Search: a093178 -id:a093178
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1, 2, 3, 6, 7, 12, 13, 20, 21, 30, 31, 42, 43, 56, 57, 72, 73, 90, 91, 110, 111, 132, 133, 156, 157, 182, 183, 210, 211, 240, 241, 272, 273, 306, 307, 342, 343, 380, 381, 420, 421, 462, 463, 506, 507, 552, 553, 600, 601, 650, 651, 702, 703, 756, 757, 812, 813
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OFFSET
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1,2
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COMMENTS
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Numbers n such that floor(n/2) is a positive triangular number. - Bruno Berselli, Sep 15 2014
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LINKS
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FORMULA
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a(1) = 1; for n > 1, a(n) = a(n-1)+1 if n is odd, a(n) = a(n-1)+(n-1) if n is even.
a(n) = A002061((n+1)/2) = (n^2+3)/4 if n is odd, a(n) = A002378(n/2) = (n^2+2*n)/4) if n is even.
G.f.: x*(1+x-x^2+x^3)/((1-x)^3*(1+x)^2).
a(1) = 1; a(n) = a(n-1) + n^(n mod 2) = (1/4)*(n^2 + 2n + 4 + (n mod 2)*(2n-1)). - Rolf Pleisch, Feb 04 2008
a(n) = (2*(n-1)*(n+2) + (2*n-3)*(-1)^n+7)/8. - Bruno Berselli, Mar 31 2011
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MATHEMATICA
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Accumulate[Join[{1}, Riffle[Range[1, 85, 2], 1]]] (* or *) LinearRecurrence[ {1, 2, -2, -1, 1}, {1, 2, 3, 6, 7}, 90] (* Harvey P. Dale, Jun 01 2016 *)
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PROG
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(PARI) {s=0; for(n=1, 57, s=s+if(n%2>0, 1, n-1); print1(s, ", "))}
(PARI) {for(n=1, 57, print1(if(n%2>0, (n^2+3)/4, (n^2+2*n)/4), ", "))}
(Magma) &cat[ [ n^2-n+1, n*(n+1) ]: n in [1..29] ];
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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1, 1, 1, 1, 2, 3, 1, 3, 9, 1, 1, 4, 18, 4, 5, 1, 5, 30, 10, 25, 1, 1, 6, 45, 20, 75, 6, 7, 1, 7, 63, 35, 175, 21, 49, 1, 1, 8, 84, 56, 350, 56, 196, 8, 9, 1, 9, 108, 84, 630, 126, 588, 36, 81, 1
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OFFSET
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0,5
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COMMENTS
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Row sums = A133936: (1, 2, 6, 14, 32, 72, 160, ...).
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LINKS
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FORMULA
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Binomial transform of an infinite lower triangular matrix with A093178: (1, 1, 3, 1, 5, 1, 7, ...) in the main diagonal and the rest zeros.
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EXAMPLE
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First few rows of the triangle:
1;
1, 1;
1, 2, 3;
1, 3, 9, 1;
1, 4, 18, 4, 5;
1, 5, 30, 10, 25, 1;
1, 6, 45, 20, 75, 6, 7;
1, 7, 63, 35, 175, 21, 49, 1;
...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A133080
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Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
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+10
29
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1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1
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OFFSET
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1,1
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COMMENTS
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A133080 * [1,2,3,...] = A114753: (1, 3, 3, 7, 5, 11, 7, 15, ...).
Inverse of A133080: subdiagonal changes to (-1, 0, -1, 0, -1, ...); main diagonal unchanged.
A133080^(-1) * [1,2,3,...] = A093178: (1, 1, 3, 1, 5, 1, 7, 1, 9, ...).
In A133081, diagonal terms are switched with subdiagonal terms.
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LINKS
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FORMULA
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Infinite lower triangular matrix, (1,1,1,...) in the main diagonal and (1,0,1,0,1,...) in the subdiagonal.
Odd rows, (n-1) zeros followed by "1". Even rows, (n-2) zeros followed by "1, 1".
T(n,n)=1. T(n,k)=0 if 1 <= k < n-1. T(n,n-1)=1 if n even. T(n,n-1)=0 if n odd. - R. J. Mathar, Feb 14 2015
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EXAMPLE
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First few rows of the triangle are:
1;
1, 1;
0, 0, 1;
0, 0, 1, 1;
0, 0, 0, 0, 1;
0, 0, 0, 0, 1, 1;
0, 0, 0, 0, 0, 0, 1;
...
