Displaying 1-10 of 20 results found.
Number T(n,k) of partitions of n with k parts in which no part occurs more than twice; triangle T(n,k), n>=0, 0<=k<= A055086(n), read by rows.
+20
11
1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 5, 3, 0, 1, 4, 6, 4, 1, 0, 1, 5, 8, 6, 2, 0, 1, 5, 10, 8, 3, 0, 1, 6, 11, 12, 5, 1, 0, 1, 6, 14, 14, 8, 1, 0, 1, 7, 16, 19, 11, 3, 0, 1, 7, 18, 23, 16, 5
EXAMPLE
T(8,3) = 5: [6,1,1], [5,2,1], [4,3,1], [4,2,2], [3,3,2].
T(8,4) = 3: [4,2,1,1], [3,3,1,1], [3,2,2,1].
T(9,3) = 6: [7,1,1], [6,2,1], [5,3,1], [4,4,1], [5,2,2], [4,3,2].
T(9,4) = 4: [5,2,1,1], [4,3,1,1], [4,2,2,1], [3,3,2,1].
T(9,5) = 1: [3,2,2,1,1].
Triangle begins:
1;
0, 1;
0, 1, 1;
0, 1, 1;
0, 1, 2, 1;
0, 1, 2, 2;
0, 1, 3, 2, 1;
0, 1, 3, 4, 1;
0, 1, 4, 5, 3;
0, 1, 4, 6, 4, 1;
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(expand(b(n-i*j, i-1)*x^j), j=0..min(2, n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..20);
MATHEMATICA
max = 15; g = -1+Product[1+t*x^j+t^2*x^(2j), {j, 1, max}]; t[n_, k_] := SeriesCoefficient[g, {x, 0, n}, {t, 0, k}]; t[0, 0] = 1; Table[Table[t[n, k], {k, 0, n}] /. {a__, 0 ..} -> {a}, {n, 0, max}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)
CROSSREFS
Columns k=0-10 give: A000007, A057427, A004526, A230059 (conjectured), A320592, A320593, A320594, A320595, A320596, A320597, A320598.
0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 3, 0, 4, 2, 2, 2, 5, 1, 5, 2, 4, 4, 6, 0, 6, 5, 5, 3, 7, 2, 8, 4, 6, 6, 6, 2, 9, 7, 7, 3, 9, 4, 10, 6, 6, 8, 10, 2, 10, 7, 9, 7, 11, 5, 9, 6, 10, 10, 12, 2, 12, 10, 8, 8, 11, 7, 13, 9, 11, 7, 13, 4, 14, 12, 10, 10, 12, 8, 14, 6, 12, 13, 15, 5, 13, 13, 13, 9, 15, 6, 14
COMMENTS
It appears that the indices of the zeros in the sequence are in A018253. - Omar E. Pol, Oct 22 2013
EXAMPLE
n=15: sequence of D1 = {floor(15/j)} = {15,7,5,3,3,2,2,1,1,1,1,1,1,1,1}, Union(D1) = {15,7,5,3,2,1} = divisors(15) and {7,2}, a(15)=2 the number of terms beyond divisors.
MATHEMATICA
Table[Length[Union[Table[Floor[w/j], {j, 1, w}]]] -DivisorSigma[0, w], {w, 1, 128}]
1, 2, 4, 8, 20, 52, 140, 400, 1203, 3808, 12634, 43771, 157821, 590518, 2287070, 9148286, 37719353, 160029667, 697553152, 3119552536, 14295585635, 67052240163, 321571247089, 1575370942644, 7876854713228, 40164235888490, 208699491618146, 1104333907134455, 5947020092025820
MATHEMATICA
Table[ Length[ Union[ Table[ Floor[w!/j], {j, 1, w!}]]], {w, 1, 11}]
a[n_] := Floor[Sqrt[4*n! + 1]] - 1; Array[a, 30] (* Amiram Eldar, Aug 18 2024 *)
Triangle read by rows: T(n,k), n>=1, k>=1, in which row n lists m terms, where m = A055086(n). If k divides n and k < n^(1/2) then T(n,k) = k and T(n,m-k+1) = n/T(n,k). Also, if k^2 = n then T(n,k) = k. Other terms are zeros.
+20
4
1, 1, 2, 1, 3, 1, 2, 4, 1, 0, 5, 1, 2, 3, 6, 1, 0, 0, 7, 1, 2, 4, 8, 1, 0, 3, 0, 9, 1, 2, 0, 5, 10, 1, 0, 0, 0, 11, 1, 2, 3, 4, 6, 12, 1, 0, 0, 0, 0, 13, 1, 2, 0, 0, 7, 14, 1, 0, 3, 5, 0, 15, 1, 2, 0, 4, 0, 8, 16, 1, 0, 0, 0, 0, 0, 17, 1, 2, 3, 0, 6, 9, 18
COMMENTS
The number of positive terms of row n is A000005(n). The positive terms of row n are the divisors of n in increasing order.
