A144130 -id:A144130 - cp's OEIS frontend

A188644 Array of (k^n + k^(-n))/2 where k = (sqrt(x^2-1) + x)^2 for integers x >= 1.

1, 1, 1, 1, 7, 1, 1, 97, 17, 1, 1, 1351, 577, 31, 1, 1, 18817, 19601, 1921, 49, 1, 1, 262087, 665857, 119071, 4801, 71, 1, 1, 3650401, 22619537, 7380481, 470449, 10081, 97, 1, 1, 50843527, 768398401, 457470751, 46099201, 1431431, 18817, 127, 1

Offset: 0

Charles L. Hohn, Apr 06 2011

Conjecture: Given the function f(x,y) = (sqrt(x^2+y) + x)^2 and constant k=f(x,y), then for all integers x >= 1 and y=[+-]1, k may be irrational, but (k^n + k^(-n))/2 always produces integer sequences; y=-1 results shown here; y=1 results are A188645.

Also square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{2*k}(x), evaluated at x=n. - Seiichi Manyama, Dec 30 2018

			Row 2 gives {( (2+sqrt(3))^(2*n) + (2-sqrt(3))^(2*n) )/2}.
Square array begins:
     | 0    1       2          3             4
-----+---------------------------------------------
   1 | 1,   1,      1,         1,            1, ...
   2 | 1,   7,     97,      1351,        18817, ...
   3 | 1,  17,    577,     19601,       665857, ...
   4 | 1,  31,   1921,    119071,      7380481, ...
   5 | 1,  49,   4801,    470449,     46099201, ...
   6 | 1,  71,  10081,   1431431,    203253121, ...
   7 | 1,  97,  18817,   3650401,    708158977, ...
   8 | 1, 127,  32257,   8193151,   2081028097, ...
   9 | 1, 161,  51841,  16692641,   5374978561, ...
  10 | 1, 199,  79201,  31521799,  12545596801, ...
  11 | 1, 241, 116161,  55989361,  26986755841, ...
  12 | 1, 287, 164737,  94558751,  54276558337, ...
  13 | 1, 337, 227137, 153090001, 103182433537, ...
  14 | 1, 391, 305761, 239104711, 186979578241, ...
  15 | 1, 449, 403201, 362074049, 325142092801, ...
  ...

Row 2 is A011943, row 3 is A056771, row 8 is A175633, (row 2)*2 is A067902, (row 9)*2 is A089775.
Column 0-5 give A000012, A056220, A144130, A243132, A243134, A243136.
(column 1)*2 is A060626.
Cf. A188645 (f(x, y) as above with y=1).
Diagonals give A173129, A322899.
Cf. A188646, A322836.

max = 9; y = -1; t = Table[k = ((x^2 + y)^(1/2) + x)^2; ((k^n) + (k^(-n)))/2 // FullSimplify, {n, 0, max - 1}, {x, 1, max}]; Table[ t[[n - k + 1, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 17 2013 *)

A(n,k) = (A188646(n,k-1) + A188646(n,k))/2.

A(n,k) = Sum_{j=0..k} binomial(2*k,2*j)*(n^2-1)^(k-j)*n^(2*j). - Seiichi Manyama, Jan 01 2019

Edited by Seiichi Manyama, Dec 30 2018

More terms from Seiichi Manyama, Jan 01 2019

A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Dec 28 2018

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 1,    2,     3,      4,      5,       6, ...
  -1, 1,    7,    17,     31,     49,      71, ...
   0, 1,   26,    99,    244,    485,     846, ...
   1, 1,   97,   577,   1921,   4801,   10081, ...
   0, 1,  362,  3363,  15124,  47525,  120126, ...
  -1, 1, 1351, 19601, 119071, 470449, 1431431, ...

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

Mirror of A101124.
Columns 0-20 give A056594, A000012, A001075, A001541, A001091, A001079, A023038, A011943(n+1), A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203, A322888, A056771, A322889, A078986, A322890.
Rows 0-10 give A000012, A001477, A056220, A144129, A144130, A243131, A243132, A243133, A243134, A243135, A243136.
Main diagonal gives A115066.
Cf. A323182 (Chebyshev polynomial of the second kind).

