A001835 -id:A001835 - cp's OEIS frontend

A334178 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 2^n * sqrt(Resultant(U_{2n}(x/2), T_{k}(ix/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

1, 1, 2, 1, 1, 4, 1, 3, 1, 8, 1, 4, 11, 1, 16, 1, 7, 19, 41, 1, 32, 1, 11, 71, 91, 153, 1, 64, 1, 18, 176, 769, 436, 571, 1, 128, 1, 29, 539, 2911, 8449, 2089, 2131, 1, 256, 1, 47, 1471, 17753, 48301, 93127, 10009, 7953, 1, 512, 1, 76, 4271, 79808, 603126, 801701, 1027207, 47956, 29681, 1, 1024

Offset: 0

Seiichi Manyama, Apr 17 2020

			Square array begins:
   1, 1,    1,     1,       1,        1,         1, ...
   2, 1,    3,     4,       7,       11,        18, ...
   4, 1,   11,    19,      71,      176,       539, ...
   8, 1,   41,    91,     769,     2911,     17753, ...
  16, 1,  153,   436,    8449,    48301,    603126, ...
  32, 1,  571,  2089,   93127,   801701,  20721019, ...
  64, 1, 2131, 10009, 1027207, 13307111, 714790675, ...

Rows 0..1 give A000012, A000032.
Columns 0..15 give A000079, A000012, A001835(n+1), A004253(n+1), A334135, A003729, A334179, A028478, A334180, A028480, A334181, A028482, A334182, A028484, A334183, A028486.
Cf. A103997.

T[n_, k_] := 2^n * Sqrt[Resultant[ChebyshevU[2*n, x/2], ChebyshevT[k, I*x/2], x]]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 04 2021 *)

{T(n, k) = sqrtint(4^n*polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(k, 1, I*x/2)))}

T(n,2*k) = A103997(n,k) for k > 0.

A103772 Larger of two sides in a (k,k,k-1)-integer-sided triangle with integer area.

Original entry on oeis.org

1, 17, 241, 3361, 46817, 652081, 9082321, 126500417, 1761923521, 24540428881, 341804080817, 4760716702561, 66308229755041, 923554499868017, 12863454768397201, 179164812257692801, 2495443916839302017, 34757050023492535441, 484103256412056194161

Offset: 1

Zak Seidov, Feb 23 2005

Corresponding areas are 0, 120, 25080, 4890480, 949077360, 184120982760, ...
Values of (x^2 + y^2)/2, where the pair (x, y) satisfies x^2 - 3*y^2 = -2, i.e., a(n) = {(A001834(n))^2 + (A001835(n))^2}/2 = {(A001834(n))^2 + A046184(n)}/2. - Lekraj Beedassy, Jul 13 2006
The heights of these triangles are given in A028230. (A028230(n), A045899(n), A103772(n)) forms a primitive Pythagorean triple.
Shortest side of (k,k+2,k+3) triangle such that median to longest side is integral. Sequence of such medians is A028230. - James R. Buddenhagen, Nov 22 2013
Numbers n such that (n+1)*(3n-1) is a square. - James R. Buddenhagen, Nov 22 2013

Colin Barker, Table of n, a(n) for n = 1..850
J. B. Cosgrave, The Gauss-Factorial Motzkin connection (Maple worksheet, change suffix to .mw)
J. B. Cosgrave and K. Dilcher, An Introduction to Gauss Factorials, The American Mathematical Monthly, 118 (Nov. 2011), 812-829.
Project Euler, Problem 94: Almost Equilateral Triangles.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A102341, A103974, A016064, A011945, A028230.

