A008865 -id:A008865 - cp's OEIS frontend

A147973 a(n) = -2n^2 + 12n - 14.

-4, 2, 4, 2, -4, -14, -28, -46, -68, -94, -124, -158, -196, -238, -284, -334, -388, -446, -508, -574, -644, -718, -796, -878, -964, -1054, -1148, -1246, -1348, -1454, -1564, -1678, -1796, -1918, -2044, -2174, -2308, -2446, -2588, -2734, -2884, -3038, -3196, -3358

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 18 2008

-a(n+3) = 2*n^2 - 4, n >= 0, [-4,-2, 4, 14, ...] appears as the first member of the quartet for the square of [n, n+1, n+2, n+3], for n >= 0, in the Clifford algebra Cl_2. The other members are given in A046092(n), A054000(n+1) and A139570(n). The basis of Cl_2 is <1, s1, s2, s12> with s1.s1 = s2.s2 = 1, s12.s12 = -1, s1.s2 = -s2.s1 = s12. See e.g., pp. 5-6, eqs. (2.4)-(2.13) of the S. Gull et al. reference. - Wolfdieter Lang, Oct 15 2014
Related to the previous comment: if one uses the exterior (Grassmann) product with s1.s1 = s2.s2 = s12.s12 = 0 and s1.s2 = -s2.s1 = s12, then the four components of the square of [n, n+1, n+2, n+3] are [A000290(n), A046092(n), A054000(n+1), A139570(n)], n >= 0. - Wolfdieter Lang, Nov 13 2014
2 - a(n)/2 is a square. - Bruno Berselli, Apr 10 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. Gull, A. Lasenby and C. Doran, Imaginary Numbers are not Real - the Geometric Algebra of Spacetime, Found. Phys., Vol. 23(9) (1993), pp. 1175-1201.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A008865, A046092, A054000, A139570.

[-2*n^2+12*n-14: n in [1..50]]; // Vincenzo Librandi, Jul 10 2012

[-2*n^2+12*n-14$n=1..50]; # Muniru A Asiru, Feb 12 2019

lst={};Do[k=n^2-((n-1)^2+(n-2)^2+(n-3)^2);AppendTo[lst,k],{n,5!}];lst
Table[-2n^2+12n-14,{n,1,50}] (* Vincenzo Librandi, Jul 10 2012 *)
LinearRecurrence[{3,-3,1},{-4,2,4},50] (* Harvey P. Dale, Mar 02 2020 *)

a(n)=-2*n^2+12*n-14 \\ Charles R Greathouse IV, Sep 24 2015

Vec(-2*x*(2 - 7*x + 7*x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, Feb 12 2019

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 10 2012
a(n) = -2*A008865(n-3). - J. M. Bergot, Jun 25 2018
G.f.: -2*x*(2 - 7*x + 7*x^2)/(1 - x)^3. - Colin Barker, Feb 12 2019
E.g.f.: -2*(exp(x)*(x^2 - 5*x + 7) - 7). - Elmo R. Oliveira, Nov 17 2024

A289192 A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 7, 6, 1, 4, 14, 34, 24, 1, 5, 23, 86, 209, 120, 1, 6, 34, 168, 648, 1546, 720, 1, 7, 47, 286, 1473, 5752, 13327, 5040, 1, 8, 62, 446, 2840, 14988, 58576, 130922, 40320, 1, 9, 79, 654, 4929, 32344, 173007, 671568, 1441729, 362880

Offset: 0

Alois P. Heinz, Jun 28 2017

			Square array A(n,k) begins:
:   1,    1,    1,     1,     1,     1, ...
:   1,    2,    3,     4,     5,     6, ...
:   2,    7,   14,    23,    34,    47, ...
:   6,   34,   86,   168,   286,   446, ...
:  24,  209,  648,  1473,  2840,  4929, ...
: 120, 1546, 5752, 14988, 32344, 61870, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Columns k=0-10 give: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216.
Rows n=0-2 give: A000012, A000027(k+1), A008865(k+2).
Main diagonal gives A277373.
Cf. A253286, A341014.

