.
(4*A078989(n))^2 - 17*a(n)^2 = -1 (Pell -1 equation, see A077232-3).
Starting with a(1), hypotenuses of primitive Pythagorean triples in A195619 and A195620. - Clark Kimberling, Sep 22 2011

			(x,y) = (4,1), (268,65), (17684,4289), ... give the positive integer solutions to x^2 - 17*y^2 =-1.

Colin Barker, Table of n, a(n) for n = 0..549
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.
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 (66,-1).

Row 66 of array A094954.
Cf. A097316 for S(n, 66).
Row 4 of array A188647.

a:=[1,65];; for n in [3..20] do a[n]:=66*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1, 65]; [n le 2 select I[n] else 66*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

CoefficientList[Series[(1-x)/(1-66x+x^2), {x,0,20}], x] (* Michael De Vlieger, Apr 15 2019 *)
LinearRecurrence[{66,-1}, {1,65}, 21] (* G. C. Greubel, Aug 01 2019 *)

Vec((1-x)/(1-66*x+x^2) + O(x^20)) \\ Colin Barker, Jun 15 2015

((1-x)/(1-66*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

G.f.: (1-x)/(1-66*x+x^2).
a(n) = T(2*n+1, sqrt(17))/sqrt(17) = ((-1)^n)*S(2*n, 8*i) = S(n, 66) - S(n-1, 66) with i^2=-1 and T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310.
a(n) = A041025(2*n).
a(n) = 66*a(n-1) - a(n-2) for n>1 ; a(0)=1, a(1)=65. - Philippe Deléham, Nov 18 2008

A098493 Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.

Original entry on oeis.org

1, 0, -1, -1, -1, 1, -1, 1, 2, -1, 0, 3, 0, -3, 1, 1, 2, -5, -2, 4, -1, 1, -2, -7, 6, 5, -5, 1, 0, -5, 0, 15, -5, -9, 6, -1, -1, -3, 12, 9, -25, 1, 14, -7, 1, -1, 3, 15, -18, -29, 35, 7, -20, 8, -1, 0, 7, 0, -42, 14, 63, -42, -20, 27, -9, 1, 1, 4, -22, -24, 85, 14, -112, 42

Offset: 0

Ralf Stephan, Sep 12 2004

Also, coefficients of polynomials that have values in A098495 and A094954.

			Triangle begins:
   1;
   0, -1;
  -1, -1, 1;
  -1,  1, 2, -1;
   0,  3, 0, -3, 1;
  ...

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.

Columns include A010892, -A076118. Diagonals include A033999, A038608, (-1)^n*A000096. Row sums are in A057077.

Cf. A098494 (diagonal polynomials), A085478, A244419.

A098493 := proc (n, k)
add((-1)^(k+binomial(n-j+1,2))*binomial(floor((1/2)*n+(1/2)*j),j)* binomial(j,k), j = k..n);
end proc:
seq(seq(A098493(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021

T(n,k)=if(k>n||k<0||n<0,0,if(k>=n-1,(-1)^n*if(k==n,1,-k),if(n==1,0,if(k==0,T(n-1,0)-T(n-2,0),T(n-1,k)-T(n-2,k)-T(n-1,k-1)))))

T(n, k) = A098489[n(n+1)/2, k] - A098490[n(n+1)/2, k].
Recurrence: T(n, k) = T(n-1, k)-T(n-1, k-1)-T(n-2, k); T(n, k)=0 for n<0, k>n, k<0; T(n, n)=(-1)^n; T(n, n-1)=(-1)^n*(1-n).
G.f.: (1-x)/(1+(y-1)*x+x^2). [Vladeta Jovovic, Dec 14 2009]
From Peter Bala, Jul 13 2021: (Start)
Riordan array ( (1 - x)/(1 - x + x^2), -x/(1 - x + x^2) ).
T(n,k) = (-1)^k * the (n,k)-th entry of Q^(-1)*P = Sum_{j = k..n} (-1)^(k+binomial(n-j+1,2))*binomial(floor((1/2)*n+(1/2)*j),j)* binomial(j,k), where P denotes Pascal's triangle A007318 and Q denotes triangle A061554 (formed from P by sorting the rows into descending order). (End)
From Peter Bala, Jun 26 2025: (Start)
n-th row polynomial R(n, x) = Sum_{k = 0..n} (-1)^k * binomial(n+k, 2*k) * (1 + x)^k.
R(n, 2*x + 1) = (-1)^n * Dir(n, x), where Dir(n,x) denotes the n-th row polynomial of the triangle A244419.
R(n, -1 - x) = b(n, x), where b(n, x) denotes the n-th row polynomial of the triangle A085478. (End)

