A269254 -id:A269254 - cp's OEIS frontend

A333859 Integers k such that A269254(k) = -1.

7, 14, 23, 34, 47, 62, 79, 98, 110, 119, 123, 142, 167, 194, 223, 254, 287, 322, 359, 398, 439, 482, 488, 527, 574, 623, 674, 702, 727, 782, 839, 898, 959, 1022, 1087, 1154, 1223, 1294, 1298, 1367, 1442, 1519

Offset: 1

Jinyuan Wang, Apr 08 2020

A008865(m) = m^2 - 2 is a term for any m > 2, see A269254 for a proof. Also note that 110, 123, 488, 702, 1298, ... are not of the form m^2 - 2.

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A008865, A269254.

A298675 Rectangular array A: first differences of row entries of array A294099, read by antidiagonals.

Original entry on oeis.org

1, 2, -1, 3, 2, -2, 4, 7, 2, -1, 5, 14, 18, 2, 1, 6, 23, 52, 47, 2, 2, 7, 34, 110, 194, 123, 2, 1, 8, 47, 198, 527, 724, 322, 2, -1, 9, 62, 322, 1154, 2525, 2702, 843, 2, -2, 10, 79, 488, 2207, 6726, 12098, 10084, 2207, 2, -1, 11, 98, 702, 3842, 15127, 39202, 57965, 37634, 5778, 2, 1

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Jan 24 2018

From a problem in A269254. For detailed theory, see [Hone].
From Charles L. Hohn, Sep 28 2024: (Start)
For rows n >= 3, values x >= 3 where (x^2-4)/(n^2-4) is a square.
For rows n >= 3, Lim_{k->oo}(T(n, k+1)/T(n, k)) = (sqrt(n^2-4)+n)/2. (End)

			Array begins:
   1 -1  -2   -1     1      2       1       -1        -2         -1
   2  2   2    2     2      2       2        2         2          2
   3  7  18   47   123    322     843     2207      5778      15127
   4 14  52  194   724   2702   10084    37634    140452     524174
   5 23 110  527  2525  12098   57965   277727   1330670    6375623
   6 34 198 1154  6726  39202  228486  1331714   7761798   45239074
   7 47 322 2207 15127 103682  710647  4870847  33385282  228826127
   8 62 488 3842 30248 238142 1874888 14760962 116212808  914941502
   9 79 702 6239 55449 492802 4379769 38925119 345946302 3074591599
  10 98 970 9602 95050 940898 9313930 92198402 912670090 9034502498

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Main diagonal gives A343259.

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.

t[n_, 0] := 2; t[n_, 1] := n; t[n_, k_] := n*t[n, k - 1] - t[n, k - 2]; Table[t[n, k], {n, 10}, {k, 10}] // Grid

A(n,k) = T_k(n), n >= 1, k >= 1, where T_j(x) = x*T_{j-1}(x) - T_{j-2}(x), j >= 2, T_0(x) = 2, T_1(x) = x, (dilated Chebyshev polynomials of the first kind).

A294099 Rectangular array read by (upward) antidiagonals: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))n^(k-j), n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 4, 5, -1, 1, 5, 11, 7, -2, 1, 6, 19, 29, 9, -1, 1, 7, 29, 71, 76, 11, 1, 1, 8, 41, 139, 265, 199, 13, 2, 1, 9, 55, 239, 666, 989, 521, 15, 1, 1, 10, 71, 377, 1393, 3191, 3691, 1364, 17, -1, 1, 11, 89, 559, 2584, 8119, 15289, 13775, 3571, 19, -2

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Oct 22 2017

This array is used to compute A269254: A269254(n) = least k such that A(n,k) is a prime, or -1 if no such k exists.
For detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018
The array can be extended to k<0 with A(n, k) = -A(n, -k-1) for all k in Z. - Michael Somos, Jun 19 2023

			Array begins:
  1   2    1    -1     -2      -1        1         2          1          -1
  1   3    5     7      9      11       13        15         17          19
  1   4   11    29     76     199      521      1364       3571        9349
  1   5   19    71    265     989     3691     13775      51409      191861
  1   6   29   139    666    3191    15289     73254     350981     1681651
  1   7   41   239   1393    8119    47321    275807    1607521     9369319
  1   8   55   377   2584   17711   121393    832040    5702887    39088169
  1   9   71   559   4401   34649   272791   2147679   16908641   133121449
  1  10   89   791   7030   62479   555281   4935050   43860169   389806471
  1  11  109  1079  10681  105731  1046629  10360559  102558961  1015229051

Robert Price, Table of n, a(n) for n = 1..5050
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.
Rows: A057079, A005408, A002878, A001834, A030221, A002315, A033890, A057080, A057081, A054320, ...
Columns: A000012, A000027, A028387, ...