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MAPLE
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if n = k then
1;
elif k=n-1 and type(n, even) then
1;
else
0 ;
end if;
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MATHEMATICA
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T[n_, k_] := If[k == n, 1, If[k == n - 1, (1 + (-1)^n)/2 , 0]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)
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PROG
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(PARI) T(n, k) = if (k==n, 1, if (k == (n-1), 1 - (n % 2), 0)); \\ Michel Marcus, Feb 13 2014
(PARI) firstrows(n) = {my(res = vector(binomial(n + 1, 2)), t=0); for(i=1, n, t+=i; res[t] = 1; if(i%2==0, res[t-1]=1)) ; res} \\ David A. Corneth, Oct 21 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A049471
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Decimal expansion of tan(1).
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+10
14
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1, 5, 5, 7, 4, 0, 7, 7, 2, 4, 6, 5, 4, 9, 0, 2, 2, 3, 0, 5, 0, 6, 9, 7, 4, 8, 0, 7, 4, 5, 8, 3, 6, 0, 1, 7, 3, 0, 8, 7, 2, 5, 0, 7, 7, 2, 3, 8, 1, 5, 2, 0, 0, 3, 8, 3, 8, 3, 9, 4, 6, 6, 0, 5, 6, 9, 8, 8, 6, 1, 3, 9, 7, 1, 5, 1, 7, 2, 7, 2, 8, 9, 5, 5, 5, 0, 9, 9, 9, 6, 5, 2, 0, 2, 2, 4, 2, 9, 8
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Equals Sum_{k>=1} (-1)^(k+1) * B(2*k) * 2^(2*k) * (2^(2*k) - 1) / (2*k)!, where B(k) is the k-th Bernoulli number. - Amiram Eldar, May 15 2021
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EXAMPLE
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1.5574077246549022305...
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MATHEMATICA
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RealDigits[Tan[1], 10, 100][[1]] (* Amiram Eldar, May 15 2021 *)
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PROG
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(PARI) default(realprecision, 20080); x=tan(1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b049471.txt", n, " ", d)); \\
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A243366
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Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2)-1), read by rows.
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+10
13
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1, 1, 2, 5, 13, 1, 37, 5, 112, 19, 1, 352, 70, 7, 1136, 259, 34, 1, 3742, 962, 149, 9, 12529, 3585, 627, 54, 1, 42513, 13399, 2584, 279, 11, 145868, 50201, 10529, 1334, 79, 1, 505234, 188481, 42606, 6092, 474, 13, 1764157, 709001, 171563, 27048, 2561, 109, 1
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OFFSET
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0,3
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COMMENTS
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Conjecture: Generally, column k is asymptotic to c(k) * d^n * n^(k-3/2), where d = 3.8821590268628506747194368909643384... is the root of the equation d^8 - 2*d^7 - 10*d^6 + 12*d^5 - 5*d^4 - 2*d^3 - 5*d^2 - 8*d - 3 = 0, and c(k) are specific constants (independent on n). - Vaclav Kotesovec, Jun 05 2014
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LINKS
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EXAMPLE
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T(4,1) = 1: UDUUDUDD.
T(5,1) = 5: UDUDUUDUDD, UDUUDUDDUD, UDUUDUDUDD, UDUUDUUDDD, UUDUUDUDDD.
T(6,1) = 19: UDUDUDUUDUDD, UDUDUUDUDDUD, UDUDUUDUDUDD, UDUDUUDUUDDD, UDUUDUDDUDUD, UDUUDUDDUUDD, UDUUDUDUDDUD, UDUUDUDUDUDD, UDUUDUDUUDDD, UDUUDUUDDDUD, UDUUDUUDDUDD, UDUUDUUUDDDD, UUDDUDUUDUDD, UUDUDUUDUDDD, UUDUUDUDDDUD, UUDUUDUDDUDD, UUDUUDUDUDDD, UUDUUDUUDDDD, UUUDUUDUDDDD.
T(6,2) = 1: UDUUDUUDUDDD.
T(7,2) = 7: UDUDUUDUUDUDDD, UDUUDUDUUDUDDD, UDUUDUUDUDDDUD, UDUUDUUDUDDUDD, UDUUDUUDUDUDDD, UDUUDUUDUUDDDD, UUDUUDUUDUDDDD.
T(8,3) = 1: UDUUDUUDUUDUDDDD.