Column k starts in row A002620(k+1). The number of zeros in row n equals A078152(n). It appears that there are only eight rows that do not contain zeros. The indices of these rows are 1, 2, 3, 4, 6, 8, 12, 24, the divisors of 24, see A018253.
EXAMPLE
For n = 60 the 60th row of triangle is [1, 2, 3, 4, 5, 6, 0, 0, 10, 12, 15, 20, 30, 60]. The row length is A055086(60) = 14. The number of zeros is A078152(60) = 2. The number of positive terms is A000005(60) = 12. The positive terms are the divisors of 60. The row sum is A000203(60) = 168. Triangle begins:
1;
1, 2;
1, 3;
1, 2, 4;
1, 0, 5;
1, 2, 3, 6;
1, 0, 0, 7;
1, 2, 4, 8;
1, 0, 3, 0, 9;
1, 2, 0, 5, 10;
1, 0, 0, 0, 11;
1, 2, 3, 4, 6, 12;
1, 0, 0, 0, 0, 13;
1, 2, 0, 0, 7, 14;
1, 0, 3, 5, 0, 15;
1, 2, 0, 4, 0, 8, 16;
1, 0, 0, 0, 0, 0, 17;
1, 2, 3, 0, 6, 9, 18;
1, 0, 0, 0, 0, 0, 19;
1, 2, 0, 4, 5, 0, 10, 20;
1, 0, 3, 0, 0, 7, 0, 21;
1, 2, 0, 0, 0, 0, 11, 22;
1, 0, 0, 0, 0, 0, 0, 23;
1, 2, 3, 4, 6, 8, 12, 24;
...
CROSSREFS
Cf. A000005, A000203, A002620, A004526, A018253, A027750, A055086, A078152, A127093, A147861, A161904, A196020, A210959, A212119, A212120, A221645, A228813, A228814.
2, 3, 4, 7, 10, 15, 21, 31, 44, 63, 89, 127, 180, 255, 361, 511, 723, 1023, 1447, 2047, 2895, 4095, 5791, 8191, 11584, 16383, 23169, 32767, 46339, 65535, 92680, 131071, 185362, 262143, 370726, 524287, 741454, 1048575, 1482909, 2097151, 2965819, 4194303, 5931640, 8388607
COMMENTS
a(2n-2)=2^n-1 and a(n)=2a(n-2)+1 generally.
MATHEMATICA
Table[ Length[ Union[ Table[ Floor[2^w/j], {j, 1, 2^w}]]], {w, 1, 25}]
PROG
(PARI) a(n) = ceil(2*sqrt(2^n+1)) - 2; \\ Michel Marcus, Sep 02 2019
2, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 19, 19, 20, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 29, 29, 30, 30, 31, 31, 31, 31, 32, 32, 32, 33, 34, 34, 34, 34, 35, 35, 36, 36, 36, 36
EXAMPLE
n=25:p(25)=97, sequence Union[Floor[97/j]]= {1,2,3,4,5,6,7,8,9,10,12,13,16,19,24,32,48,97}, a[25]=18.
MATHEMATICA
Table[Length[Union[Table[Floor[Prime[w]/j], {j, 1, Prime[w]}]]], {w, 1, 100}]
Integer part of square root of 4n+1.
+10
25
1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17
COMMENTS
1^1, 2^1, 3^2, 4^2, 5^3, 6^3, 7^4, 8^4, 9^5, 10^5, ...
Start with n, repeatedly subtract the square root of the previous term; a(n) gives number of steps to reach 0. - Robert G. Wilson v, Jul 22 2002 Every number k is present consecutively (floor((2*k+3)/4)) times. - Bernard Schott, Jun 08 2019
REFERENCES
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 20.
Bruce C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, 1994, see p. 77, Entry 23.
LINKS
S. Ramanujan, Question 723, J. Ind. Math. Soc., Vol. 7 (1915), p. 240, Vol. 10 (1918), pp. 357-358.
FORMULA
a(n) = 1 + a(n - floor(n^(1/2))), if n>0. - Michael Somos, Jul 22 2002 a(n) = floor( 1 / ( sqrt(n + 1) - sqrt(n) ) ). - Robert A. Stump (bob_ess107(AT)yahoo.com), Apr 07 2003
a(n) = ceiling(2*sqrt(n+1) - 1). - Mircea Merca, Feb 03 2012 a(n) = floor( sqrt(4*n + k) ) where k = 1, 2, or 3. - Michael Somos, Mar 11 2015 G.f.: (Sum_{k>0} x^floor(k^2 / 4)) / (1 - x). - Michael Somos, Mar 11 2015 a(n) = floor(sqrt(n+1)+1/2) + floor(sqrt(n)). - Ridouane Oudra, Jun 07 2019 Sum_{k>=0} (-1)^k/a(k) = Pi/8 + log(2)/4. - Amiram Eldar, Jan 26 2024
EXAMPLE
1;
2, 3;
3, 4, 5;
4, 5, 6, 7;
5, 6, 7, 8, 9;
6, 7, 8, 9, 10, 11; ...
