| Displaying 1-9 of 9 results found.
page
1
1, 3, 11, 17, 33, 43, 67, 81, 113, 131, 171, 193, 241, 267, 323, 353, 417, 451, 523, 561, 641, 683, 771, 817, 913, 963, 1067, 1121, 1233, 1291, 1411, 1473, 1601, 1667, 1803, 1873, 2017, 2091, 2243, 2321, 2481, 2563, 2731, 2817, 2993, 3083, 3267, 3361, 3553, 3651
FORMULA
G.f.: ( -1-2*x-6*x^2-2*x^3-x^4 ) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Feb 28 2011 a(n) = 3*(1+2*n+2*n^2)/4 + (-1)^n*(1+2*n)/4. - R. J. Mathar, Feb 28 2011 Appears to be the sequence of exponents in the following series expansion:
Sum_{n >= 0} (-1)^n * x^n/Product_{k = 1..n} 1 - x^(2*k-1) = 1 - x - x^3 + x^11 + x^17 - x^33 - x^43 + + - - .... Cf. A053253. More generally, for nonnegative integer N, we appear to have the identity
Product_{j = 1..N} 1/(1 + x^(2*j-1))*( P(N,x) + Sum_{n >= 1} (-1)^n * x^((2*N+1)*n-N)/Product_{k = 1..n} 1 - x^(2*k-1) ) = 1 - x - x^3 + x^11 + x^17 - x^33 - x^43 + + - - ..., where P(N,x) is a polynomial in x of degree N^2 - 1, with the first few values given empirically by
P(0,x) = 0, P(1,x) = 1, P(2,x) = 1 - x^2 + x^3, P(3,x) = 1 - x^2 + x^5 - x^7 + x^8 and P(4,x) = 1 - x^2 - x^4 + x^5 + x^8 - x^9 + x^12 - x^14 + x^15. Cf. A203568. (End) E.g.f.: ((2 + 5*x + 3*x^2)*cosh(x) + (1 + 7*x + 3*x^2)*sinh(x))/2. - Stefano Spezia, May 08 2021
PROG
(Haskell)
a186424 n = a186424_list !! n
a186424_list = filter odd a186423_list
(Python)
def A186424(n): return (n*(3*n + 2) + 1 if n&1 else n*(3*n + 4) + 2)>>1 # Chai Wah Wu, Jan 31 2023
Irregular triangle read by rows in which row n lists 2n-1 copies of 2n-1 and n copies of 2n, for n >= 1.
+10
8
1, 2, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14
COMMENTS
Sequence of successive positive integers k in which if k is odd then k appears k times, otherwise if k is even then k appears k/2 times.
Note that an arrangement of the blocks of this sequence shows the growth of the generalized pentagonal numbers A001318 (see example). The sums of each block give the positive integers of A129194: 1, 2, 9, 8, 25, 18, 49,... Partial sums of A080995. - Paolo P. Lava, Aug 23 2011. Also a(n) = number of squares in the arithmetic progression {24k + 1: 0 <= k <= n-1} [Granville]. - N. J. A. Sloane, Dec 13 2017
FORMULA
a(n) = sqrt(8n/3) plus or minus 1 [Granville] - N. J. A. Sloane, Dec 13 2017
EXAMPLE
a) If written as a triangle the initial rows are
1, 2,
3, 3, 3, 4, 4,
5, 5, 5, 5, 5, 6, 6, 6,
7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8,
9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10,
...
b) An application using the blocks of this sequence: the illustration of the growth of an arrangement which represents the generalized pentagonal numbers A001318. For example; the first 9 positive initial terms: 1, 2, 5, 7, 12, 15, 22, 26, 35. .
. 9
. 8 9
. 8 7 9
. 8 6 7 9
. 8 6 5 7 9
. 6 4 5 7 9
. 4 3 5 7 9
. 2 3 5 7 9
. 1 3 5 7 9
...