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 5;
: 4 : 13, 1;
: 5 : 37, 5;
: 6 : 112, 19, 1;
: 7 : 352, 70, 7;
: 8 : 1136, 259, 34, 1;
: 9 : 3742, 962, 149, 9;
: 10 : 12529, 3585, 627, 54, 1;
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MAPLE
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b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 5, 2, 4][t])*
`if`(t=6, z, 1) +b(x-1, y-1, [1, 3, 1, 3, 6, 1][t]))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..20);
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MATHEMATICA
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b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, {2, 2, 4, 5, 2, 4}[[t]]]*If[t == 6, z, 1] + b[x-1, y-1, {1, 3, 1, 3, 6, 1}[[t]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)
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CROSSREFS
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Column k=0-10 give: A243412, A243413, A243414, A243415, A243416, A243417, A243418, A243419, A243420, A243421, A243422.
T(n,floor(n/2)-1) gives A093178(n) for n>3.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A124625
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Even numbers sandwiched between 1's.
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+10
11
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1, 0, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12, 1, 14, 1, 16, 1, 18, 1, 20, 1, 22, 1, 24, 1, 26, 1, 28, 1, 30, 1, 32, 1, 34, 1, 36, 1, 38, 1, 40, 1, 42, 1, 44, 1, 46, 1, 48, 1, 50, 1, 52, 1, 54, 1, 56, 1, 58, 1, 60, 1, 62, 1, 64, 1, 66, 1, 68, 1, 70, 1, 72, 1, 74, 1, 76, 1, 78, 1, 80, 1, 82, 1, 84
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OFFSET
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0,4
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COMMENTS
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Created to simplify the definition of A129952.
Starting (1, 2, 1, 4,...): square (1 + x - x^2 - x^3 + x^4 + x^5 - ...) = (1 + 2x - x^2 - 4x^3 + x^4 + 6x^5 - ...).
With a(3) taken as 0, a(n+2) = n^k+1 mod 2*n, n>=1, for any k>=2, also for k=n. - Wolfdieter Lang, Dec 21 2011
Greatest common divisor of n-1 and (n-1) mod 2. - Bruno Berselli, Mar 07 2017
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REFERENCES
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Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.
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LINKS
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FORMULA
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a(n) = 1 for even n, a(n) = n-1 for odd n.
a(2*k) = 1, a(2*k+1) = 2*k.
G.f.: (1 - x^2 + 2*x^3)/((1 - x)^2*(1 + x)^2).
E.g.f.: 1 + x^2*U(0)/2 where U(k)= 1 + 2*x*(k+1)/(2*k + 3 - x*(2*k+3)/(x + 4*(k+2)*(k+1)/U(k+1)) (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Oct 20 2012
a(n) = (n-1) - a(a(n-1))*a(n-1), a(0) = 0. - Eli Jaffe, Jun 07 2016
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MAPLE
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MATHEMATICA
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PROG
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(PARI) {for(n=0, 85, print1(if(n%2>0, n-1, 1), ", "))}
(Magma) &cat[[1, 2*k]: k in [0..42]];
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A241269
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Denominator of c(n) = (n^2+n+2)/((n+1)*(n+2)*(n+3)).
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+10
8
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3, 6, 15, 60, 105, 21, 126, 360, 495, 330, 429, 1092, 1365, 420, 1020, 2448, 2907, 1710, 1995, 4620, 5313, 759, 3450, 7800, 8775, 4914, 5481, 12180, 13485, 3720, 8184, 17952, 19635, 10710, 11655, 25308, 27417, 3705, 15990, 34440, 37023, 19866, 21285, 45540
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OFFSET
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0,1
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COMMENTS
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All terms are multiples of 3.
Difference table of c(n):
1/3, 1/6, 2/15, 7/60, 2/21,...
-1/6, -1/30, -1/60, -1/84, -1/105,...
2/15, 1/60, 1/210, 1/420, 1/630,...
-7/60, -1/84, -1/420, -1/1260, -1/2520,... .
This is an autosequence of the second kind; the inverse binomial transform is the signed sequence. The main diagonal is the first upper diagonal multiplied by 2.
Denominators of the main diagonal: A051133(n+1).
Denominators of the first upper diagonal; A000911(n).
Based on the Akiyama-Tanigawa transform applied to 1/(n+1) which yields the Bernoulli numbers A164555(n)/A027642(n).
Are the numerators of the main diagonal (-1)^n? If yes, what is the value of 1/3 - 1/30 + 1/210,... or 1 - 1/10 + 1/70 - 1/420, ... , from A002802(n)?
