A001541 -id:A001541 - cp's OEIS frontend

A101124 Number triangle associated to Chebyshev polynomials of first kind.

1, 0, 1, -1, 1, 1, 0, 1, 2, 1, 1, 1, 7, 3, 1, 0, 1, 26, 17, 4, 1, -1, 1, 97, 99, 31, 5, 1, 0, 1, 362, 577, 244, 49, 6, 1, 1, 1, 1351, 3363, 1921, 485, 71, 7, 1, 0, 1, 5042, 19601, 15124, 4801, 846, 97, 8, 1, -1, 1, 18817, 114243, 119071, 47525, 10081, 1351, 127, 9, 1, 0, 1, 70226, 665857, 937444, 470449, 120126, 18817, 2024, 161

Offset: 0

Paul Barry, Dec 02 2004

			As a number triangle, rows begin:
  {1},
  {0,1},
  {-1,1,1},
  {0,1,2,1},
  ...
As a square array, rows begin
   1, 1,  1,   1,    1, ...
   0, 1,  2,   3,    4, ...
  -1, 1,  7,  17,   31, ...
   0, 1, 26,  99,  244, ...
   1, 1, 97, 577, 1921, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns include A001075, A001541, A001091, A001079, A023038, A011943.
Row sums are A101125.
Diagonal sums are A101126.
Main diagonal gives A115066.
Mirror of A322836.
Cf. A053120.

T[n_, k_] := SeriesCoefficient[x^k (1 - k x)/(1 - 2 k x + x^2), {x, 0, n}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 12 2017 *)

Number triangle S(n, k)=T(n-k, k), k

Columns have g.f. x^k(1-kx)/(1-2kx+x^2).

Also, square array if(n=0, 1, T(n, k)) read by antidiagonals.

A232948 T(n,k)=Number of nXk 0..3 arrays with no element x(i,j) adjacent to value 3-x(i,j) horizontally, vertically or antidiagonally, and top left element zero.

Original entry on oeis.org

1, 3, 3, 9, 17, 9, 27, 99, 99, 27, 81, 577, 1163, 577, 81, 243, 3363, 13707, 13707, 3363, 243, 729, 19601, 161621, 328989, 161621, 19601, 729, 2187, 114243, 1905737, 7915555, 7915555, 1905737, 114243, 2187, 6561, 665857, 22471303, 190528543

Offset: 1

R. H. Hardin, Dec 02 2013

Table starts
.....1........3...........9.............27.................81
.....3.......17..........99............577...............3363
.....9.......99........1163..........13707.............161621
....27......577.......13707.........328989............7915555
....81.....3363......161621........7915555..........389997789
...243....19601.....1905737......190528543........19245039957
...729...114243....22471303.....4586348177.......950054488141
..2187...665857...264968059...110402289283.....46905123394619
..6561..3880899..3124343613..2657600587981...2315805078150557
.19683.22619537.36840376337.63973693477099.114336803085406731

			Some solutions for n=3 k=4
..0..2..3..2....0..2..3..2....0..2..3..3....0..1..1..0....0..2..3..3
..2..3..3..3....0..2..2..0....2..3..3..1....1..1..1..1....0..2..2..2
..2..2..3..2....2..0..2..2....2..2..3..1....3..3..1..0....0..2..0..2

R. H. Hardin, Table of n, a(n) for n = 1..264

Column 1 is A000244(n-1)

Column 2 is A001541

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 6*a(n-1) -a(n-2)
k=3: a(n) = 13*a(n-1) -15*a(n-2) +9*a(n-3) -2*a(n-4) for n>5, A232943.
k=4: [order 10] for n>11, A232944.
k=5: [order 21] for n>23, A232945.
k=6: [order 57] for n>59

