A114525 -id:A114525 - cp's OEIS frontend

A322091 Digits of one of the two 13-adic integers sqrt(-3).

6, 3, 12, 6, 10, 7, 4, 4, 9, 8, 9, 2, 8, 5, 12, 3, 5, 4, 0, 6, 5, 1, 2, 6, 5, 9, 4, 9, 1, 1, 4, 6, 11, 3, 1, 12, 5, 2, 2, 6, 3, 11, 11, 8, 4, 5, 10, 10, 7, 9, 5, 7, 7, 7, 8, 0, 1, 0, 7, 7, 0, 9, 12, 10, 8, 1, 6, 1, 2, 10, 2, 9, 7, 2, 1, 10, 11, 4, 3, 5, 6

Offset: 0

Jianing Song, Nov 26 2018

This square root of -3 in the 13-adic field ends with digit 6. The other, A322092, ends with digit 7.

			...36225C13B64119495621560453C582989447A6C36.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Peter Bala, Using Lucas polynomials to find the p-adic square roots of -1, -2 and -3, Dec 2022.
Wikipedia, p-adic number

Cf. A114525, A286838, A286839, A322087, A322088, A322089, A322092.

a(n) = truncate(sqrt(-3+O(13^(n+1))))\13^n

a(n) = (A322089(n+1) - A322089(n))/13^n.
For n > 0, a(n) = 12 - A322092(n).
Equals A286838*A322088 = A286839*A322087.
This 13-adic integer is the 13-adic limit as n -> oo of the integer sequence {L(13^n,6)}, where L(n,x) denotes the n-th Lucas polynomial, the n-th row polynomial of A114525. - Peter Bala, Dec 05 2022

A322092 Digits of one of the two 13-adic integers sqrt(-3).

Original entry on oeis.org

7, 9, 0, 6, 2, 5, 8, 8, 3, 4, 3, 10, 4, 7, 0, 9, 7, 8, 12, 6, 7, 11, 10, 6, 7, 3, 8, 3, 11, 11, 8, 6, 1, 9, 11, 0, 7, 10, 10, 6, 9, 1, 1, 4, 8, 7, 2, 2, 5, 3, 7, 5, 5, 5, 4, 12, 11, 12, 5, 5, 12, 3, 0, 2, 4, 11, 6, 11, 10, 2, 10, 3, 5, 10, 11, 2, 1, 8, 9, 7, 6

Offset: 0

Jianing Song, Nov 26 2018

This square root of -3 in the 13-adic field ends with digit 7. The other, A322091, ends with digit 6.

			...96AA70B9168BB38376AB76C879074A34388526097.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Peter Bala, Using Lucas polynomials to find the p-adic square roots of -1, -2 and -3, Dec 2022.
Wikipedia, p-adic number

Cf. A114525, A286838, A286839, A322087, A322088, A322090, A322091.

a(n) = truncate(-sqrt(-3+O(13^(n+1))))\13^n

a(n) = (A322090(n+1) - A322090(n))/13^n.
For n > 0, a(n) = 12 - A322091(n).
Equals A286838*A322087 = A286839*A322088.
This 13-adic integer is the 13-adic limit as n -> oo of the integer sequence {L(13^n,7)}, where L(n,x) denotes the n-th Lucas polynomial, the n-th row polynomial of A114525. - Peter Bala, Dec 05 2022

A117938 Triangle, columns generated from Lucas Polynomials.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 4, 11, 14, 7, 1, 5, 18, 36, 34, 11, 1, 6, 27, 76, 119, 82, 18, 1, 7, 38, 140, 322, 393, 198, 29, 1, 8, 51, 234, 727, 1364, 1298, 478, 47, 1, 9, 66, 364, 1442, 3775, 5778, 4287, 1154, 76, 1, 10, 83, 536, 2599, 8886, 19602, 24476, 14159, 2786, 123

Offset: 1

Gary W. Adamson, Apr 03 2006

Companion triangle using Fibonacci polynomial generators = A073133. Inverse binomial transforms of the columns defines rows of A117937 (with some adjustments of offset).