(* Array: *)
Grid[Table[LinearRecurrence[{n, -1}, {1, 1 + n}, 10], {n, 10}]]
(* Array antidiagonals flattened (gives this sequence): *)
A294099[n_, k_] := Sum[(-1)^(Floor[j/2]) Binomial[k - Floor[(j + 1)/2], Floor[j/2]] n^(k - j), {j, 0, k}]; Flatten[Table[A294099[n - k, k], {n, 11}, {k, 0, n - 1}]]

{A(n, k) = sum(j=0, k, (-1)^(j\2)*binomial(k-(j+1)\2, j\2)*n^(k-j))}; /* Michael Somos, Jun 19 2023 */

A(n,0) = 1, A(n,1) = n + 1, A(n,k) = n*A(n,k-1) - A(n,k-2), n >= 1, k >= 2.
G.f. for row n: (1 + x)/(1 - n*x + x^2), n >= 1.
A(n, k) = B(-n, k) where B = A299045. - Michael Somos, Jun 19 2023

A299109 Probable primes in A030221.

Original entry on oeis.org

29, 139, 3191, 15289, 350981, 1681651, 20344613659, 2237722536169, 5650248517599839, 1464318118372209903213451940281613111, 471219735266432821374794400248484805597413615642086220363989152195627985749609

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 02 2018

From a problem in A269254. For detailed theory, see [Hone].

Subsequent terms have too many digits to display.

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Positions of primes in A030221, A294099 (row 5).

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A269254, A298675, A298677, A298878, A299045, A299071.

a(n) = A030221(A299101(n)).

A285992 Primes in the bisected Lucas sequence A002878.

Original entry on oeis.org

11, 29, 199, 521, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149, 412670427844921037470771, 258899611203303418721656157249445530046830073044201152332257717521

Offset: 1

R. J. Mathar, Apr 30 2017

nonn

Subsequence of A005479.

Cf. A001606, A002878, A005479, A121534, A180363, A269254, A294099 (row 3).

select(isprime, [seq(combinat:-fibonacci(2*n)+combinat:-fibonacci(2*n+2), n=1..200)]); # Robert Israel, May 01 2017

Select[LucasL[Range[1, 400, 2]], PrimeQ] (* Vincenzo Librandi, May 01 2017 *)
Select[LinearRecurrence[{3,-1},{1,4},160],PrimeQ] (* Harvey P. Dale, Sep 01 2024 *)

A002878 INTERSECT A000040.

A000032(k) for odd k in A001606. - Robert Israel, May 01 2017

A298677 a(n) = 110*a(n-1) - a(n-2) for n >= 2, a(0)=1, a(1)=111.

Original entry on oeis.org

1, 111, 12209, 1342879, 147704481, 16246150031, 1786928798929, 196545921732159, 21618264461738561, 2377812544869509551, 261537761671184312049, 28766775971285404815839, 3164083819079723345430241, 348020453322798282592510671, 38279085781688731361830743569, 4210351415532437651518789281919

Offset: 0

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Jan 24 2018

Sequence {s_k(110)} of A269254.
The sequence contains no primes; see A269254 for a proof by N. J. A. Sloane.
For detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018

Colin Barker, Table of n, a(n) for n = 0..450
Index entries for linear recurrences with constant coefficients, signature (110,-1).
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.

s[0, n_] := 1; s[1, n_] := n + 1; s[k_, n_] := n*s[k - 1, n] - s[k - 2, n]; Table[s[k, 110], {k, 0, 15}]
LinearRecurrence[{110, -1}, {1, 111}, 15]
CoefficientList[Series[(1 + x)/(1 - 110*x + x^2), {x, 0, 14}], x]

Vec((1 + x)/(1 - 110*x + x^2) + O(x^20)) \\ Colin Barker, Jan 25 2018

G.f.: (1 + x)/(1 - 110*x + x^2).