Table[ChebyshevT[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 28 2018 *)

T(n,k) = polchebyshev(n,1,k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Dec 28 2018

T(n, k) = round(cos(n*acos(k)));\\ Seiichi Manyama, Mar 05 2021

T(n, k) = if(n==0, 1, n*sum(j=0, n, (2*k-2)^j*binomial(n+j, 2*j)/(n+j))); \\ Seiichi Manyama, Mar 05 2021

A(0,k) = 1, A(1,k) = k and A(n,k) = 2 * k * A(n-1,k) - A(n-2,k) for n > 1.
A(n,k) = cos(n*arccos(k)). - Seiichi Manyama, Mar 05 2021
A(n,k) = n * Sum_{j=0..n} (2*k-2)^j * binomial(n+j,2*j)/(n+j) for n > 0. - Seiichi Manyama, Mar 05 2021

A144131 Primes of the form T_4(n), where T_4(x) = 8x^4 - 8x^2 + 1 is the fourth Chebyshev polynomial (of the first kind).

Original entry on oeis.org

97, 577, 4801, 32257, 79201, 305761, 665857, 1039681, 7380481, 8380417, 10681441, 11995201, 18495361, 42448897, 49980001, 54100801, 63101377, 68001121, 96911041, 110736961, 227143297, 266851201, 296071777, 398240641, 479694337

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

nonn

Sequence is infinite under Bunyakovsky's conjecture. - Charles R Greathouse IV, May 29 2013

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A144130.

T4:= unapply(orthopoly[T](4,x),x):
select(isprime, map(T4, [$0..300])); # Robert Israel, Apr 27 2020

lst={};Do[p=ChebyshevT[4,n];If[PrimeQ[p],AppendTo[lst,p]],{n,9^3}];lst

select(isprime,vector(100,n,polchebyshev(4,1,n))) \\ Charles R Greathouse IV, May 29 2013

A239609 Sin(arcsin(n)- 3 arccos(n))^2.

Original entry on oeis.org

1, 1, 9409, 332929, 3690241, 23049601, 101626561, 354079489, 1040514049, 2687489281, 6272798401, 13493377921, 27138279169, 51591216769, 93489789121, 162571046401, 272735662081, 443365544449, 700932305089, 1080936581761, 1630220793601, 2409700487041

Offset: 0

José María Grau Ribas, Mar 22 2014

The terms are integers.

This is assuming the "standard branch" of arcsin and arccos, so that sin(arccos(n)) = cos(arcsin(n)) = sqrt(1-n^2). - Robert Israel, May 25 2014

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A239607, A239608, A239610.

G[n_, a_, b_] := G[n, a, b] = Sin[a ArcSin[ n] + b ArcCos[n]]^2 // ComplexExpand // FullSimplify; Table[G[n, 1, -3], {n, 0, 43}]

vector(100, n, round(sin(asin(n-1) - 3*acos(n-1))^2)) \\ Colin Barker, May 24 2014

G.f.: -(x^8 +9400*x^7 +248284*x^6 +1032520*x^5 +1032646*x^4 +248200*x^3 +9436*x^2 -8*x +1) / (x -1)^9. - Colin Barker, May 24 2014

a(n) = A144130(n)^2. - Robert Israel, May 25 2014

A243132 32n^6 - 48n^4 + 18*n^2 - 1.

Original entry on oeis.org

-1, 1, 1351, 19601, 119071, 470449, 1431431, 3650401, 8193151, 16692641, 31521799, 55989361, 94558751, 153090001, 239104711, 362074049, 533729791, 768398401, 1083358151, 1499219281, 2040327199, 2735188721, 3616921351, 4723725601, 6099380351, 7793761249

Offset: 0

Vincenzo Librandi, May 31 2014

Chebyshev polynomial of the first kind T(6,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A056220, A144129, A144130.

Magma
```
[32*n^6-48*n^4+18*n^2-1: n in [0..40]];
```

Table[ChebyshevT[6, n], {n, 0, 40}] (* or *) Table[32 n^6 - 48 n^4 + 18 n^2 - 1, {n, 0, 20}]

G.f.: (-1 + 8*x + 1323*x^2 + 10200*x^3 + 10165*x^4 + 1344*x^5 + x^6)/(1 - x)^7.

a(n) = (2*n^2 - 1)*(16*n^4 - 16*n^2 + 1).