MATHEMATICA
Array[Join @@ MapIndexed[ConstantArray[#, #/(1 + Boole[First@ #2 == 2])] &, {2 # - 1, 2 #}] &, 7] // Flatten (* or *)
Table[If[k <= 2 n - 1, 2 n - 1, 2 n], {n, 7}, {k, 3 n - 1}] // Flatten (* Michael De Vlieger, Dec 14 2017 *)
PROG
(Haskell)
a193832 n k = a193832_tabf !! (n-1) !! (k-1)
a193832_row n = a193832_tabf !! (n-1)
a193832_tabf = zipWith (++) a001650_tabf a111650_tabl
a193832' n = a193832_list !! (n - 1)
a193832_list = concat a193832_tabf
1, 2, 13, 34, 65, 106, 157, 218, 289, 370, 461, 562, 673, 794, 925, 1066, 1217, 1378, 1549, 1730, 1921, 2122, 2333, 2554, 2785, 3026, 3277, 3538, 3809, 4090, 4381, 4682, 4993, 5314, 5645, 5986, 6337, 6698, 7069, 7450, 7841, 8242, 8653, 9074
COMMENTS
For n >= 2, hypotenuses of primitive Pythagorean triangles with m = 2*n-1, where the sides of the triangle are a = m^2 - n^2, b = 2*n*m, c = m^2 + n^2; this sequence is the c values, short sides (a) are A045944(n-1), and long sides (b) are A002939(n).
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=2, a(2)=13.
G.f.: (1 - x + 10*x^2)/(1-x)^3. (End)
MATHEMATICA
Table[5*n^2 - 4*n + 1, {n, 0, 100}]
LinearRecurrence[{3, -3, 1}, {1, 2, 13}, 100] (* or *) CoefficientList[ Series[ (-10 x^2+x-1)/(x-1)^3, {x, 0, 100}], x] (* Harvey P. Dale, May 24 2011 *)
PROG
(SageMath) [5*n^2-4*n+1 for n in range(41)] # G. C. Greubel, Dec 03 2023
The number of different classes of 2-dimensional convex lattice polytopes having volume n/2 up to unimodular equivalence.
+10
6
1, 2, 3, 7, 6, 13, 13, 27, 26, 44, 43, 83, 81, 122, 136, 208, 215, 317, 341, 490, 542, 710, 778, 1073, 1186, 1519, 1708, 2178, 2405, 3042, 3408, 4247, 4785, 5782, 6438, 7870, 8833, 10560, 11857, 14131, 15733, 18636, 20773, 24381, 27353, 31764, 35284, 41081, 45791, 52762
COMMENTS
Lattice polytopes up to the equivalence relation used here are also called toric diagrams, see references below. - Andrey Zabolotskiy, May 10 2019 Liu & Zong give a(7) = 11, and others use their list, but their list lacks polygons No. 3 and 4 from Balletti's file 2-polytopes/v7.txt. - Andrey Zabolotskiy, Dec 28 2021
LINKS
Gabriele Balletti, Dataset of "small" lattice polytopes. Beware that the vertices are not always listed in sorted order around the polygon boundary (clockwise or counterclockwise).
EXTENSIONS
Name edited, a(7) corrected, a(9)-a(50) added using Balletti's data by Andrey Zabolotskiy, Dec 28 2021
a(n) = floor((3*n + 2)^2/24 + 1/3).
+10
5
0, 1, 3, 5, 8, 12, 17, 22, 28, 35, 43, 51, 60, 70, 81, 92, 104, 117, 131, 145, 160, 176, 193, 210, 228, 247, 267, 287, 308, 330, 353, 376, 400, 425, 451, 477, 504, 532, 561, 590, 620, 651, 683, 715, 748, 782, 817, 852, 888, 925, 963
COMMENTS
List of quadruples: 2*n*(3*n+1), (2*n+1)*(3*n+1), 6*n^2+8*n+3, (n+1)*(6*n+5). These terms belong to the sequences A033580, A033570, A126587 and A049452, respectively. See links for all the permutations. It seems that a(n) is the smallest number of cells that need to be painted in a (n+1) X (n+1) grid, such that it has no unpainted hexominoes (see link to Kamenetsky and Pratt). - Rob Pratt, Dmitry Kamenetsky, Aug 30 2020
FORMULA
G.f.: x*(1 + x + x^3)/((1 + x)*(1 + x^2)*(1 - x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>5.
a(n) = floor((3*n + 2)^2/24 + 2/3).
a(n) = (6*n^2 + 8*n + 3 + (-1)^n - 2*((-1)^((2*n - 1 + (-1)^n)/4) + (-1)^((2*n + 1 - (-1)^n)/4)))/16. Therefore:
a(2*k) = (6*k^2 + 4*k + 1 - (-1)^k)/4,
a(2*k+1) = (k + 1)*(3*k + 2)/2.
a(n) = (6*n^2 + 8*n + 3 + cos(n*Pi) - 4*cos(n*Pi/2))/16.
a(n) = (3*n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n)*(1 + 2*i^((n+1)*(n+2))))/16, where i=sqrt(-1).