Is a(n+40) - a(n) divisible by 10?
Are the common divisors to A014206(n) and A007531(n+3) of period 16: repeat 2, 4, 4, 2, 2, 16, 4, 2, 2, 4, 4, 2, 2, 8, 4, 2?
Reduce c(n) = f(n) = b(n)/a(n) = 1/3, 1/6, 2/15, 7/60, 11/105, 2/21, 11/126, 29/360, ... .
Consider the successively interleaved autosequences (also called eigensequences) of the second kind and of the first kind
1, 1/2, 1/3, 1/4, 1/5, 1/6, ...
0, 1/6, 1/6, 3/20, 2/15, 5/42, ...
1/3, 1/6, 2/15, 7/60, 11/105, 2/21, ...
0, 1/10, 1/10, 13/140, 3/35, 5/63, ...
1/5, 1/10, 3/35, 11/140, 23/315, 43/630, ...
0, 1/14, 1/14, 17/252, 4/63, ...
This array is Au1(m,n). Au1(0,0)=1, Au1(0,1)=1/2.
Au1(m+1,n) = 2*Au1(m,n+1) - Au1(m,n).
First row: see A003506, Leibniz's Harmonic Triangle.
a(n) is the denominator of the third row f(n).
The first column is 1, 0, 1/3, 0, 1/5, 0, 1/7, 0, ... . Numerators: A093178(n+1). This incites, considering tan(1), to introduce before the first row
Ta0(n) = 0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, ... .
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LINKS
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FORMULA
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The sum of the difference table main diagonal is 1/3 - 1/30 + 1/210 - ... = 10*A086466-4 = 4*(sqrt(5)*log(phi)-1) = 0.3040894... - Jean-François Alcover, Apr 22 2014
a(n) = (n+1)*(n+2)*(n+3)/gcd(4*n - 4, n^2 + n + 2), where gcd(4*n - 4, n^2 + n + 2) is periodic with period 16. - Robert Israel, Jul 17 2023
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MAPLE
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seq(denom((n^2+n+2)/((n+1)*(n+2)*(n+3))), n=0..1000);
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MATHEMATICA
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Denominator[Table[(n^2+n+2)/Times@@(n+{1, 2, 3}), {n, 0, 50}]] (* Harvey P. Dale, Mar 27 2015 *)
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PROG
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(PARI) for(n=0, 100, print1(denominator((n^2+n+2)/((n+1)*(n+2)*(n+3))), ", ")) \\ Colin Barker, Apr 18 2014
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A009001
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Expansion of e.g.f: (1+x)*cos(x).
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+10
6
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1, 1, -1, -3, 1, 5, -1, -7, 1, 9, -1, -11, 1, 13, -1, -15, 1, 17, -1, -19, 1, 21, -1, -23, 1, 25, -1, -27, 1, 29, -1, -31, 1, 33, -1, -35, 1, 37, -1, -39, 1, 41, -1, -43, 1, 45, -1, -47, 1, 49, -1, -51, 1, 53, -1, -55, 1, 57, -1, -59, 1, 61, -1, -63, 1, 65, -1, -67, 1, 69, -1, -71, 1, 73, -1, -75, 1, 77, -1, -79, 1, 81, -1, -83, 1, 85
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OFFSET
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0,4
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COMMENTS
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If signs are ignored, continued fraction for tan(1) (cf. A093178).
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LINKS
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FORMULA
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a(n) = (-1)^(n/2) if n is even, n*(-1)^((n-1)/2) if n is odd.
a(n) = -a(n-2) if n is even, 2*a(n-1) - a(n-2) if n is odd. - Michael Somos, Jan 26 2014
a(n) = (n^n mod (n+1))*(-1)^floor(n/2) for n > 0.
a(n) = (-1)^n*(a(n-2) - a(n-1)) - a(n-3) for n > 2. (End)
G.f.: (1+x+x^2-x^3)/(1+x^2)^2.
a(0) = 1; for n > 1,
a(n) = a(n-1) * n if n mod 4 = 1,
a(n-1) - n if n mod 4 = 2,
a(n-1) * n if n mod 4 = 3,
a(n-1) + n if n mod 4 = 4. (End)
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EXAMPLE
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tan(1) = 1.557407724654902230... = 1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + ...)))). - Harry J. Smith, Jun 15 2009
G.f. = 1 + x - x^2 - 3*x^3 + x^4 + 5*x^5 - x^6 - 7*x^7 + x^8 + 9*x^9 - x^10 + ...