A243134 128n^8 - 256n^6 + 160n^4 - 32n^2 + 1.

Original entry on oeis.org

1, 1, 18817, 665857, 7380481, 46099201, 203253121, 708158977, 2081028097, 5374978561, 12545596801, 26986755841, 54276558337, 103182433537, 186979578241, 325142092801, 545471324161, 886731088897, 1401864610177, 2161873163521, 3260441587201, 4819400974081

Offset: 0

Vincenzo Librandi, May 31 2014

Chebyshev polynomial of the first kind T(8,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A056220, A144129, A144130.

[128*n^8-256*n^6+160*n^4-32*n^2+1: n in [0..40]];

Table[ChebyshevT[8, n], {n, 0, 40}] (* or *) Table[128 n^8 - 256 n^6 + 160 n^4 - 32 n^2 + 1, {n, 0, 20}]
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,1,18817,665857,7380481,46099201,203253121,708158977,2081028097},30] (* Harvey P. Dale, Nov 01 2015 *)

G.f.: (1 - 8*x + 18844*x^2 + 496456*x^3 + 2065222*x^4 + 2065096*x^5 + 496540*x^6 + 18808*x^7 + x^8)/(1 - x)^9.

a(0)=1, a(1)=1, a(2)=18817, a(3)=665857, a(4)=7380481, a(5)=46099201, a(6)=203253121, a(7)=708158977, a(8)=2081028097, a(n)=9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)- 9*a(n-8)+ a(n-9). - Harvey P. Dale, Nov 01 2015

A243136 a(n) = 512n^10 - 1280n^8 + 1120n^6 - 400n^4 + 50*n^2 - 1.

Original entry on oeis.org

-1, 1, 262087, 22619537, 457470751, 4517251249, 28860511751, 137379191137, 528572943487, 1730726404001, 4993116004999, 13007560326001, 31154649926687, 69544807113937, 146217791079751, 291977237261249, 557471159562751, 1023286908188737, 1814011722210887

Offset: 0

Vincenzo Librandi, May 31 2014

Chebyshev polynomial of the first kind T(10,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A056220, A144129, A144130.

[512*n^10-1280*n^8+1120*n^6-400*n^4+50*n^2-1: n in [0..30]];

a:= n-> simplify(ChebyshevT(10, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, May 31 2014

Table[ChebyshevT[10, n], {n, 0, 30}] (* or *) Table[512 n^10 - 1280 n^8 + 1120 n^6 - 400 n^4 + 50 n^2 - 1, {n, 0, 30}]

G.f.: (-1 + 12*x + 262021*x^2 + 19736800*x^3 + 223070134*x^4 + 685903960*x^5 + 685903498*x^6 + 223070464*x^7 + 19736635*x^8 + 262076*x^9 + x^10)/(1 - x)^11.
a(n) = (2*n^2 - 1)*(256*n^8 - 512*n^6 + 304*n^4 - 48*n^2 + 1).
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). - Wesley Ivan Hurt, May 04 2024

A243131 a(n) = 16n^5 - 20n^3 + 5*n.

Original entry on oeis.org

0, 1, 362, 3363, 15124, 47525, 120126, 262087, 514088, 930249, 1580050, 2550251, 3946812, 5896813, 8550374, 12082575, 16695376, 22619537, 30116538, 39480499, 51040100, 65160501, 82245262, 102738263, 127125624, 155937625, 189750626, 229188987

Offset: 0

Vincenzo Librandi, May 31 2014

Chebyshev polynomial of the first kind T(5,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A056220, A144129, A144130.

Magma
```
[16*n^5-20*n^3+5*n: n in [0..40]];
```

a:= n-> simplify(ChebyshevT(5, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, May 31 2014

Table[ChebyshevT[5, n], {n, 0, 40}] (* or *) Table[16*n^5 - 20*n^3 + 5*n, {n, 0, 20}]
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,362,3363,15124,47525},30] (* Harvey P. Dale, Aug 03 2023 *)

apply(x->polchebyshev(5,1,x), vector(30,i,i-1)) \\ Hugo Pfoertner, Oct 18 2022

a(n) = n*(16*n^4-20*n^2+5) = (-1/4)*n *(-8*n^2+5+sqrt(5))*(8*n^2-5+sqrt(5)).