A309220 is another version of the same triangle (except it omits the last diagonal), and perhaps has a clearer definition. - N. J. A. Sloane, Aug 13 2019

			First few rows of the triangle are:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  6,   4;
  1, 4, 11,  14,   7;
  1, 5, 18,  36,  34,  11;
  1, 6, 27,  76, 119,  82,  18;
  1, 7, 38, 140, 322, 393, 198, 29;
  ...
For example, T(7,4) = 76 = f(4), x^3 + 3*x = 64 + 12 = 76.

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A104509, A114525, A117936, A117937, A118980, A118981, A309220.

Cf. A000204 (diagonal), A059100 (column 3), A061989 (column 4).

Lucas := proc(n,x) # see A114525
    option remember;
    if  n=0 then
        2;
    elif n =1 then
        x ;
    else
        x*procname(n-1,x)+procname(n-2,x) ;
    end if;
    expand(%) ;
end proc:
A117938 := proc(n::integer,k::integer)
    if k = 1 then
        1;
    else
        subs(x=n-k+1,Lucas(k-1,x)) ;
    end if;
end proc:
seq(seq(A117938(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Aug 16 2019

T[n_, k_]:= LucasL[k-1, n-k+1] - Boole[k==1];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 28 2021 *)

def A117938(n,k): return 1 if (k==1) else round(2^(1-k)*( (n-k+1 + sqrt((n-k)*(n-k+2) + 5))^(k-1) + (n-k+1 - sqrt((n-k)*(n-k+2) + 5))^(k-1) ))
flatten([[A117938(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 28 2021

Columns are f(x), x = 1, 2, 3, ..., of the Lucas Polynomials: (1, defined different from A034807 and A114525); (x); (x^2 + 2); (x^3 + 3*x); (x^4 + 4*x^2 + 2); (x^5 + 5*x^3 + 5*x); (x^6 + 6*x^4 + 9*x^2 + 2); (x^7 + 7*x^5 + 14*x^3 + 7*x); ...

Terms a(51) and a(52) corrected by G. C. Greubel, Oct 28 2021

A108045 Triangle read by rows: lower triangular matrix obtained by inverting the lower triangular matrix in A108044.

Original entry on oeis.org

1, 0, 1, -2, 0, 1, 0, -3, 0, 1, 2, 0, -4, 0, 1, 0, 5, 0, -5, 0, 1, -2, 0, 9, 0, -6, 0, 1, 0, -7, 0, 14, 0, -7, 0, 1, 2, 0, -16, 0, 20, 0, -8, 0, 1, 0, 9, 0, -30, 0, 27, 0, -9, 0, 1, -2, 0, 25, 0, -50, 0, 35, 0, -10, 0, 1, 0, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 2, 0, -36, 0, 105, 0, -112, 0, 54, 0, -12, 0, 1

Offset: 0

N. J. A. Sloane, Jun 02 2005

Signed version of A114525. - Eric W. Weisstein, Apr 07 2017
For n >= 3, also the coefficients of the matching polynomial for the n-cycle graph C_n. - Eric W. Weisstein, Apr 07 2017
This triangle describes the Chebyshev transform of A100047 and following. Chebyshev transform of sequence b is c(n) = Sum_{k=1..n} a(n,k)*b(k). - Christian G. Bower, Jun 12 2005

			Triangle begins:
   1;
   0,  1;
  -2,  0,  1;
   0, -3,  0,  1;
   2,  0, -4,  0,  1;

G. C. Greubel, Rows n=0..100 of triangle, flattened
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Paul Barry and Aoife Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 5.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Matching Polynomial

Cf. A114525 (unsigned version).
Cf. A007318, A108044.
Cf. A127672.