A:= (n,k)-> n! * add(binomial(n, i)*k^i/i!, i=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := n! * LaguerreL[n, -k];
Table[A[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 05 2019 *)

{T(n, k) = if(n<2, k*n+1, (2*n+k-1)*T(n-1, k)-(n-1)^2*T(n-2, k))} \\ Seiichi Manyama, Feb 03 2021

T(n, k) = n!*pollaguerre(n, 0, -k); \\ Michel Marcus, Feb 05 2021

from sympy import binomial, factorial as f
def A(n, k): return f(n)*sum(binomial(n, i)*k**i/f(i) for i in range(n + 1))
for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Jun 28 2017

A(n,k) = n! * Sum_{i=0..n} k^i/i! * binomial(n,i).
E.g.f. of column k: exp(k*x/(1-x))/(1-x).
A(n, k) = (-1)^n*KummerU(-n, 1, -k). - Peter Luschny, Feb 12 2020
A(n, k) = (2*n+k-1)*A(n-1, k) - (n-1)^2*A(n-2, k) for n > 1. - Seiichi Manyama, Feb 03 2021

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 *)

A298878 Union_{p prime, n >= 0} {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

-2, -1, 0, 1, 2, 7, 14, 18, 23, 34, 47, 52, 62, 79, 98, 110, 119, 123, 142, 167, 194, 198, 223, 254, 287, 322, 359, 398, 439, 482, 488, 527, 574, 623, 674, 702, 724, 727, 782, 839, 843, 898, 959, 970, 1022, 1087, 1154, 1223, 1294, 1298, 1367, 1442, 1519, 1598

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Jan 27 2018

sign

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

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)), A299071.

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

A159016 Square array A, read by antidiagonals, with entry n in row k given by A(k,0) = A028387(k) and A(k,n) = A(k,n-1) + floor(sqrt(A(k,n-1))), n>0, k>=0.

Original entry on oeis.org

1, 2, 5, 3, 7, 11, 4, 9, 14, 19, 6, 12, 17, 23, 29, 8, 15, 21, 27, 34, 41, 10, 18, 25, 32, 39, 47, 55, 13, 22, 30, 37, 45, 53, 62, 71, 16, 26, 35, 43, 51, 60, 69, 79, 89, 20, 31, 40, 49, 58, 67, 77, 87, 98, 109, 24, 36, 46, 56, 65, 75, 85, 96, 107, 119, 131

Offset: 0

Philippe Deléham, Apr 02 2009

Permutation of positive integers.

			Array begins:
...1....2....3....4....6....8...10...13...16...20..
...5....7....9...12...15...18...22...26...31...36..
..11...14...17...21...25...30...35...40...46...52..
..19...23...27...32...37...43...49...56...63...70..
..29...34...39...45...51...58...65...73...81...90..
..41...47...53...60...67...75...83...92..101..111..
..55...62...69...77...85...94..103..113..123..134..
..71...79...87...96..105..115..125..136..147..159..
..89...98..107..117..127..138..149..161..173..186..
.109..119..129..140..151..163..175..188..201..215..

Cf. A002984 (= row 0), A028387, A008865, A014209, A028392.

Definition corrected by Philippe Deléham, Apr 04 2009

Edited by L. Edson Jeffery, May 28 2013

A101986 Maximum sum of products of successive pairs in a permutation of order n+1.

Original entry on oeis.org

0, 2, 9, 23, 46, 80, 127, 189, 268, 366, 485, 627, 794, 988, 1211, 1465, 1752, 2074, 2433, 2831, 3270, 3752, 4279, 4853, 5476, 6150, 6877, 7659, 8498, 9396, 10355, 11377, 12464, 13618, 14841, 16135, 17502, 18944, 20463, 22061, 23740, 25502