I:=[1,17]; [n le 2 select I[n] else 14*Self(n-1)-Self(n-2)+4: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016

a[1] = 1; a[2] = 17; a[3] = 241; a[n_] := a[n] = 15a[n - 1] - 15a[n - 2] + a[n - 3]; Table[ a[n] - 1, {n, 17}] (* Robert G. Wilson v, Mar 24 2005 *)
LinearRecurrence[{15,-15,1},{1,17,241},20] (* Harvey P. Dale, Jan 02 2016 *)
RecurrenceTable[{a[1] == 1, a[2] == 17, a[n] == 14 a[n-1] - a[n-2] + 4}, a, {n, 20}] (* Vincenzo Librandi, Mar 05 2016 *)

Vec(x*(1+x)^2/((1-x)*(1-14*x+x^2)) + O(x^25)) \\ Colin Barker, Mar 05 2016

a(n) = (4*A001570(n+1) - 1)/3, n > 0. - Ralf Stephan, May 20 2007
a(n) = A052530(n-1)*A052530(n) + 1. - Johannes Boot, May 21 2011
G.f.: x*(1+x)^2/((1-x)*(1-14*x+x^2)). - Colin Barker, Apr 09 2012
a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3); a(1)=1, a(2)=17, a(3)=241. - Harvey P. Dale, Jan 02 2016
a(n) = (-1+(7-4*sqrt(3))^n*(2+sqrt(3))-(-2+sqrt(3))*(7+4*sqrt(3))^n)/3. - Colin Barker, Mar 05 2016
a(n) = 14*a(n-1) - a(n-2) + 4. - Vincenzo Librandi, Mar 05 2016
a(n) = A001353(n)^2 + A001353(n-1)^2. - Antonio Alberto Olivares, Apr 06 2020

More terms from Robert G. Wilson v, Mar 24 2005

A147720 Riordan array (1, x(1-x)/(1-3x)).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 18, 16, 6, 1, 0, 54, 60, 30, 8, 1, 0, 162, 216, 134, 48, 10, 1, 0, 486, 756, 558, 248, 70, 12, 1, 0, 1458, 2592, 2214, 1168, 410, 96, 14, 1, 0, 4374, 8748, 8478, 5160, 2150, 628, 126, 16

Offset: 0

Paul Barry, Nov 11 2008

Array [0,2,1,0,0,0,....] DELTA [1,0,0,0,......] for Deléham DELTA as in A084938.
Row sums are A001835. Diagonal sums are related to A030186.
Row sums of inverse are essentially A091593. A147720*A007318 is A147721.

			Triangle begins
1;
0,   1;
0,   2,   1;
0,   6,   4,   1;
0,  18,  16,   6,   1;
0,  54,  60,  30,   8,   1;
0, 162, 216, 134,  48,  10,   1;

Indranil Ghosh, Rows 0..100, flattened

nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[(1-3*x)/(1-(3+y)*x+y*x^2), {x, 0, nmax}],x],{y,0,nmax}],y]] (* Indranil Ghosh, Mar 10 2017, after Philippe Deléham *)

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001835(n), A147722(n), A084120(n) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Nov 15 2008

G.f.: (1-3*x)/(1-(3+y)*x+y*x^2). - Philippe Deléham, Feb 15 2012

A147721 a(n) = C(2,n) DELTA C(0,n).

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 11, 17, 7, 1, 41, 72, 40, 10, 1, 153, 301, 208, 72, 13, 1, 571, 1244, 1021, 446, 113, 16, 1, 2131, 5093, 4819, 2525, 813, 163, 19, 1, 7953, 20688, 22104, 13452, 5218, 1336, 222, 22, 1, 29681, 83481, 99192, 68568, 30986, 9586, 2042, 290, 25, 1

Offset: 0

Paul Barry, Nov 11 2008

Triangle T equal to [1,2,1,0,0,0,...] DELTA [1,0,0,0,...] for Deléham DELTA as in A084938.

T = A147720*A007318. Row sums are A147722.