A162997 Array A(n,k) read by antidiagonals downward (n >= 0, k >= 1): the bottom-right element of the 2 X 2 matrix [1,n; 1,n+1] raised to k-th power.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 13, 11, 4, 1, 34, 41, 19, 5, 1, 89, 153, 92, 29, 6, 1, 233, 571, 436, 169, 41, 7, 1, 610, 2131, 2089, 985, 281, 55, 8, 1, 1597, 7953, 10009, 5741, 1926, 433, 71, 9

Offset: 0

Gary W. Adamson, Jul 19 2009

With k=0 column added, becomes A094954.
Also, A(n,k) is the top-left element of the same 2 X 2 matrix raised to (k+1)-th power.
Also, A(n,k) is the denominator of the rational number which has continued fraction expansion consisting of k repeats of [1, n]. Example: the row (3, 11, 41, ...) is extracted from denominators of the continued fractions [0; 1, 2], [0; 1, 2, 1, 2], ... = 2/3, 8/11, ...
Also, A(n,k)=Product_{i=1..k} (n+2+2*cos(2*Pi*i/(2*k+1))). This is somehow connected to the diagonal product formulas for (2*k+1)-gons found by Steinbach.
Row sums of the triangle = A162998: (1, 3, 9, 29, 100, 369, 1458, ...).

			The array begins:
1,...1,...1,....1,....1,.....1,.....1,...
2,...5,..13,...34,...89,...233....610,...
3,..11,..41,..153,..571,..2131,..........
4,..19,..91,..436,.2089,.................
5,..29,.169,..985,.......................
6,..41,.281,.............................
7,..55,..................................
8,.......................................
...

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Cf. A028387, A094954, A162998, A152063.

Spelling corrected by Jason G. Wurtzel, Aug 22 2010

Edited by Andrey Zabolotskiy, Sep 18 2017

A098495 Array T(r,c) read by antidiagonals: values of triangle A098493 interpreted as polynomials, r >= 0.

Original entry on oeis.org

1, 1, 0, 1, -1, -1, 1, -2, -1, -1, 1, -3, 1, 1, 0, 1, -4, 5, 1, 1, 1, 1, -5, 11, -7, -2, -1, 1, 1, -6, 19, -29, 9, 1, -1, 0, 1, -7, 29, -71, 76, -11, 1, 1, -1, 1, -8, 41, -139, 265, -199, 13, -2, 1, -1, 1, -9, 55, -239, 666, -989, 521, -15, 1, -1, 0, 1, -10, 71, -377, 1393, -3191, 3691, -1364, 17, 1, -1, 1, 1, -11, 89, -559

Offset: 0

Ralf Stephan, Sep 12 2004

			Array begins
  1,  0, -1,  -1,   0,    1,    1,   0, -1, ...
  1, -1, -1,   1,   1,   -1,   -1,   1,  1, ...
  1, -2,  1,   1,  -2,    1,    1,  -2,  1, ...
  1, -3,  5,  -7,   9,  -11,   13, -15, ...
  1, -4, 11, -29,  76, -199,  521, ...
  1, -5, 19, -71, 265, -989, 3691, ...
  ...

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.

See A094954 (with negative k) for negative r and more formulas and programs.

Rows include (-1)^c times A005408, A002878, A001834, A030221, A002315. Columns include A028387. Antidiagonal sums are in A098496.