A322790 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..k} binomial(2k,2j)(n+1)^(k-j)n^j.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 17, 5, 1, 1, 99, 49, 7, 1, 1, 577, 485, 97, 9, 1, 1, 3363, 4801, 1351, 161, 11, 1, 1, 19601, 47525, 18817, 2889, 241, 13, 1, 1, 114243, 470449, 262087, 51841, 5291, 337, 15, 1, 1, 665857, 4656965, 3650401, 930249, 116161, 8749, 449, 17, 1

Offset: 0

Seiichi Manyama, Dec 26 2018

			Square array begins:
   1,  1,   1,    1,      1,       1,         1, ...
   1,  3,  17,   99,    577,    3363,     19601, ...
   1,  5,  49,  485,   4801,   47525,    470449, ...
   1,  7,  97, 1351,  18817,  262087,   3650401, ...
   1,  9, 161, 2889,  51841,  930249,  16692641, ...
   1, 11, 241, 5291, 116161, 2550251,  55989361, ...
   1, 13, 337, 8749, 227137, 5896813, 153090001, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Columns 0-3 give A000012, A005408, A069129(n+1), A322830.
Rows 0-9 give A000012, A001541, A001079, A011943(n+1), A023039, A077422, A097308, A068203, A056771, A078986.
Main diagonal gives A173174.
A(n-1,n) gives A173148(n).
Cf. A322699, A322747.

A[0, k_] := 1; A[n_, k_] := Sum[Binomial[2 k, 2 j]*(n + 1)^(k - j)*n^j, {j, 0, k}]; Table[A[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 26 2018 *)

a(n) = 2 * A322699(n) + 1.
A(n,k) + sqrt(A(n,k)^2 - 1) = (sqrt(n+1) + sqrt(n))^(2*k).
A(n,k) - sqrt(A(n,k)^2 - 1) = (sqrt(n+1) - sqrt(n))^(2*k).
A(n,0) = 1, A(n,1) = 2*n+1 and A(n,k) = (4*n+2) * A(n,k-1) - A(n,k-2) for k > 1.
A(n,k) = T_{k}(2*n+1) where T_{k}(x) is a Chebyshev polynomial of the first kind.
T_1(x) = x. So A(n,1) = 2*n+1.

A099142 a(n) = 6^n * T(n, 4/3) where T is the Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 8, 92, 1184, 15632, 207488, 2757056, 36643328, 487039232, 6473467904, 86042074112, 1143628341248, 15200538791936, 202038000386048, 2685388609667072, 35692849740775424, 474411605904392192

Offset: 0

Paul Barry, Sep 30 2004

In general, r^n * T(n,(r+2)/r) has g.f. (1-(r+2)*x)/(1-2*(r+2)*x + r^2*x^2), e.g.f. exp((r+2)*x)*cosh(2*sqrt(r+1)*x), a(n) = Sum_{k=0..n} (r+1)^k*binomial(2n,2k) and a(n) = (1+sqrt(r+1))^(2*n)/2 + (1-sqrt(r+1))^(2*n)/2.

Seiichi Manyama, Table of n, a(n) for n = 0..500
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (16,-36).

Column k=7 of A333988.

Cf. A001541, A081294, A083884, A090965, A099140, A099141.

LinearRecurrence[{16,-36},{1,8},20] (* Harvey P. Dale, Mar 09 2018 *)

a(n) = 6^n*polchebyshev(n, 1, 4/3); \\ Michel Marcus, Sep 08 2019

G.f.: (1-8*x)/(1-16*x+36*x^2);
E.g.f.: exp(8*x)*cosh(2*sqrt(7)*x).
a(n) = 6^n * T(n, 8/6) where T is the Chebyshev polynomial of the first kind.
a(n) = Sum_{k=0..n} 7^k * binomial(2n, 2k).
a(n) = (1+sqrt(7))^(2*n)/2 + (1-sqrt(7))^(2*n)/2.
a(0)=1, a(1)=8, a(n) = 16*a(n-1) - 36*a(n-2) for n > 1. - Philippe Deléham, Sep 08 2009

A106525 Values of x in x^2 - 49 = 2*y^2.