			Triangle begins
    1;
    1,   1;
    3,   4,   1;
   11,  17,   7,   1;
   41,  72,  40,  10,   1;
  153, 301, 208,  72,  13,   1;

Indranil Ghosh, Rows 0..100, flattened

Cf. A129267, A007318, A147703.

nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[(1 - 3*x)/(1 - 4*x + (1 + y)*x^2 - y*x), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 10 2017, after Philippe Deléham *)

Riordan array ((1-3x)/(1-4x+x^2), x(1-x)/(1-4x+x^2)).
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), n > 1. - Philippe Deléham, Feb 13 2012
G.f.: (1-3*x)/(1-4*x+(1+y)*x^2-y*x). - Philippe Deléham, Feb 13 2012
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001835(n), A147722(n), A084120(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Feb 13 2012

A227418 Array A(n,k) with all numbers m such that 3*m^2 +- 3^k is a square and their corresponding square roots, read by downward antidiagonals.

Original entry on oeis.org

0, 1, 1, 0, 2, 4, 3, 3, 7, 15, 0, 6, 12, 26, 56, 9, 9, 21, 45, 97, 209, 0, 18, 36, 78, 168, 362, 780, 27, 27, 63, 135, 291, 627, 1351, 2911, 0, 54, 108, 234, 504, 1086, 2340, 5042, 10864, 81, 81, 189, 405, 873, 1881, 4053, 8733, 18817, 40545

Offset: 0

Richard R. Forberg, Sep 02 2013

Array is analogous to A228405 in goal and structure, with key differences.
Left column is A001353.  Top row (not in OEIS) interleaves 0 with the powers of 3, as: 0, 1, 0, 3, 0, 9, 0, 27, 0, 81.
Either or both may be used as initializing values. See Formula section.
The left column is the second binomial transform of the top row. The intermediate transform sequence is A002605, not present in this array.
The columns of the array hold all values, in sequential order, of numbers m such that 3*m^2 + 3^k or 3*m^2 - 3^k are squares, and their corresponding square roots in the next column, which then form the "next round" of m values for column k+1.
For example: A(n,0) are numbers such that 3*m^2 + 1 are squares, the integer square roots of each are in A(n,1), which are then numbers m such that 3*m^2 - 3 are squares, with those square roots in A(n,2), etc.  The sign alternates for each increment of k, etc. No integer square roots exist for the opposite sign in a given column, regardless of n.
Also, A(n,1) are values of m such that floor(m^2/3) is square, with the corresponding square roots given by A(n,0).
A(n,k)/A(n,k-2) = 3; A(n,k)/A(n,k-1) converges to sqrt(3) for large n.
A(n,k)/A(n-1,k) converges to 2 + sqrt(3) for large n.
Several ways of combining the first few columns give OEIS sequences:
  A(n,0) + A(n,1) = A001835; A(n,1) + A(n,2)= A001834; A(n,2) + A(n,3) = A082841;
  A(n,0)*A(n,1)/2 = A007655(n); A(n+2,0)*A(n+1,1) = A001922(n);
  A(n,0)*A(n+1,1) = A001921(n); A(n,0)^2 + A(n,1)^2 = A103974(n);
  A(n,1)^2 - A(n,0)^2 = A011922(n); (A(n+2,0)^2 + A(n+1,1)^2)/2 = A122770(n) = 2*A011916(n).
The main diagonal (without initial 0) = 2*A090018.  The first subdiagonal =  abs(A099842). First superdiagonal = A141041.
A001353 (in left column) are the only initializing set of numbers where the recursive square root equation (see below) produces exclusively integer values, for all iterations of k.  For any other initial values only even iterations (at k = 2, 4, ...) produce integers.

			The array, A(n, k), begins as:
    0,    1,    0,    3,    0,     9,     0,    27, ... see A000244;
    1,    2,    3,    6,    9,    18,    27,    54, ... A038754;
    4,    7,   12,   21,   36,    63,   108,   189, ... A228879;
   15,   26,   45,   78,  135,   234,   405,   702, ...
   56,   97,  168,  291,  504,   873,  1512,  2619, ...
  209,  362,  627, 1086, 1881,  3258,  5643,  9774, ...
  780, 1351, 2340, 4053, 7020, 12159, 21060, 36477, ...
Antidiagonal triangle, T(n, k), begins as:
   0;
   1,  1;
   0,  2,   4;
   3,  3,   7,  15;
   0,  6,  12,  26,  56;
   9,  9,  21,  45,  97,  209;
   0, 18,  36,  78, 168,  362,  780;
  27, 27,  63, 135, 291,  627, 1351, 2911;
   0, 54, 108, 234, 504, 1086, 2340, 5042, 10864;
  81, 81, 189, 405, 873, 1881, 4053, 8733, 18817, 40545;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A001353, A001075, A005320, A038754, A090018, A099842, A141041, A151961.