T[r_, 1] := 1; T[r_, 2] := -r - 1; T[r_, c_] := -r*T[r, c - 1] - T[r, c - 2]; Flatten[ Table[ T[n - i, i], {n, 0, 11}, {i, n + 1}]] (* Robert G. Wilson v, May 10 2005 *)

{ t(r,c)=if(c>r||c<0||r<0,0,if(c>=r-1,(-1)^r*if(c==r,1,-c),if(r==1,0,if(c==0,t(r-1,0)-t(r-2,0),t(r-1,c)-t(r-2,c)-t(r-1,c-1))))) }
T(r,c)=sum(i=0,c,t(c,i)*r^i);
matrix(5,5,n,k,T(n-1,k-1))

Recurrence: T(r, 1) = 1, T(r, 2) = -r-1, T(r, c) = -rT(r, c-1) - T(r, c-2). (Corrected Oct 19 2004)

More terms from Robert G. Wilson v, May 10 2005

A073134 Table by antidiagonals of T(n,k)=n*T(n,k-1)-T(n,k-2) starting with T(n,1)=1.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, -1, 3, 3, 1, -1, 4, 8, 4, 1, 0, 5, 21, 15, 5, 1, 1, 6, 55, 56, 24, 6, 1, 1, 7, 144, 209, 115, 35, 7, 1, 0, 8, 377, 780, 551, 204, 48, 8, 1, -1, 9, 987, 2911, 2640, 1189, 329, 63, 9, 1, -1, 10, 2584, 10864, 12649, 6930, 2255, 496, 80, 10, 1, 0, 11, 6765, 40545, 60605, 40391, 15456, 3905, 711, 99, 11, 1, 1, 12

Offset: 1

Henry Bottomley, Jul 16 2002

			Rows start:
  1, 1,  0, -1,  -1,   0,    1, ...;
  1, 2,  3,  4,   5,   6,    7, ...;
  1, 3,  8, 21,  55, 144,  377, ...;
  1, 4, 15, 56, 209, 780, 2911, ...;
  ...

Shmuel T. Klein, Combinatorial Representation of Generalized Fibonacci Numbers, Fib. Quarterly 29 (2) (1991) 124-131, variable U_n^m. [From _R. J. Mathar_, Feb 19 2010]

Rows include A010892, A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190. Columns include (with some gaps) A000012, A000027, A005563, A057722.

Cf. A094954.

T(n,k) = sum(j=0,k-1,A049310(k-1,j)*n^j) \\ Jason Yuen, Aug 20 2024

T(n, k) = A073133(n, k)-2*A073135(n, k-2).

T(n, k) = Sum_{j=0..k-1} A049310(k-1, j)*n^j.

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

A121872 Triangle T(n, k) = (kChebyshevU(n, (k+2)/2) + 2ChebyshevT(n+1, (k+2)/2))/2.

Original entry on oeis.org

5, 13, 41, 34, 153, 436, 89, 571, 2089, 5741, 233, 2131, 10009, 33461, 90481, 610, 7953, 47956, 195025, 620166, 1663585, 1597, 29681, 229771, 1136689, 4250681, 13097377, 34988311, 4181, 110771, 1100899, 6625109, 29134601, 103115431, 310957991, 828931049

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 09 2006

			Triangle begins as:
    5;
   13,   41;
   34,  153,   436;
   89,  571,  2089,  5741;
  233, 2131, 10009, 33461, 90481;

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A094954, A162997.

Cf. A053117, A053120.

T:= func< n,k | ( k*Sinh((n+1)*Argcosh((k+2)/2))/Sinh(Argcosh((k+2)/2)) + 2*Cosh((n+1)*Argcosh((k+2)/2)) )/2 >;
[Round(T(n,k)): k in [1..n], n in [1..10]]; // G. C. Greubel, Oct 08 2019

seq(seq(simplify(( k*ChebyshevU(n, (k+2)/2) + 2*ChebyshevT(n+1, (k+2)/2) )/2), k=1..n), n=1..10); # G. C. Greubel, Oct 09 2019

f[k_]:= Sqrt[k*(k+4)]; T[n_, m_]:= T[n, m]= FullSimplify[((m+f[m])*(m+2 - f[m])^(n+2) - (m-f[m])*(m+2 + f[m])^(n+2))/(2^(n+3)*f[m])]; Table[T[n, m], {n,10}, {m,n}]//Flatten (* modified by G. C. Greubel, Oct 08 2019 *)
T[n_, k_]:= T[n, k]= (k*ChebyshevU[n, (k+2)/2] + 2*ChebyshevT[n+1, (k+ 2)/2])/2; Table[T[n, k], {n,10}, {k,n}]/Flatten (* G. C. Greubel, Oct 08 2019 *)