EXAMPLE
Rectangular array with four columns:
. 0, 1, 3, 5;
. 8, 12, 17, 22;
. 28, 35, 43, 51;
. 60, 70, 81, 92;
. 104, 117, 131, 145, etc.
For n = 3, painting only 2 cells would leave an unpainted hexomino, but painting the following 3 cells avoids all unpainted hexominoes:
. . .
. . X
X X .
(End)
MATHEMATICA
Table[Floor[(3 n + 2)^2/24 + 1/3], {n, 0, 50}] (* or *) CoefficientList[Series[x (1 + x + x^3)/((1 + x) (1 + x^2) (1 - x)^3), {x, 0, 50}], x] (* or *) Table[(6 n^2 + 8 n + 3 + Cos[n Pi] - 4 Cos[n Pi/2])/16, {n, 0, 50}] (* or *) Table[(3 n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n) (1 + 2 I^((n + 1) (n + 2))))/16, {n, 0, 50}] (* Michael De Vlieger, Feb 17 2017 *) LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 1, 3, 5, 8, 12}, 60] (* Harvey P. Dale, Aug 10 2024 *)
PROG
(Magma) [(3*n^2+4*n+4) div 8: n in [0..50]]; // Bruno Berselli, Feb 17 2017
CROSSREFS
Cf. A033436: floor((3*n)^2/24 + 1/3). Cf. A000326, A000567, A025767, A033570, A033580, A049452, A064412, A126587, A222017, A269064, A274221.
Four-column table read by rows: row n is the unique primitive Pythagorean quadruple (a,b,c,d) such that a < (a + b + c - d)/2 = 2n(n + 1) and b = c.
+10
2
1, 12, 12, 17, 7, 30, 30, 43, 17, 56, 56, 81, 31, 90, 90, 131, 49, 132, 132, 193, 71, 182, 182, 267, 97, 240, 240, 353, 127, 306, 306, 451, 161, 380, 380, 561, 199, 462, 462, 683, 241, 552, 552, 817, 287, 650, 650, 963, 337, 756, 756, 1121, 391, 870, 870, 1291, 449, 992, 992, 1473
COMMENTS
A Pythagorean quadruple is a quadruple (a,b,c,d) of positive integers such that a^2 + b^2 + c^2 = d^2 with a <= b <= c. Its inradius is (a+b+c-d)/2, which is a positive integer.
REFERENCES
Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
FORMULA
Row n = (a, b, c, d) = (2n^2 - 1, 4n^2 + 6n + 2, 4n^2 + 6n + 2, 6n^2 + 8n + 3).
EXAMPLE
Table begins:
n=1: 1, 12, 12, 17;
n=2: 7, 30, 30, 43;
n=3: 17, 56, 56, 81;
n=4: 31, 90, 90, 131;
n=5: 49, 132, 132, 193;
MATHEMATICA
cuaternas={}; Do[cuaternas=Join[cuaternas, {2n^2-1, 4n^2+6n+2, 4n^2+6n+2, 6n^2+8n+3}], {n, 1, 35}]; cuaternas
Array T(j,k) of counts of internal lattice points within all Pythagorean triangles (see comments for array order).
+10
1
3, 22, 17, 49, 103, 43, 69, 217, 244, 81, 156, 305, 505, 445, 131, 187, 671, 709, 913, 706, 193, 190, 793, 1546, 1281, 1441, 1027, 267, 295, 799, 1819, 2781, 2021, 2089, 1408, 353, 465, 1249, 1828, 3265, 4376, 2929, 2857, 1849, 451, 498, 1937, 2863, 3277, 5131, 6331, 4005, 3745, 2350, 561
COMMENTS
The array of counts of internal lattice points within all Pythagorean triangles T(j,k) is arranged so that its first column is the ordered counts of internal lattice points within the k-th primitive Pythagorean triangle (PPT) A225414(k) and the j-th column is j multiples of these PPT side lengths. Let the k-th PPT have integer perpendicular sides a, b then its j-th multiple has area A = j^2*a*b/2 and the count of lattice points intersected by its boundary is B = j*(a+b+1) by the application of Pick's theorem the count of internal lattice points within it is I = (j^2*a*b-j*(a+b+1)+2)/2.