a(n) = (1/18)*((55+12*sqrt(21))^(-n)*(9-2*sqrt(21) + (9+2*sqrt(21))*(55+12*sqrt(21))^(2*n))). - Colin Barker, Jan 25 2018

A117522 Numbers k such that L(2*k + 1) is prime, where L(m) is a Lucas number.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 15, 18, 20, 23, 26, 30, 35, 39, 56, 156, 176, 251, 306, 308, 431, 548, 680, 2393, 2396, 2925, 3870, 4233, 5345, 6125, 6981, 7224, 9734, 17724, 18389, 22253, 25584, 28001, 40835, 44924, 47411, 70028, 74045, 79760, 91544, 96600, 101333, 172146, 193716, 221804, 266138, 287109, 308393, 315590, 318875, 325910, 346073, 450828, 525924

Offset: 1

Parthasarathy Nambi, Apr 26 2006

For n = 24..43, we can only claim that L(2*a(n) + 1) is a probable prime. Sequence arises in a study of A269254; for detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018

			If k = 56, then L(2*k + 1) is a prime with twenty-four digits.

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A000032, A001606, A269251, A269252, A269253, A269254.

Cf. A294099, A298675, A298677, A298878, A299045, A299071, A285992, A299107, A299109, A088165, A299100, A299101, A113501.

Values beyond 680 from L. Edson Jeffery, et al., Feb 02 2018
a(44)-a(56) from Robert Price, Jun 12 2025
a(57)-a(59) (using data in A001606) from Alois P. Heinz, Jun 12 2025

A269253 Smallest prime in the sequence s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n + 1 (or -1 if no such prime exists).

Original entry on oeis.org

2, 3, 11, 5, 29, 7, -1, 71, 89, 11, 131, 13, 181, -1, 239, 17, 5167, 19, 379, 419, 461, 23, -1, 599, 251894449, 701, 20357, 29, 25171, 31, 991, 36002209323169, 47468744103199, -1, 1259, 37, 2625505273, 1481, 1559, 41, 1721, 43, 150103799, 1979, 2069, 47, -1, 2351, 287762399

Offset: 1

Arkadiusz Wesolowski, Jul 09 2016

sign

For n >= 3, smallest prime of the form (x^y - 1/x^y)/(x - 1/x), where x = (sqrt(n+2) +/- sqrt(n-2))/2 and y is an odd positive integer, or -1 if no such prime exists.
When n=7, the sequence {s(k)} is A033890, which is Fibonacci(4i+2), and since x|y <=> F_x|F_y, and 2i+1|4i+2, A033890 is never prime, and so a(7) = -1. What is the proof for the other entries that are -1? Answer: See the Comments in A269254. - N. J. A. Sloane, Oct 22 2017
For detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018

Hans Havermann, Table of n, a(n) for n = 1..300
C. K. Caldwell, Top Twenty page, Lehmer number
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.
Wikipedia, Lehmer number

Cf. A117522, A269251, A269252, A269254.

Cf. A294099, A298675, A298677, A298878, A299045, A299071, A285992, A299107, A299109, A088165, A299100, A299101, A113501.

lst:=[]; for n in [1..49] do if n gt 2 and IsSquare(n+2) then Append(~lst, -1); else a:=n+1; c:=1; if IsPrime(a) then Append(~lst, a); else repeat b:=n*a-c; c:=a; a:=b; until IsPrime(a); Append(~lst, a); end if; end if; end for; lst;

terms = 172;
kmax = 120;
a[n_] := Module[{s, k}, s[k_] := s[k] = n s[k-1] - s[k-2]; s[0] = 1; s[1] = n+1; For[k = 1, k <= kmax, k++, If[PrimeQ[s[k]], Return[s[k]]]]];
Array[a, terms] /. Null -> -1 (* Jean-François Alcover, Aug 30 2018 *)

If n is prime then a(n-1) = n.