Read by diagonals:
1;
2;
3, 3;
4, 4;
5, 5, 5;
6, 6, 6;
7, 7, 7, 7;
8, 8, 8, 8;
9, 9, 9, 9, 9;
10, 10, 10, 10, 10; (End)
G.f. = 1 + 2*x + 3*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 5*x^6 + 5*x^7 + 5*x^8 + 6*x^9 + ...
MATHEMATICA
Table[Floor[Sqrt[4*n + 1]], {n, 0, 100}] (* T. D. Noe, Jun 19 2012 *)
PROG
(PARI) {a(n) = if( n<0, 0, sqrtint(4*n + 1))};
(Haskell)
Triangle read by rows T(n,k) in which column k lists 1's interleaved with A004526(k-1) zeros starting from the row A002620(k+1), with n>=1, k>=1.
+10
13
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0
COMMENTS
The sum of row n equals the number of divisors of n.
The number of zeros in row n equals A078152(n). It appears that there are only eight rows that do not contain zeros. The indices of these rows are 1, 2, 3, 4, 6, 8, 12, 24, the divisors of 24, see A018253. It appears that A066522 gives the indices of the rows in which the elements are in nonincreasing order.
EXAMPLE
For n = 10, row 10 is [1, 1, 1, 1, 0], and the sum of row 10 is 1+1+1+1+0 = 4. On the other hand, 10 has four divisors: 1, 2, 5, and 10. Note that the sum of row 10 is also A000005(10) = 4, the number of divisors of 10. Triangle begins:
1;
1, 1;
1, 1;
1, 1, 1;
1, 1, 0;
1, 1, 1, 1;
1, 1, 0, 0;
1, 1, 1, 1;
1, 1, 0, 0, 1;
1, 1, 1, 1, 0;
1, 1, 0, 0, 0;
1, 1, 1, 1, 1, 1;
1, 1, 0, 0, 0, 0;
1, 1, 1, 1, 0, 0;
1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 0, 0, 1;
1, 1, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 0;
1, 1, 0, 0, 0, 0, 0;
1, 1, 1, 1, 0, 0, 1, 1;
1, 1, 0, 0, 1, 1, 0, 0;
1, 1, 1, 1, 0, 0, 0, 0;
1, 1, 0, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, 1;
...
CROSSREFS
Cf. A002620, A004526, A018253, A027750, A066522, A078152, A147861, A161904, A196020, A210959, A212119, A212120, A221645, A228812, A228814.
Triangle T(n,k) read by rows: number of 1-dimensional sandpiles (see A186085) with n grains and base length k.
+10
6
1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 1, 6, 5, 1, 0, 0, 0, 0, 1, 4, 10, 6, 1, 0, 0, 0, 0, 0, 3, 10, 15, 7, 1, 0, 0, 0, 0, 0, 2, 8, 20, 21, 8, 1, 0, 0, 0, 0, 0, 1, 7, 19, 35, 28, 9, 1, 0, 0, 0, 0, 0, 0, 5, 18, 40, 56, 36, 10, 1, 0, 0, 0, 0, 0, 0, 3, 16, 41, 76, 84, 45, 11, 1, 0, 0, 0, 0, 0, 0, 1, 12, 41, 86, 133, 120, 55, 12, 1
COMMENTS
Compositions of n into k nonzero parts such that the first and last parts are 1 and the absolute difference between consecutive parts is <=1.
Column sums are the Motzkin numbers ( A001006). First nonzero entry in row n appears in column A055086(n). The transposed triangle (with zeros omitted) is A129181. For large k, the columns end in reverse([1, 1, 3, 5, 9, 14, 24, 35, ...]) for k even (cf. A053993) and reverse([1, 2, 3, 6, 10, 16, 26, 40, 60, 90, ...]) for k odd (cf. A201077). The diagonals below the main diagonal are (apart from leading zeros), n, n*(n+1)/2, n*(n+1)*(n+2)/6, and the e-th diagonal appears to have a g.f. of the form f(x)/(1-x)^e.