f:=(1-x^2)/(1+x^2): g:=x/(1+x^2): G:=simplify(f/(1-t*g)): Gser:=simplify(series(G,x=0,14)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser,x^n) od: for n from 0 to 12 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form # Emeric Deutsch, Jun 06 2005

a[n_, k_] := SeriesCoefficient[(1-x^2)/(1+x^2-t*x), {x, 0, n}, {t, 0, k}]; a[0, 0] = 1; Table[a[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 08 2014, after Emeric Deutsch *)
Flatten[{{1}, CoefficientList[Table[I^n LucasL[n, -I x], {n, 10}], x]}] (* Eric W. Weisstein, Apr 07 2017 *)
Flatten[{{1}, CoefficientList[LinearRecurrence[{x, -1}, {x, -2 + x^2}, 10], x]}] (* Eric W. Weisstein, Apr 07 2017 *)

Riordan array ( (1-x^2)/(1+x^2), x/(1+x^2)).
G.f.: (1-x^2)/(1+x^2-tx). - Emeric Deutsch, Jun 06 2005
From Peter Bala, Jun 29 2015: (Start)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x/(1 + x^2) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = ( 1 + sqrt(1 - 4*x^2) )/2.
In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

More terms from Emeric Deutsch and Christian G. Bower, Jun 06 2005

A322089 One of the two successive approximations up to 13^n for 13-adic integer sqrt(-3). Here the 6 (mod 13) case (except for n = 0).

Original entry on oeis.org

0, 6, 45, 2073, 15255, 300865, 2899916, 22207152, 273201220, 7614777709, 92450772693, 1333177199334, 4917497987408, 191302178967256, 1705677711928521, 48954194340319989, 202511873382592260, 3529594919298491465, 38131258596823843197, 38131258596823843197, 8809653000849500507259

Offset: 0

Jianing Song, Nov 26 2018

For n > 0, a(n) is the unique solution to x^2 == -3 (mod 13^n) in the range [0, 13^n - 1] and congruent to 6 modulo 13.

A322090 is the approximation (congruent to 7 mod 13) of another square root of -3 over the 13-adic field.

			6^2 = 36 = 3*13 - 3.
45^2 = 2025 = 12*13^2 - 3.
2073^2 = 4297329 = 1956*13^3 - 3.

Wikipedia, p-adic number

Cf. A114525, A286840, A286841, A322085, A322086, A322090, A322091.

PARI
```
a(n) = truncate(sqrt(-3+O(13^n)))
```

For n > 0, a(n) = 13^n - A322090(n).
a(n) = Sum_{i=0..n-1} A322091(i)*13^i.
a(n) = A286840(n)*A322086(n) mod 13^n = A286841(n)*A322085(n) mod 13^n.
a(n) == L(13^n,6) (mod 13^n) == (3 + sqrt(10))^(13^n) + (3 - sqrt(10))^(13^n) (mod 13^n), where L(n,x) denotes the n-th Lucas polynomial, the n-th row polynomial of A114525. - Peter Bala, Dec 05 2022

A322090 One of the two successive approximations up to 13^n for 13-adic integer sqrt(3). Here the 7 (mod 13) case (except for n = 0).

Original entry on oeis.org

0, 7, 124, 124, 13306, 70428, 1926893, 40541365, 542529501, 2989721664, 45407719156, 458983194703, 18380587135073, 111572927624997, 2231698673770768, 2231698673770768, 462904735800587581, 5120821000082846468, 74324148355133549932, 1423789031778622267480, 10195310774031298931542

Offset: 0

Jianing Song, Nov 26 2018

For n > 0, a(n) is the unique solution to x^2 == -3 (mod 13^n) in the range [0, 13^n - 1] and congruent to 7 modulo 13.

A322089 is the approximation (congruent to 6 mod 13) of another square root of -3 over the 13-adic field.

			7^2 = 49 = 4*13 - 3.
124^2 = 15376 = 91*13^2 - 3 = 7*13^3 - 3.
13306^2 = 177049636 = 6199*13^4 - 3.

Wikipedia, p-adic number

Cf. A114525, A286840, A286841, A322085, A322086, A322089, A322092.