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Jan 29 2005

1 3 5 4 2 is the 11th permutation, in lexical order. of order 5. Its reverse 2 4 5 3 1 is the 41st. The earliest permutation of order 6 is the 41st, 1 3 5 6 4 2. This pattern continues as far as I have looked, so its reversal 2 4 6 5 3 1 is the 191st and the earliest permutation of order 7 is the 191st, et cetera.
Comments from Dmitry Kamenetsky, Dec 15 2006: (Start)
This sequence is related to A026035, except here we take the maximum sum of products of successive pairs. Here is a method for generating such permutations. Start with two lists, the first has numbers 1 to n, while the second is empty.
Repeat the following operations until the first list is empty: 1. Move the smallest number of the first list to the leftmost available position in the second list. The move operation removes the original number from the first list. 2. Move the smallest number of the first list to the rightmost available position in the second list. For example when n=8, the permutation is 1, 3, 5, 7, 8, 6, 4, 2. (End)
Convolution of odd numbers and integers greater than 1. - Reinhard Zumkeller, Mar 30 2012
For n>0, a(n) is row 2 of the convolution array A213751. - Clark Kimberling, Jun 20 2012

			The permutations of order 5 with maximum sum of products is 1 3 5 4 2 and its reverse, since (1*3)+(3*5)+(5*4)+(4*2) is 46. All others are empirically less than 46. So a(4) = 46.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Leroy Quet, Max/Min Sums From Permutations - a message on Jan 28 2005 to the sequence list, asking for someone to provide more values and to submit these to OEIS.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Pairwise sums of A005581.

Cf. A000290, A000330, A008865, A026035, A160805.

a101986 n = sum $ zipWith (*) [1,3..] (reverse [2..n+1])
-- Reinhard Zumkeller, Mar 30 2012

0 1 9 2 & p. % 6 & p. (A) NB. the polynomial P such that P(n) is a(n).
NB. where 0 1 9 2 are the coefficients in ascending order of the numerator of a rational polynomial and 6 is the (constant) coefficient of its denominator. J's primitive function p. produces a polynomial with these coefficients. Division is indicated by % . Thus the J expression (A) is equivalent to the formula above.

a:=n->add((n+j^2),j=1..n): seq(a(n),n=0..41); # Zerinvary Lajos, Jul 27 2006

Table[(n + 9 n^2 + 2 n^3)/6, {n, 0, 41}] (* Robert G. Wilson v, Feb 04 2005 *)

a(n)=n*(2*n^2+9*n+1)/6 \\ Charles R Greathouse IV, Jan 17 2012

a(n) = n*(2*n^2 + 9*n + 1)/6.
a(n+1) = a(n) + A008865(n+2); a(n) = A160805(n) - 4. [Reinhard Zumkeller, May 26 2009]
G.f.: x*(1+x)*(2-x)/(1-x)^4. - L. Edson Jeffery, Jan 17 2012
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3, a(0)=0, a(1)=2, a(2)=9, a(3)=23. - L. Edson Jeffery, Jan 17 2012
a(n) = A000330(n) + A005449(n) - A000217(n). - Richard R. Forberg, Aug 07 2013
a(n) = 1 + sum( A008865(i), i=1..n+1 ). [Bruno Berselli, Jan 13 2015]
a(n) = A000290(n) + A000330(n). - J. M. Bergot, Apr 26 2018

Edited by Bruno Berselli, Jan 13 2015

Name edited by Alois P. Heinz, Feb 02 2019

A229012 T(n,k) = number of arrays of median of three adjacent elements of some length n+2 0..k array, with no adjacent equal elements in the latter.

Original entry on oeis.org

2, 3, 2, 4, 7, 2, 5, 14, 15, 2, 6, 23, 46, 31, 2, 7, 34, 101, 130, 57, 2, 8, 47, 186, 359, 332, 105, 2, 9, 62, 307, 794, 1145, 830, 193, 2, 10, 79, 470, 1527, 3002, 3527, 2054, 353, 2, 11, 98, 681, 2666, 6635, 10860, 10735, 5108, 653, 2, 12, 119, 946, 4335, 13040, 27379

Offset: 1

R. H. Hardin, Sep 10 2013

Table starts
.2...3.....4......5......6.......7.......8........9.......10.......11........12
.2...7....14.....23.....34......47......62.......79.......98......119.......142
.2..15....46....101....186.....307.....470......681......946.....1271......1662
.2..31...130....359....794....1527....2666.....4335.....6674.....9839.....14002
.2..57...332...1145...3002....6635...13040....23515....39698....63605.....97668
.2.105...830...3527..10860...27379...60180...119653...220318...381749....629586
.2.193..2054..10735..38768..111311..273124...597477..1197190..2238005...3954490
.2.353..5108..32907.139456..456029.1248872..3004839..6549040.13200731..24974126
.2.653.12790.101635.506236.1888383.5780144.15315095.36345246.79063593.160271154

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

R. H. Hardin, Table of n, a(n) for n = 1..925

Row 2 is A008865(n+1).