A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 01 2018

This array is used to compute A269252: A269252(n) = least k such that |A(n,k)| is a prime, or -1 if no such k exists.
For detailed theory, see [Hone].
The array can be extended to k<0 with A(n, k) = -A(n, -k-1) for all k in Z. - Michael Somos, Jun 19 2023

			Array begins:
1   0  -1     1     0      -1       1         0        -1           1
1  -1   1    -1     1      -1       1        -1         1          -1
1  -2   5   -13    34     -89     233      -610      1597       -4181
1  -3  11   -41   153    -571    2131     -7953     29681     -110771
1  -4  19   -91   436   -2089   10009    -47956    229771    -1100899
1  -5  29  -169   985   -5741   33461   -195025   1136689    -6625109
1  -6  41  -281  1926  -13201   90481   -620166   4250681   -29134601
1  -7  55  -433  3409  -26839  211303  -1663585  13097377  -103115431
1  -8  71  -631  5608  -49841  442961  -3936808  34988311  -310957991
1  -9  89  -881  8721  -86329  854569  -8459361  83739041  -828931049

Robert Price, Table of n, a(n) for n = 1..5050
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269251, A269252, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.
Cf. A094954 (unsigned version of this array, but missing the first row).
Cf. Rows: A057078, A033999, A099496, A079935 (or A001835), A004253, A001653, A049685, A070997, A070998, A138288 (or A072256), ...
Cf. Columns: A000012, A001477 (A000027), A110331 (A165900), A123972, A192398, ...

(* Array: *)
Grid[Table[LinearRecurrence[{-n, -1}, {1, 1 - n}, 10], {n, 10}]]
(*Array antidiagonals flattened (gives this sequence):*)
A299045[n_, k_] := Sum[(-1)^(Floor[j/2]) Binomial[k - Floor[(j + 1)/2], Floor[j/2]] (-n)^(k - j), {j, 0, k}]; Flatten[Table[A299045[n - k, k], {n, 11}, {k, 0, n - 1}]]

{A(n, k) = sum(j=0, k, (-1)^(j\2)*binomial(k-(j+1)\2, j\2)*(-n)^(k-j))}; /* Michael Somos, Jun 19 2023 */

G.f. for row n: (1 + x)/(1 + n*x + x^2), n >= 1.

A(n, k) = B(-n, k) where B = A294099. - Michael Somos, Jun 19 2023

A299071 Union_{odd primes p, n >= 3} {T_p(n)}, where T_m(x) = x*T_{m-1}(x) - T_{m-2}(x), m >= 2, T_0(x) = 2, T_1(x) = x (dilated Chebyshev polynomials of the first kind).

Original entry on oeis.org

18, 52, 110, 123, 198, 488, 702, 724, 843, 970, 1298, 1692, 2158, 2525, 3330, 4048, 4862, 5778, 6726, 6802, 7940, 9198, 10084, 10582, 13752, 15550, 17498, 19602, 21868, 24302, 26910, 29698, 30248, 32672, 35838, 39603, 42770, 46548, 50542

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 01 2018

nonn

From a problem in A269254. For detailed theory, see [Hone].

Sequence avoids numbers of the form T_p(T_2(j)).

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A008865 (T_2(n)), A298878 (T_p(n), p prime).

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A269254, A294099, A298675, A298677, A299045, A299071.

maxT = 55000; maxn = 12;
T[0][] = 2; T[1][x] = x;
T[m_][x_] := T[m][x] = x T[m-1][x] - T[m-2][x];
TT = Table[T[p][n], {p, Prime[Range[2, maxn]]}, {n, 3, Prime[maxn]}] // Flatten // Union // Select[#, # <= maxT&]&;
avoid = Table[T[p][T[2][n]], {p, Prime[Range[2, maxn]]}, {n, 3, Prime[maxn] }] // Flatten // Union // Select[#, # <= maxT&]&;
Complement[TT, avoid] (* Jean-François Alcover, Nov 03 2018 *)

cp's OEIS Frontend

A333859 Integers k such that A269254(k) = -1.

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A298675 Rectangular array A: first differences of row entries of array A294099, read by antidiagonals.

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A294099 Rectangular array read by (upward) antidiagonals: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)*binomial(k-floor((j+1)/2), floor(j/2))*n^(k-j), n >= 1, k >= 0.

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A299109 Probable primes in A030221.

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A285992 Primes in the bisected Lucas sequence A002878.

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A298677 a(n) = 110*a(n-1) - a(n-2) for n >= 2, a(0)=1, a(1)=111.

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A117522 Numbers k such that L(2*k + 1) is prime, where L(m) is a Lucas number.

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A269253 Smallest prime in the sequence s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n + 1 (or -1 if no such prime exists).

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A294099 Rectangular array read by (upward) antidiagonals: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))n^(k-j), n >= 1, k >= 0.

A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.