PARI
```
a(n) = truncate(-sqrt(-3+O(13^n)))
```

For n > 0, a(n) = 13^n - A322089(n).
a(n) = Sum_{i=0..n-1} A322092(i)*13^i.
a(n) = A286840(n)*A322085(n) mod 13^n = A286841(n)*A322086(n) mod 13^n.
a(n) == L(13^n,7) (mod 13^n) == ((7 + sqrt(53))/2)^(13^n) + ((7 - sqrt(53))/2)^(13^n) (mod 13^n), where L(n,x) denotes the n-th Lucas polynomial, the n-th row polynomial of A114525. - Peter Bala, Dec 05 2022

A352362 Array read by ascending antidiagonals. T(n, k) = L(k, n) where L are the Lucas polynomials.

Original entry on oeis.org

2, 2, 0, 2, 1, 2, 2, 2, 3, 0, 2, 3, 6, 4, 2, 2, 4, 11, 14, 7, 0, 2, 5, 18, 36, 34, 11, 2, 2, 6, 27, 76, 119, 82, 18, 0, 2, 7, 38, 140, 322, 393, 198, 29, 2, 2, 8, 51, 234, 727, 1364, 1298, 478, 47, 0, 2, 9, 66, 364, 1442, 3775, 5778, 4287, 1154, 76, 2

Offset: 0

Peter Luschny, Mar 18 2022

			Array starts:
n\k 0, 1,  2,   3,    4,     5,      6,       7,        8, ...
--------------------------------------------------------------
[0] 2, 0,  2,   0,    2,     0,      2,       0,        2, ... A010673
[1] 2, 1,  3,   4,    7,    11,     18,      29,       47, ... A000032
[2] 2, 2,  6,  14,   34,    82,    198,     478,     1154, ... A002203
[3] 2, 3, 11,  36,  119,   393,   1298,    4287,    14159, ... A006497
[4] 2, 4, 18,  76,  322,  1364,   5778,   24476,   103682, ... A014448
[5] 2, 5, 27, 140,  727,  3775,  19602,  101785,   528527, ... A087130
[6] 2, 6, 38, 234, 1442,  8886,  54758,  337434,  2079362, ... A085447
[7] 2, 7, 51, 364, 2599, 18557, 132498,  946043,  6754799, ... A086902
[8] 2, 8, 66, 536, 4354, 35368, 287298, 2333752, 18957314, ... A086594
[9] 2, 9, 83, 756, 6887, 62739, 571538, 5206581, 47430767, ... A087798
A007395|A059100|
    A001477 A079908

Eric Weisstein's World of Mathematics, Lucas Polynomial

Rows: A010673, A000032, A002203, A006497, A014448, A087130, A085447, A086902, A086594, A087798.
Columns: A007395, A001477, A059100, A079908.
Cf. A320570 (main diagonal), A114525, A309220 (variant), A117938 (variant), A352361 (Fibonacci polynomials), A350470 (Jacobsthal polynomials).

T := (n, k) -> (n/2 + sqrt((n/2)^2 + 1))^k + (n/2 - sqrt((n/2)^2 + 1))^k:
seq(seq(simplify(T(n - k, k)), k = 0..n), n = 0..10);

Table[LucasL[k, n], {n, 0, 9}, {k, 0, 9}] // TableForm
(* or *)
T[ 0, k_] := 2 Mod[k+1, 2]; T[n_, 0] := 2;
T[n_, k_] := n^k Hypergeometric2F1[1/2 - k/2, -k/2, 1 - k, -4/n^2];
Table[T[n, k], {n, 0, 9}, {k, 0, 8}] // TableForm

T(n, k) = ([0, 1; 1, k]^n*[2; k])[1, 1] ;
export(T)
for(k = 0, 9, print(parvector(10, n, T(n - 1, k))))

T(n, k) = Sum_{j=0..floor(k/2)} binomial(k-j, j)*(k/(k-j))*n^(k-2*j) for k >= 1.
T(n, k) = (n/2 + sqrt((n/2)^2 + 1))^k + (n/2 - sqrt((n/2)^2 + 1))^k.
T(n, k) = [x^k] ((2 - n*x)/(1 - n*x - x^2)).
T(n, k) = n^k*hypergeom([1/2 - k/2, -k/2], [1 - k], -4/n^2) for n,k >= 1.