Cf. A000244, A084156, A228879.

function A(n,k)
  if k lt 0 then return 0;
  elif n eq 0 then return Round((1/2)*(1-(-1)^k)*3^((k-1)/2));
  elif k eq 0 then return Evaluate(ChebyshevSecond(n), 2);
  else return 2*A(n, k-1) - A(n-1, k-1);
  end if; return A;
end function;
A227418:= func< n,k | A(k, n-k) >;
[A227418(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Oct 09 2022

A[n_, k_]:= If[k<0, 0, If[k==0, ChebyshevU[n-1, 2], 2*A[n, k-1] - A[n-1, k-1]]];
T[n_, k_]:= A[k, n-k];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 09 2022 *)

def A(n,k):
    if (k<0): return 0
    elif (k==0): return chebyshev_U(n-1,2)
    else: return 2*A(n, k-1) - A(n-1, k-1)
def A227418(n, k): return A(k, n-k)
flatten([[A227418(n,k) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Oct 09 2022

If using the left column and top row to initialize, then: A(n,k) = 2*A(n, k-1) - A(n-1, k-1).
If using only the top row to initialize, then: A(n,k) = 4*A(n-1,k) - A(n-2,k).
If using the left column to initialize, then: A(n,k) = sqrt(3*A(n,k-1) + (-3)^(k-1)), for all n, k > 0.
Other internal relationships that apply are: A(2*n-1, 2*k) = A(n,k)^2 - A(n-1,k)^2;
A(n+1,k) * A(n,k+1) - A(n+1, k+1) * A(n,k) = (-3)^k, for all n, k > 0.
A(n, 0) = A001353(n).
A(n, 1) = A001075(n).
A(n, 2) = A005320(n).
A(n, 3) = A151961(n).
A(1, k) = A038754(k).
A(n, n) = 2*A090018(n), for n > 0 (main diagonal).
A(n, n+1) = A141041(n-1) (superdiagonal).
A(n+1, n) = abs(A099842(n)) (subdiagonal).
From G. C. Greubel, Oct 09 2022: (Start)
T(n, 0) = (1/2)*(1-(-1)^n)*3^((n-1)/2).
T(n, 1) = A038754(n-1).
T(n, 2) = A228879(n-2).
T(2*n-1, n-1) = A141041(n-1).
T(2*n, n) = 2*A090018(n-1), n > 0.
T(n, n-4) = 3*A005320(n-4).
T(n, n-3) = 3*A001075(n-3).
T(n, n-2) = 3*A001353(n-2).
T(n, n-1) = A001075(n-1).
T(n, n) = A001353(n).
Sum_{k=0..n-1} T(n, k) = A084156(n).
Sum_{k=0..n} T(n, k) = A084156(n) + A001353(n). (End)

Offset corrected by G. C. Greubel, Oct 09 2022

A123519 Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).

Original entry on oeis.org

1, 1, 2, 1, 6, 4, 1, 12, 20, 8, 1, 20, 60, 56, 16, 1, 30, 140, 224, 144, 32, 1, 42, 280, 672, 720, 352, 64, 1, 56, 504, 1680, 2640, 2112, 832, 128, 1, 72, 840, 3696, 7920, 9152, 5824, 1920, 256, 1, 90, 1320, 7392, 20592, 32032, 29120, 15360, 4352, 512, 1, 110, 1980, 13728, 48048, 96096, 116480, 87040, 39168, 9728, 1024

Offset: 0

Emeric Deutsch, Oct 16 2006

Sum of terms in row n = A001835(n+1). Sum(k*T(n,k), k=0..n)=A123520(n) (n>=1).