(End)
FORMULA
G.f. A(x,y) satisfies: A(x,y) = 1/(1 - x*y - x^3*y^2*A(x, x*y) ). [ Paul D. Hanna, Feb 22 2011] G.f.: (formatting to make the structure apparent)
A(x,y) = 1 /
(1 - x^1*y / (1 - x^2*y / (1 - x^5*y^2 /
(1 - x^3*y / (1 - x^4*y / (1 - x^9*y^2 /
(1 - x^5*y / (1 - x^6*y / (1 - x^13*y^2 /
(1 - x^7*y / (1 - x^8*y / (1 - x^17*y^2 / (1 -...)))))))))))))
G.f.: A(x,y) = 1/(1-x*y - x^3*y^2/(1-x^2*y - x^5*y^2/(1-x^3*y - x^7*y^2/(1 -...)))) (continued fraction). [ Paul D. Hanna]
EXAMPLE
Triangle begins:
1;
0,1;
0,0,1;
0,0,1,1;
0,0,0,2,1;
0,0,0,1,3,1;
0,0,0,0,3,4,1;
0,0,0,0,1,6,5,1;
0,0,0,0,1,4,10,6,1;
0,0,0,0,0,3,10,15,7,1;
0,0,0,0,0,2,8,20,21,8,1;
0,0,0,0,0,1,7,19,35,28,9,1;
MAPLE
b:= proc(n, i) option remember; `if`(n=0, `if`(i=1, 1, 0),
`if`(n<0 or i<1, 0, expand(x*add(b(n-i, i+j), j=-1..1)) ))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, If[i == 1, 1, 0], If[n<0 || i<1, 0, Expand[ x*Sum[b[n-i, i+j], {j, -1, 1}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
PROG
(PARI) {T(n, k)=local(A=1+x*y); for(i=1, n, A=1/(1-x*y-x^3*y^2*subst(A, y, x*y+x*O(x^n)))); polcoeff(polcoeff(A, n, x), k, y)}
Triangle read by rows T(n,k), n>=1, k>=1, in which column k starts in row A002620(k+1). If k is odd the column k lists j's interleaved with (k-1)/2 zeros, where j = (k+1)/2. Otherwise, if k is even the column k lists the positive integers but starting from k/2+1, interleaved with (k-2)/2 zeros.
+10
6
1, 1, 2, 1, 3, 1, 4, 2, 1, 5, 0, 1, 6, 2, 3, 1, 7, 0, 0, 1, 8, 2, 4, 1, 9, 0, 0, 3, 1, 10, 2, 5, 0, 1, 11, 0, 0, 0, 1, 12, 2, 6, 3, 4, 1, 13, 0, 0, 0, 0, 1, 14, 2, 7, 0, 0, 1, 15, 0, 0, 3, 5, 1, 16, 2, 8, 0, 0, 4, 1, 17, 0, 0, 0, 0, 0, 1, 18, 2, 9, 3, 6, 0
COMMENTS
The number of positive terms of row n is A000005(n). The positive terms of row n are the divisors of n.
The number of zeros in row n equals A078152(n). It appears that there are only eight rows that do not contain zeros. The indices of these rows are 1, 2, 3, 4, 6, 8, 12, 24, the divisors of 24, see A018253.
EXAMPLE
For n = 60 the 60th row of triangle is [1, 60, 2, 30, 3, 20, 4, 15, 5, 12, 6, 10, 0, 0]. The row length is A055086(60) = 14. The number of zeros is A078152(60) = 2. The number of positive terms is A000005(60) = 12. The row sum is A000203(60) = 168. Triangle begins:
1;
1, 2;
1, 3;
1, 4, 2;
1, 5, 0;
1, 6, 2, 3;
1, 7, 0, 0;
1, 8, 2, 4;
1, 9, 0, 0, 3;
1, 10, 2, 5, 0;
1, 11, 0, 0, 0;
1, 12, 2, 6, 3, 4;
1, 13, 0, 0, 0, 0;
1, 14, 2, 7, 0, 0;
1, 15, 0, 0, 3, 5;
1, 16, 2, 8, 0, 0, 4;
1, 17, 0, 0, 0, 0, 0;
1, 18, 2, 9, 3, 6, 0;
1, 19, 0, 0, 0, 0, 0;
1, 20, 2, 10, 0, 0, 4, 5;
1, 21, 0, 0, 3, 7, 0, 0;
1, 22, 2, 11, 0, 0, 0, 0;
1, 23, 0, 0, 0, 0, 0, 0;
1, 24, 2, 12, 3, 8, 4, 6;
...
CROSSREFS
Cf. A000005, A000203, A002620, A004526, A018253, A027750, A055086, A078152, A147861, A161904, A196020, A210959, A212119, A212120, A221645, A228812, A228813, A229940, A229942, A228944, A229950, A228951.
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