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: [order 13]
k=3: [order 27]
k=4: [order 46]
k=5: [order 69]
k=6: [order 95] for n>97
Empirical for row n:
n=1: a(n) = 1*n + 1
n=2: a(n) = 1*n^2 + 2*n - 1
n=3: a(n) = 1*n^3 + 3*n^2 - 3*n + 1
n=4: a(n) = (2/3)*n^4 + (10/3)*n^3 - (5/3)*n^2 + (2/3)*n - 1
n=5: [polynomial of degree 5]
n=6: [polynomial of degree 6]
n=7: [polynomial of degree 7]

A057897 Numbers which can be written as m^k-k, with m, k > 1.

Original entry on oeis.org

2, 5, 7, 12, 14, 23, 24, 27, 34, 47, 58, 61, 62, 77, 79, 98, 119, 121, 122, 142, 167, 194, 213, 223, 238, 248, 252, 254, 287, 322, 340, 359, 398, 439, 482, 503, 509, 527, 574, 621, 623, 674, 723, 726, 727, 782, 839, 898, 959, 997, 1014, 1019, 1022, 1087, 1154

Offset: 1

Henry Bottomley, Sep 26 2000

nonn

It may be that positive integers can be written as m^k-k (with m and k > 1) in at most one way [checked up to 10000].

All numbers < 10^16 of this form have a unique representation. The uniqueness question leads to a Pillai-like exponential Diophantine equation a^x-b^y = x-y for x > y > 1 and b > a > 1, which appears to have no solutions. - T. D. Noe, Oct 06 2004

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000325, A008865, A024024, A024037, A024050, A057895, A057898, A057899.

Cf. A099225 (numbers of the form m^k+k, with m and k > 1), A074981 (n such that there is no solution to Pillai's equation), A099226 (numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1).

nLim=1000; lst={}; Do[k=2; While[n=m^k-k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Union[lst] (* T. D. Noe, Oct 06 2004 *)

ok(n)={my(e=2); while(2^e <= n+e, if(ispower(n+e, e), return(1)); e++); 0} \\ Andrew Howroyd, Oct 20 2020

upto(lim)={my(p=logint(lim,2)); while(logint(lim+p+1,2)>p, p++); Vec(Set(concat(vector(p-1, e, e++; vector(sqrtnint(lim+e,e)-1, m, (m+1)^e-e)))))} \\ Andrew Howroyd, Oct 20 2020

A212835 T(n,k)=Number of 0..k arrays of length n+1 with 0 never adjacent to k.

Original entry on oeis.org

2, 7, 2, 14, 17, 2, 23, 50, 41, 2, 34, 107, 178, 99, 2, 47, 194, 497, 634, 239, 2, 62, 317, 1106, 2309, 2258, 577, 2, 79, 482, 2137, 6306, 10727, 8042, 1393, 2, 98, 695, 3746, 14407, 35954, 49835, 28642, 3363, 2, 119, 962, 6113, 29114, 97127, 204994, 231521

Offset: 1

R. H. Hardin May 28 2012

Table starts
.2.....7......14.......23........34.........47.........62..........79
.2....17......50......107.......194........317........482.........695
.2....41.....178......497......1106.......2137.......3746........6113
.2....99.....634.....2309......6306......14407......29114.......53769
.2...239....2258....10727.....35954......97127.....226274......472943
.2...577....8042....49835....204994.....654797....1758602.....4159927
.2..1393...28642...231521...1168786....4414417...13667858....36590017
.2..3363..102010..1075589...6663906...29760487..106226618...321839625
.2..8119..363314..4996919..37994674..200635007..825593474..2830847119
.2.19601.1293962.23214443.216628994.1352612477.6416514026.24899654327