A277282 Max coefficient in n-th Lucas polynomial.

Original entry on oeis.org

2, 1, 2, 3, 4, 5, 9, 14, 20, 30, 50, 77, 112, 182, 294, 450, 672, 1122, 1782, 2717, 4290, 7007, 11011, 16744, 27456, 44200, 68952, 107406, 176358, 281010, 436050, 700910, 1136960, 1797818, 2778446, 4576264, 7354710, 11560835, 18349630, 29910465, 47720400

Offset: 0

Vladimir Reshetnikov, Oct 08 2016

nonn

			For n = 6, L_6(x) = x^6 + 6*x^4 + 9*x^2 + 2, so a(6) = 9.

Vaclav Kotesovec, Table of n, a(n) for n = 0..4000
Vaclav Kotesovec, Plot of a(n) / (goldenratio^n / sqrt(n)) for n = 1..10000
Eric Weisstein's World of Mathematics, Lucas Polynomial.

Cf. A114525, A073028.

Table[Max@CoefficientList[LucasL[n, x], x], {n, 0, 40}]

A123185 Triangular array from a zero coefficient sum of recursive polynomials: p(k, x) = x*p(k - 1, x) + p(k - 2, x).

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -1, 3, -3, 1, 2, -4, 4, -3, 1, -1, 5, -7, 5, -3, 1, 2, -5, 9, -10, 6, -3, 1, -1, 7, -12, 14, -13, 7, -3, 1, 2, -6, 16, -22, 20, -16, 8, -3, 1, -1, 9, -18, 30, -35, 27, -19, 9, -3, 1, 2, -7, 25, -40, 50, -51, 35, -22, 10, -3, 1

Offset: 0

Roger L. Bagula, Oct 02 2006

Except for p(0,x)=1 all sum to zero: Table[Sum[CoefficientList[p[n, x], x][[m]], {m, 1, n + 1}], {n, 0, 12}] {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

			1
-1, 1
2, -3, 1
-1, 3, -3, 1
2, -4, 4, -3, 1
-1, 5, -7, 5, -3, 1
2, -5, 9, -10, 6, -3, 1

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

Cf. A114525, A168561.

S:= series((1 - t + (1-2*x)*t^2)/(1 - t*x - t^2), t, 21):
R:= [seq(coeff(S,t,n),n=0..19)]:
seq(seq(coeff(R[n],x,j),j=0..n-1),n=1..20); # Robert Israel, Jul 12 2016

p[0, x] = 1; p[1, x] = x - 1; p[2, x] = x^2 - 3x + 2; p[k_, x_] := p[k, x] = x*p[k - 1, x] + p[k - 2, x] ; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]

p(k, x) = x*p(k - 1, x) + p(k - 2, x)  for k >= 3 with p(0,x) = 1, p(1,x) = x-1 and p(2,x) = x^2-3x+2. (Corrected by Robert Israel, Jul 12 2016)
G.f.: g(t,x) = (1 - t + (1-2x) t^2)/(1 - t x - t^2). - Robert Israel, Jul 12 2016
For n >= 1, T(n,k) = A114525(n,k) + 2*A114525(n-1,k) - 5*A168561(n-1,k), where we take A114525(n-1,n) = A168561(n-1,n) = 0. - Robert Israel, Jul 13 2016

Edited by Robert Israel, Jul 13 2016

cp's OEIS Frontend

A322091 Digits of one of the two 13-adic integers sqrt(-3).

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A322092 Digits of one of the two 13-adic integers sqrt(-3).

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A117938 Triangle, columns generated from Lucas Polynomials.

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A108045 Triangle read by rows: lower triangular matrix obtained by inverting the lower triangular matrix in A108044.

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A322089 One of the two successive approximations up to 13^n for 13-adic integer sqrt(-3). Here the 6 (mod 13) case (except for n = 0).

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A322090 One of the two successive approximations up to 13^n for 13-adic integer sqrt(3). Here the 7 (mod 13) case (except for n = 0).

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A352362 Array read by ascending antidiagonals. T(n, k) = L(k, n) where L are the Lucas polynomials.

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