Original entry on oeis.org

9, 11, 21, 43, 57, 119, 249, 331, 693, 1451, 1929, 4039, 8457, 11243, 23541, 49291, 65529, 137207, 287289, 381931, 799701, 1674443, 2226057, 4660999, 9759369, 12974411, 27166293, 56881771, 75620409, 158336759, 331531257, 440748043

Offset: 1

Andras Erszegi (erszegi.andras(AT)chello.hu), May 07 2005

From Wolfdieter Lang, Sep 27 2016: (Start)
These are the x members of all positive solutions (x(n), y(n)), proper and improper, of the Pell equation x^2 - 2*y^2 = 7^2.
The y(n) members are given in 2*A276600(n+2).
This sequence is composed of the two y members of the two proper classes of solutions of the Pell equation x^2 - 2*y^2 = 7^2 and of 7 times the proper solutions X of the Pell equation X^2 - 2*Y^2 = +1. See A275793, A275795 and 7*A001541. See A275793 for further information, and the Nagell reference. (End)
The sums of the consecutive integers in the following sequences will be squares: for n, i >= 1, if mod(i,3)=0 then 7*n+1, 7*n+2, ..., a(i)*n + (A001541(i/3)-1)/2; otherwise, if mod(i,3)=1 or 2 then 7*n+4, 7*n+5, ..., a(i)*n + (a(i)-1)/2.

			In the following, aa(n) denotes A001541(n):
a(9)=693; as mod(9,3)=0, a(9)=aa(3)*7=99*7=693, also 693^2-49=2*490^2
a(10)=1451; as mod(10,3)=1, a(10)=(aa(5)+aa(2)+aa(3)-aa(4))/2 =(3363+17+99-577)/2=1451, also 1451^2-49=2*1026^2.
The solutions (proper and every third pair improper) of x^2 - 2*y^2 = +49 begin [9, 4], [11, 6], [21, 14], [43, 30], [57, 40], [119, 84], [249, 176], [331, 234], [693, 490], [1451, 1026], [1929, 1364], [4039, 2856], [8457, 5980], [11243, 7950], [23541, 16646], ... - _Wolfdieter Lang_, Sep 27 2016

Colin Barker, Table of n, a(n) for n = 1..1001 (offset adapted by _Georg Fischer_, Jan 31 2019).
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. 14*A001109, 7*A001541, A106525, A106526.

Cf. A275793, A275794, A275795, A275796, 2*A276600.

I:=[9,11,21,43,57,119]; [n le 6 select I[n] else 6*Self(n-3)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Oct 26 2018

LinearRecurrence[{0,0,6,0,0,-1}, {9,11,21,43,57,119}, 50] (* Vincenzo Librandi, Oct 26 2018 *)

Vec((9+11*x+21*x^2-11*x^3-9*x^4-7*x^5)/(1-6*x^3+x^6) + O(x^50)) \\ Colin Barker, Sep 28 2016

def A106525_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(9+11*x+21*x^2-11*x^3-9*x^4-7*x^5)/(1-6*x^3+x^6) ).list()
a=A106525_list(41); a[1:] # G. C. Greubel, Sep 15 2021

a(3*k)   = A001541(k)*7 for k >= 2.
a(3*k+1) = (A001541(k+2) + A001541(k-1) + A001541(k) - A001541(k+1))/2;
a(3*k+2) = (A001541(k+2) + A001541(k-1) - A001541(k) + A001541(k+1))/2.
a(3*n) = A275793(n), a(3*n+1) = A275795(n), a(3*n+2) = 7*A001541(n+1), n >= 0. - Wolfdieter Lang, Sep 27 2016
From Colin Barker, Mar 29 2012: (Start)
a(n) = 6*a(n-3) - a(n-6).
G.f.: x*(9 + 11*x + 21*x^2 - 11*x^3 - 9*x^4 - 7*x^5)/(1 - 6*x^3 + x^6). (End)

Entry revised by N. J. A. Sloane, Oct 26 2018 at the suggestion of Georg Fischer.