			Some solutions for n=5 k=4
..1....4....1....1....1....3....2....1....1....4....3....1....0....2....2....3
..1....3....0....4....2....4....1....1....2....2....3....4....1....3....3....1
..1....3....3....1....3....3....2....2....2....4....3....3....1....1....0....3
..1....3....0....4....2....3....3....3....4....2....2....0....4....4....2....1
..1....4....0....3....2....2....3....3....4....4....0....1....4....3....4....2
..1....2....0....4....3....1....0....2....4....2....3....2....1....4....3....4

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 2 is A001333(n+2)
Column 3 is A055099(n+1)
Column 4 is A126473(n+1)
Column 5 is A126501(n+1)
Column 6 is A126528(n+1)
Row 1 is A008865(n+1)

Empirical for column k: a(n) = k*a(n-1) +(k-1)*a(n-2)
Empirical for rows:
n=1: a(k) = k^2 + 2*k - 1
n=2: a(k) = k^3 + 3*k^2 - k - 1
n=3: a(k) = k^4 + 4*k^3 - 4*k + 1
n=4: a(k) = k^5 + 5*k^4 + 2*k^3 - 8*k^2 + k + 1
n=5: a(k) = k^6 + 6*k^5 + 5*k^4 - 12*k^3 - 3*k^2 + 6*k - 1
n=6: a(k) = k^7 + 7*k^6 + 9*k^5 - 15*k^4 - 13*k^3 + 15*k^2 - k - 1
n=7: a(k) = k^8 + 8*k^7 + 14*k^6 - 16*k^5 - 30*k^4 + 24*k^3 + 8*k^2 - 8*k + 1

A195437 Triangle formed by: 1 even, 2 odd, 3 even, 4 odd... starting with 2.

Original entry on oeis.org

2, 5, 7, 10, 12, 14, 17, 19, 21, 23, 26, 28, 30, 32, 34, 37, 39, 41, 43, 45, 47, 50, 52, 54, 56, 58, 60, 62, 65, 67, 69, 71, 73, 75, 77, 79, 82, 84, 86, 88, 90, 92, 94, 96, 98, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 122, 124, 126, 128, 130, 132

Offset: 0

Reinhard Zumkeller, Oct 12 2011

A127366(T(n,k)) mod 2 = 1;
T(n,k) mod 2 + A000196(T(n,k)) mod 2 = 1;
complement of A133280.

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened

Cf. A002522 (left edge), A008865 (right edge), A011379 (row sums), A002939 (central terms).

a195437 n k = a195437_tabl !! n !! k
a195437_tabl = tail $ g 1 1 [0..] where
   g m j xs = (filter ((== m) . (`mod` 2)) ys) : g (1 - m) (j + 2) xs'
     where (ys,xs') = splitAt j xs
b195437 = bFile' "A195437" (concat $ take 101 a195437_tabl) 0
-- Reinhard Zumkeller, Nov 23 2011 (fixed), Oct 12 2011

p[n_,k_]:=NestList[#+2&,n,k-1]; Module[{nn=6,ev,od},ev=p@@@Partition[Riffle[Table[ 4n^2-4n+2,{n,nn}],Range[ 1,2nn,2]],2];od=p@@@Partition[Riffle[Table[4n^2+1,{n,nn}],Range[ 2,2nn+1,2]],2];Sort[Flatten[Join[ev,od]]]] (* Harvey P. Dale, Aug 27 2023 *)

cp's OEIS Frontend

A147973 a(n) = -2*n^2 + 12*n - 14.

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A289192 A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

Mathematica

PARI

PARI

Python

Formula

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).

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Mathematica

A298878 Union_{p prime, n >= 0} {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).

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A159016 Square array A, read by antidiagonals, with entry n in row k given by A(k,0) = A028387(k) and A(k,n) = A(k,n-1) + floor(sqrt(A(k,n-1))), n>0, k>=0.

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Extensions

A101986 Maximum sum of products of successive pairs in a permutation of order n+1.

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Haskell

J

Maple

Mathematica

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Formula

Extensions

A229012 T(n,k) = number of arrays of median of three adjacent elements of some length n+2 0..k array, with no adjacent equal elements in the latter.

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Formula

A057897 Numbers which can be written as m^k-k, with m, k > 1.

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A147973 a(n) = -2n^2 + 12n - 14.