T(n,k)= ( k*sin((n+1)*acos((k+2)/2))/sin(acos((k+2)/2)) + 2*cos((n+1)*acos((k+2)/2)) )/2;
for(n=1,10, for(k=1,n, print1(round(T(n,k)), ", "))) \\ G. C. Greubel, Oct 08 2019

[[( k*chebyshev_U(n,(k+2)/2) + 2*chebyshev_T(n+1, (k+2)/2) )/2 for k in (1..n)] for n in (1..10)] # G. C. Greubel, Oct 08 2019

T(n, m) = ((m+f(m))*(m+2 - f(m))^(n+2) - (m-f(m))*(m+2 + f(m))^(n+2))/( 2^(n+3)*f(m)), where f(m) = sqrt(m*(m+4)).
From G. C. Greubel, Oct 08 2019: (Start)
T(n, k) = (k*ChebyshevU(n, (k+2)/2) + 2*ChebyshevT(n+1, (k+2)/2))/2;
T(n, k) = (k*Fibonacci(n+2, m+2, -1) + Lucas(n+2, m+2, -1))/2, where Fibonacci(n, x, y) and Lucas(n, x, y) are the bi-variate Fibonacci an Lucas polynomials, respectively. (End)

Major edit and new name, G. C. Greubel, Oct 08 2019

A218220 Array a(n,m) read by antidiagonals where a(0,m)=a(1,m)=1 and a(n,m) = m*a(n-1,m)-a(n-2,m) for n>=2.

Original entry on oeis.org

1, 1, 1, 1, 1, -1, 1, 1, 0, -1, 1, 1, 1, -1, 1, 1, 1, 2, 1, -1, 1, 1, 1, 3, 5, 1, 0, -1, 1, 1, 4, 11, 13, 1, 1, -1, 1, 1, 5, 19, 41, 34, 1, 1, 1, 1, 1, 6, 29, 91, 153, 89, 1, 0, 1, 1, 1, 7, 41, 169, 436, 571, 233, 1, -1, -1, 1, 1, 8, 55, 281, 985, 2089, 2131, 610, 1, -1, -1, 1, 1, 9, 71, 433, 1926, 5741, 10009, 7953, 1597, 1, 0, 1, 1, 1, 10, 89, 631, 3409, 13201, 33461, 47956, 29681

Offset: 0

Nico Brown, Oct 23 2012

Variant of A094954 and A162997.
a(n,0) alternates 1,1,-1,-1,1,1,...
a(n,1) alternates 1,1,0,-1,-1,0,...
a(n,2)=1
a(n,3) is alternating Fibonacci numbers.

			   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
  -1,   0,   1,   2,   3,   4,   5,   6,   7,   8,   9,...
  -1,  -1,   1,   5,  11,  19,  29,  41,  55,  71,  89,...
   1,  -1,   1,  13,  41,  91, 169, 281, 433, 631, 881,...
   1,   0,   1,  34, 153, 436, 985,1926,3409,5608,8721,...
  -1,   1,   1,  89, 571,2089,5741,13201,26839,49841,86329,...
  -1,   1,   1, 233,2131,10009,33461,90481,211303,442961,854569,...

Cf. A218219.

cp's OEIS Frontend

A078988 Chebyshev sequence with Diophantine property.

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A098493 Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.

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A162997 Array A(n,k) read by antidiagonals downward (n >= 0, k >= 1): the bottom-right element of the 2 X 2 matrix [1,n; 1,n+1] raised to k-th power.

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A098495 Array T(r,c) read by antidiagonals: values of triangle A098493 interpreted as polynomials, r >= 0.

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A073134 Table by antidiagonals of T(n,k)=n*T(n,k-1)-T(n,k-2) starting with T(n,1)=1.

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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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A121872 Triangle T(n, k) = (k*ChebyshevU(n, (k+2)/2) + 2*ChebyshevT(n+1, (k+2)/2))/2.

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A121872 Triangle T(n, k) = (kChebyshevU(n, (k+2)/2) + 2ChebyshevT(n+1, (k+2)/2))/2.