A143609 Numerators of the upper principal and intermediate convergents to 2^(1/2).

Original entry on oeis.org

2, 3, 10, 17, 58, 99, 338, 577, 1970, 3363, 11482, 19601, 66922, 114243, 390050, 665857, 2273378, 3880899, 13250218, 22619537, 77227930, 131836323, 450117362, 768398401, 2623476242, 4478554083, 15290740090, 26102926097, 89120964298, 152139002499

Offset: 1

Clark Kimberling, Aug 27 2008

The upper principal and intermediate convergents to 2^(1/2), beginning with
2/1, 3/2, 10/7, 17/12, 58/41, form a strictly decreasing sequence;
essentially, numerators=A143609 and denominators=A084068.

			2*x + 3*x^2 + 10*x^3 + 17*x^4 + 58*x^5 + 99*x^6 + 338*x^7 + 577*x^8 + ...

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

Colin Barker, Table of n, a(n) for n = 1..1000
Creighton Kenneth Dement, Comments on A143608 and A143609
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A001541, A001653, A010914, A084068.

Rest@ CoefficientList[Series[x (2 + 3 x - 2 x^2 - x^3)/(1 - 6 x^2 + x^4), {x, 0, 30}], x] (* Michael De Vlieger, Mar 27 2016 *)

{a(n) = if( n<1, 0, polcoeff( x * (2 + 3*x - 2*x^2 - x^3) / (1 - 6*x^2 + x^4) + x * O(x^n), n))} /* Michael Somos, Sep 03 2013 */

x='x+O('x^99); Vec(x*(2+3*x-2*x^2-x^3)/(1-6*x^2+x^4)) \\ Altug Alkan, Mar 27 2016

a(n) = 6 * a(n-2) - a(n-4). a(2*n) = A001541(n) if n>0. a(2*n + 1) = 2 * A001653(n + 1).- Michael Somos, Sep 03 2013
G.f.: x * (2 + 3*x - 2*x^2 - x^3) / (1 - 6*x^2 + x^4). - Michael Somos, Sep 03 2013
a(n) = (2+sqrt(2)+(-1)^n*(-2+sqrt(2)))*((-1+sqrt(2))^n+(1+sqrt(2))^n)/(4*sqrt(2)). - Colin Barker, Mar 27 2016

A228564 Largest odd divisor of n^2 + 1.

Original entry on oeis.org

1, 1, 5, 5, 17, 13, 37, 25, 65, 41, 101, 61, 145, 85, 197, 113, 257, 145, 325, 181, 401, 221, 485, 265, 577, 313, 677, 365, 785, 421, 901, 481, 1025, 545, 1157, 613, 1297, 685, 1445, 761, 1601, 841, 1765, 925, 1937, 1013, 2117, 1105, 2305, 1201, 2501, 1301, 2705

Offset: 0

Jeremy Gardiner, Aug 25 2013

From Lamine Ngom, Jan 04 2023: (Start)
For n>2, a(n) = hypotenuse c of the primitive Pythagorean triple (a, b, c) such that n*a = b + c.
Terms that appear twice (1, 5, 145, 4901, ...) are the positive terms of A076218. Equivalently, the products of two consecutive terms of A001653, or one more than the squares of A001542.
These duplicated terms appear at indices i and j (i>j) such that (i^2-1)/2 = j^2 (A001541). In addition, they are hypotenuse in two primitive Pythagorean triples: (i, j^2, a(i)) and (2*j, j^2-1, a(i)). (End)

			A002522(3) = 3^2+1 = 10 => a(3) = 10/2 = 5.