			T(1,1)=2 because a 2 X 3 grid can be tiled in 2 ways with dominoes so that exactly 2 dominoes are in vertical position: place a horizontal domino above or below two adjacent vertical dominoes.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

Cf. A001835, A123520. A085478, A211955, A211956.

T:=(n,k)->2^k*binomial(n+k,2*k): for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Table[2^k*Binomial[n + k, 2*k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 14 2017 *)
CoefficientList[Table[Sqrt[2] Cosh[(2 n + 1) ArcSinh[Sqrt[x/2]]]/Sqrt[2 + x], {n, 0, 10}] // FunctionExpand // Simplify, x] // Flatten (* Eric W. Weisstein, Apr 04 2018 *)
CoefficientList[Table[ChebyshevT[2 n - 1, Sqrt[1 + x/2]]/Sqrt[1 + x/2], {n, 10}], x] (* Eric W. Weisstein, Apr 04 2018 *)

for(n=0,10, for(k=0,n, print1(2^k*binomial(n+k,2*k), ", "))) \\ G. C. Greubel, Oct 14 2017

T(n,k) = 2^k * binomial(n+k,2*k).
G.f.: (1-z)/(1 - 2*z + z^2 - 2*t*z).
Sum_{k=0..n} k*T(n,k) = A123520(n) (n>=1).
Row polynomials are b(n,2*x), where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k) * x^k are the Morgan-Voyce polynomials of A085478. The triangle is made up of the odd-indexed rows of A211956. - Peter Bala, May 01 2012

Terms a(57) onward added by G. C. Greubel, Oct 14 2017

A131887 Number of Khalimsky-continuous functions with a three-point codomain.

Original entry on oeis.org

3, 5, 11, 19, 41, 71, 153, 265, 571, 989, 2131, 3691, 7953, 13775, 29681, 51409, 110771, 191861, 413403, 716035, 1542841, 2672279, 5757961, 9973081, 21489003, 37220045, 80198051, 138907099, 299303201, 518408351, 1117014753, 1934726305, 4168755811, 7220496869, 15558008491, 26947261171

Offset: 1

Shiva Samieinia (shiva(AT)math.su.se), Oct 05 2007, Oct 09 2007

Neo Scott, Table of n, a(n) for n = 1..1500
Shiva Samieinia, Digital straight line segments and curves. Licentiate Thesis. Stockholm University, Department of Mathematics, Report 2007:6.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A001045, A000213, A131935, A001834 (bisection), A001835 (bisection)

LinearRecurrence[{0,4,0,-1},{3,5,11,19},40] (* Harvey P. Dale, Jan 01 2017 *)

a(2k) = a(2k-1) + a(2k-2) + a(2k-3) and a(2k-1) = a(2k-2) + 2a(2k-3).
The asymptotic behavior is a(2k) = t(2k) sqrt(3)(2 + sqrt(3))^k, a(2k-1) = t(2k-1)(2 + sqrt(3))^k where t(n) tends to 1/2 + sqrt(3)/6.
G.f.: -x*(-3-5*x+x^2+x^3) / ( 1-4*x^2+x^4 ). - R. J. Mathar, Nov 08 2013

A236330 Positive integers n such that x^2 - 14xy + y^2 + n = 0 has integer solutions.

Original entry on oeis.org

32, 48, 128, 176, 192, 288, 368, 416, 432, 512, 624, 704, 752, 768, 800, 944, 1056, 1136, 1152, 1184, 1200, 1328, 1472, 1568, 1584, 1664, 1712, 1728, 1776, 1952, 2048, 2096, 2208, 2288, 2336, 2352, 2496, 2592, 2672, 2816, 2864, 2928, 3008, 3056, 3072, 3104

Offset: 1

Colin Barker, Feb 16 2014

nonn

			48 is in the sequence because x^2 - 14xy + y^2 + 48 = 0 has integer solutions, for example (x, y) = (2, 26).