EXAMPLE
Array begins
3, 17, 43, 81, 131, ...
22, 103, 244, 445, ...
49, 217, 505, ...
69, 305, ...
156, ...
MATHEMATICA
getpairs[k_] := Reverse[Select[IntegerPartitions[k, {2}], GCD[#[[1]], #[[2]]]==1 &]]; getpptpairs[j_] := (newlist=getpairs[j]; Table[{(newlist[[m]][[1]]^2-newlist[[m]][[2]]^2-1)(2newlist[[m]][[1]]*newlist[[m]][[2]]-1)/2, newlist[[m]][[1]]^2-newlist[[m]][[2]]^2, 2newlist[[m]][[1]]*newlist[[m]][[2]]}, {m, 1, Length[newlist]}]); lexicographicLattice[{dim_, maxHeight_}] := Flatten[Array[Sort@Flatten[(Permutations[#1] &) /@ IntegerPartitions[#1 +dim-1, {dim}], 1] &, maxHeight], 1]; array[{x_, y_}] := (pptpair=table[[y]]; (x^2*pptpair[[2]]*pptpair[[3]])/2-x(pptpair[[2]]+pptpair[[3]]+1)/2+1); maxterms=20; table=Sort[Flatten[Table[getpptpairs[2p+1], {p, 1, maxterms}], 1]][[1;; maxterms]]; pairs=lexicographicLattice[{2, maxterms}]; Table[array[pairs[[n]]], {n, 1, maxterms(maxterms+1)/2}]
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=5, a(2)=9, a(3)=21, a(4)=29.
+10
1
1, 5, 9, 21, 29, 49, 61, 89, 105, 141, 161, 205, 229, 281, 309, 369, 401, 469, 505, 581, 621, 705, 749, 841, 889, 989, 1041, 1149, 1205, 1321, 1381, 1505, 1569, 1701, 1769, 1909, 1981, 2129, 2205, 2361
COMMENTS
The two bisections A136392(n+1)=1,9,29,61, ... and A201279(n)=5,21,49, ... are in the hexagonal spiral based on 2*n+1: .
67--65--63--61
/ \
69 33--31--29 59
/ / \ \
71 35 11---9 27 57
/ / / \ \ \
73 37 13 1 7 25 55
/ / / / / /
39 15 3---5 23 53
\ \ / /
41 17--19--21 51
\ /
43--45--47--49
.
FORMULA
a(-n) = a(n).
G.f.: (1 + x^2)*(1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^2). - Colin Barker, Jun 05 2019
MATHEMATICA
Table[(6 n^2 + 6 n + 5 - (2 n + 1)*(-1)^n)/4, {n, 0, 80}] (* Wesley Ivan Hurt, Jan 07 2021 *)
PROG
(PARI) Vec((1 + x^2)*(1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50)) \\ Colin Barker, Jun 05 2019 (Magma) [(6*n^2 + 6*n + 5 - (2*n + 1)*(-1)^n)/4 : n in [0..50]]; // Wesley Ivan Hurt, Jan 19 2021
Dimension of the space of weight 2n cusp forms for Gamma_0( 30 ).
+10
0
3, 14, 26, 38, 50, 62, 74, 86, 98, 110, 122, 134, 146, 158, 170, 182, 194, 206, 218, 230, 242, 254, 266, 278, 290, 302, 314, 326, 338, 350, 362, 374, 386, 398, 410, 422, 434, 446, 458, 470, 482, 494, 506, 518, 530, 542, 554, 566, 578, 590
FORMULA
G.f.: x*(3 + 8*x + x^2) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>3.
a(n) = 2*(6*n-5) for n>1.
(End)
CROSSREFS
Conjectured first differences of A126587.
Search completed in 0.009 seconds
| 