G.f.: x*(1 + 356*x + 1206*x^2 + 356*x^3 + x^4)/(1 - x)^6.

A243133 64n^7 - 112n^5 + 56n^3 - 7n.

Original entry on oeis.org

0, 1, 5042, 114243, 937444, 4656965, 17057046, 50843527, 130576328, 299537289, 628855930, 1229215691, 2265463212, 3974443213, 6686381534, 10850138895, 17062657936, 26102926097, 38970776898, 56930852179, 81562047860, 114812765781, 159062294182

Offset: 0

Vincenzo Librandi, May 31 2014

Chebyshev polynomial of the first kind T(7,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A056220, A144129, A144130.

[64*n^7-112*n^5+56*n^3-7*n: n in [0..40]];

Table[ChebyshevT[7, n], {n, 0, 40}] (* or *)  Table[64 n^7 - 112 n^5 + 56 n^3 - 7 n, {n, 0, 40}]
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,1,5042,114243,937444,4656965,17057046,50843527},40] (* Harvey P. Dale, Mar 27 2015 *)

G.f.: (x + 5034*x^2 + 73935*x^3 + 164620*x^4 + 73935*x^5 + 5034*x^6 + x^7)/(1 - x)^8.
a(n) = n*(64*n^6 - 112*n^4 + 56*n^2 - 7).
a(0)=0, a(1)=1, a(2)=5042, a(3)=114243, a(4)=937444, a(5)=4656965, a(6)=17057046, a(7)=50843527, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)- 70*a(n-4)+ 56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8). - Harvey P. Dale, Mar 27 2015

A243135 256n^9 - 576n^7 + 432n^5 - 120n^3 + 9*n.

Original entry on oeis.org

0, 1, 70226, 3880899, 58106404, 456335045, 2421980406, 9863382151, 33165873224, 96450076809, 250283080090, 592479412811, 1300371936876, 2678768828749, 5228741809214, 9743412645135, 17438019715216, 30122754096401, 50428155189474, 82094249361619

Offset: 0

Vincenzo Librandi, May 31 2014

Chebyshev polynomial of the first kind T(9,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A056220, A144129, A144130.

[256*n^9-576*n^7+432*n^5-120*n^3+9*n: n in [0..20]];

a:= n-> simplify(ChebyshevT(9, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, May 31 2014

Table[ChebyshevT[9, n], {n, 0, 20}] (* or *) Table[256 n^9 - 576 n^7 + 432 n^5 - 120 n^3 + 9 n, {n, 0, 20}]

G.f.: x *(1 + 70216*x + 3178684*x^2 + 22457464*x^3 + 41484550*x^4 + 22457464*x^5 + 3178684*x^6 + 70216*x^7 + x^8)/(1 - x)^10.

a(n) = n*(4*n^2 - 3)*(64*n^6 - 96*n^4 + 36*n^2 - 3).

cp's OEIS Frontend

A188644 Array of (k^n + k^(-n))/2 where k = (sqrt(x^2-1) + x)^2 for integers x >= 1.

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A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

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A144131 Primes of the form T_4(n), where T_4(x) = 8x^4 - 8x^2 + 1 is the fourth Chebyshev polynomial (of the first kind).

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Maple

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A239609 Sin(arcsin(n)- 3 arccos(n))^2.

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A243132 32*n^6 - 48*n^4 + 18*n^2 - 1.

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A243134 128*n^8 - 256*n^6 + 160*n^4 - 32*n^2 + 1.

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A243136 a(n) = 512*n^10 - 1280*n^8 + 1120*n^6 - 400*n^4 + 50*n^2 - 1.

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Maple

A243132 32n^6 - 48n^4 + 18*n^2 - 1.

A243134 128n^8 - 256n^6 + 160n^4 - 32n^2 + 1.

A243136 a(n) = 512n^10 - 1280n^8 + 1120n^6 - 400n^4 + 50*n^2 - 1.

A243131 a(n) = 16n^5 - 20n^3 + 5*n.

A243133 64n^7 - 112n^5 + 56n^3 - 7n.

A243135 256n^9 - 576n^7 + 432n^5 - 120n^3 + 9*n.