Cf. A001835 (n = 32), A001075 (n = 48), A237250 (n = 176), A003500 (n = 192), A082841 (n = 288), A151961 (n = 432), A077238 (n = 624).

Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236331.

A324176 Integers k such that floor(sqrt(k)) + floor(sqrt(k/3)) divides k.

Original entry on oeis.org

1, 2, 6, 15, 18, 24, 32, 36, 45, 55, 72, 78, 84, 98, 105, 112, 136, 144, 152, 180, 198, 220, 230, 275, 336, 390, 403, 462, 525, 540, 608, 663, 697, 756, 774, 792, 836, 855, 874, 940, 980, 1050, 1092, 1144, 1166, 1265, 1368, 1392, 1500, 1525, 1586, 1638, 1755, 1782, 1848, 1904

Offset: 1

Jinyuan Wang, Mar 08 2019

nonn

This sequence is infinite for the same reason that A324175 is: if x-1 > y > 1 satisfies x^2 - 3*y^2 = -2 (x=A001834(j), y=A001835(j+1), j>0), then x < 3*y. Let k = 3*y^2 + m. By the pigeonhole principle there exists a number m belonging to [0, 2*x - 2] such that x + y | 3*y^2 + m, so such a k is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001834, A001835, A324175.

Select[Range[2000],Divisible[#,Floor[Sqrt[#]]+Floor[Sqrt[#/3]]]&] (* Harvey P. Dale, Jun 19 2021 *)

is(n) = n%(floor(sqrt(n)) + floor(sqrt(n/3))) == 0;

A121338 Pentagonal numbers P(k) that are one-third of other pentagonal numbers: P(k) such that 3*P(k)=P(m) for some m>k.

Original entry on oeis.org

70, 511258935, 3732600255368600, 27250975409595074561065, 198953975772318806945317308330, 1452523584226469439408576900215922395, 10604587088767577582197244731443261336155260, 77421990626847055423676582260371016672624778798925

Offset: 1

Franz Vrabec, Aug 28 2006

The k values are (A001835(6n-2)+1)/6, the m values are (A001834(6n-3)+1)/6.

			a(1) = ((A001835(4))^2-1)/24 = (41^2-1)/24 = 70; this number and 3*70=210 are pentagonal numbers (in A000326).

Colin Barker, Table of n, a(n) for n = 1..146
Index entries for linear recurrences with constant coefficients, signature (7300803,-7300803,1).

Cf. A001834, A001835.

CoefficientList[Series[5 (x^2 + 40545 x + 14)/((1 - x) (x^2 - 7300802 x + 1)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 21 2015 *)

Vec(-5*x*(x^2+40545*x+14)/((x-1)*(x^2-7300802*x+1)) + O(x^20)) \\ Colin Barker, Jun 20 2015

a(n) = ((A001835(6n-2))^2-1)/24.
a(n) = 7300803*a(n-1)-7300803*a(n-2)+a(n-3). - Colin Barker, Jun 20 2015
G.f.: -5*x*(x^2+40545*x+14) / ((x-1)*(x^2-7300802*x+1)). - Colin Barker, Jun 20 2015

Added more terms, Colin Barker, Jun 20 2015

cp's OEIS Frontend

A334178 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{k}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

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A103772 Larger of two sides in a (k,k,k-1)-integer-sided triangle with integer area.

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A147720 Riordan array (1, x(1-x)/(1-3x)).

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A147721 a(n) = C(2,n) DELTA C(0,n).

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A227418 Array A(n,k) with all numbers m such that 3*m^2 +- 3^k is a square and their corresponding square roots, read by downward antidiagonals.

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A123519 Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).

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Maple

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A131887 Number of Khalimsky-continuous functions with a three-point codomain.

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A334178 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 2^n * sqrt(Resultant(U_{2n}(x/2), T_{k}(ix/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).