Jeremy Gardiner, Table of n, a(n) for n = 0..1000
Wikipedia, Quasi-polynomial.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000265, A002522, A164900, A191871.

for n = 0 to 45 : t=n^2+1
x: if not t mod 2 then t=t/2 : goto x
print str$(t);", "; : next n
print
end

List([0..60],n->NumeratorRat((n^2+1)/(n+1))); # Muniru A Asiru, Feb 20 2019

[(n^2+1)*(3+(-1)^n)/4: n in [0..60]]; // Bruno Berselli, Aug 26 2013

[Denominator(2*n^2/(n^2+1)): n in [0..60]]; // Vincenzo Librandi, Aug 19 2014

lod:= t -> t/2^padic:-ordp(t,2):
seq(lod(n^2+1),n=0..60); # Robert Israel, Aug 19 2014

Table[(n^2 + 1) (3 + (-1)^n)/4, {n, 0, 60}] (* Bruno Berselli, Aug 26 2013 *)

a(n)=if(n<2,n>0,m=n\4;[4*a(2*m)-3,2*a(2*m)+4*m-1,4*a(2*m)+16*m+1,2*a(2*m)+12*m+3][(n%4)+1]) \\ Ralf Stephan, Aug 26 2013

a(n)=(n^2+1)/2^valuation(n^2+1,2) \\ Ralf Stephan, Aug 26 2013

[(n^2+1)*(3+(-1)^n)/4 for n in (0..60)] # G. C. Greubel, Feb 20 2019

a(n) = A000265(A002522(n)).
From Ralf Stephan, Aug 26 2013: (Start)
a(4n) = 4*a(2n) - 3.
a(4n+1) = 2*a(2n) + 4*n - 1.
a(4n+2) = 4*a(2n) + 16*n + 1.
a(4n+3) = 2*a(2n) + 12*n + 3. (End)
From Bruno Berselli, Aug 26 2013: (Start)
G.f.: (1 + x + 2*x^2 + 2*x^3 + 5*x^4 + x^5) / (1-x^2)^3.
a(n) = 3*a(n-2) -3*a(n-4) +a(n-6).
a(n) = (n^2+1)*(3+(-1)^n)/4. (End)
From Peter Bala, Feb 14 2019: (Start)
a(n) = numerator((n^2 + 1)/(n + 1)).
a(n) is a quasi-polynomial in n: a(2*n) = 4*n^2 + 1; a(2*n + 1) = 2*n^2 + 2*n + 1. (End)
From Amiram Eldar, Aug 09 2022: (Start)
a(n) = numerator((n^2+1)/2).
Sum_{n>=0} 1/a(n) = (1 + coth(Pi/2)*Pi/2 + tanh(Pi/2)*Pi)/2. (End)
E.g.f.: ((2 + x + 2*x^2)*cosh(x) + (1 + x)^2*sinh(x))/2. - Stefano Spezia, Aug 04 2025

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cp's OEIS Frontend

A084130 a(n) = 8*a(n-1) - 8*a(n-2), a(0)=1, a(1)=4.

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A099140 a(n) = 4^n * T(n,3/2) where T is the Chebyshev polynomial of the first kind.

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A099141 a(n) = 5^n * T(n,7/5) where T is the Chebyshev polynomial of the first kind.

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A101124 Number triangle associated to Chebyshev polynomials of first kind.

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A232948 T(n,k)=Number of nXk 0..3 arrays with no element x(i,j) adjacent to value 3-x(i,j) horizontally, vertically or antidiagonally, and top left element zero.

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A322790 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..k} binomial(2*k,2*j)*(n+1)^(k-j)*n^j.

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A099142 a(n) = 6^n * T(n, 4/3) where T is the Chebyshev polynomial of the first kind.

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A106525 Values of x in x^2 - 49 = 2*y^2.

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A084130 a(n) = 8a(n-1) - 8a(n-2), a(0)=1, a(1)=4.

A322790 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..k} binomial(2k,2j)(n+1)^(k-j)n^j.