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MAPLE
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seq(coeff(series(factorial(n)*(1+x)*cos(x), x, n+1), x, n), n=0..90); # Muniru A Asiru, Jul 21 2018
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MATHEMATICA
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With[{nn=90}, CoefficientList[Series[(1+x)Cos[x], {x, 0, nn}], x]Range[0, nn]!] (* Harvey P. Dale, Jul 15 2012 *)
FoldList[If[Mod[#2, 4]==1, #1*#2, If[Mod[#2, 4]==2, #1-#2, If[Mod[#2, 4] ==3, #1*#2, #1+#2]]]&, 1, Range[1, 85]] (* James C. McMahon, Oct 12 2023 *)
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PROG
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(PARI) {a(n) = (-1)^(n\2) * if( n%2, n, 1)} /* Michael Somos, Oct 16 2006 */
(PARI) { allocatemem(932245000); default(realprecision, 79000); x=contfrac(tan(1)); for (n=0, 20000, write("b009001.txt", n, " ", (-1)^(n\2)*x[n+1])); } \\ Harry J. Smith, Jun 15 2009
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Cos(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 21 2018
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CROSSREFS
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KEYWORD
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sign,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A224344
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Number T(n,k) of compositions of n using exactly k primes (counted with multiplicity); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.
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+10
6
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1, 1, 1, 1, 1, 3, 2, 5, 1, 3, 8, 5, 5, 13, 13, 1, 7, 23, 27, 7, 11, 39, 52, 25, 1, 17, 65, 99, 66, 9, 27, 106, 186, 151, 41, 1, 40, 177, 340, 323, 133, 11, 61, 293, 608, 666, 358, 61, 1, 92, 482, 1076, 1330, 867, 236, 13, 142, 781, 1894, 2581, 1971, 737, 85, 1
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OFFSET
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0,6
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LINKS
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FORMULA
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Sum_{k=1..floor(n/2)} k * T(n,k) = A102291(n).
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EXAMPLE
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A(5,1) = 8: [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2], [3,1,1], [1,3,1], [1,1,3], [5].
Triangle T(n,k) begins:
1;
1;
1, 1;
1, 3;
2, 5, 1;
3, 8, 5;
5, 13, 13, 1;
7, 23, 27, 7;
11, 39, 52, 25, 1;
17, 65, 99, 66, 9;
27, 106, 186, 151, 41, 1;
40, 177, 340, 323, 133, 11;
...
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MAPLE
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T:= proc(n) option remember; local j; if n=0 then 1
else []; for j to n do zip((x, y)->x+y, %,
[`if`(isprime(j), 0, NULL), T(n-j)], 0) od; %[] fi
end:
seq(T(n), n=0..16);
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MATHEMATICA
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zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, Thread[f[PadRight[x, m, z], PadRight[y, m, z]]]]; T[n_] := T[n] = Module[{j, pc}, If[n == 0, {1}, pc = {}; For[j = 1, j <= n, j++, pc = zip[Plus, pc, Join[If[PrimeQ[j], {0}, {}], T[n-j]], 0]]; pc]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A019426
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Continued fraction for tan(1/3).
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+10
4
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0, 2, 1, 7, 1, 13, 1, 19, 1, 25, 1, 31, 1, 37, 1, 43, 1, 49, 1, 55, 1, 61, 1, 67, 1, 73, 1, 79, 1, 85, 1, 91, 1, 97, 1, 103, 1, 109, 1, 115, 1, 121, 1, 127, 1, 133, 1, 139, 1, 145, 1, 151, 1, 157, 1, 163, 1, 169, 1, 175, 1, 181, 1, 187, 1, 193, 1, 199, 1, 205, 1, 211, 1, 217, 1, 223, 1
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OFFSET
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0,2
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COMMENTS
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The simple continued fraction expansion of 3*tan(1/3) is [1; 25, 1, 3, 1, 61, 1, 7, 1, 97, 1, 11, 1, ..., 36*n + 25, 1, 4*n + 3, 1, ...], while the simple continued fraction expansion of (1/3)*tan(1/3) is [0; 8, 1, 1, 1, 43, 1, 5, 1, 79, 1, 9, 1, 115, 1, 13, 1, ..., 36*n + 7, 1, 4*n + 1, 1, ...]. See my comment in A019425. - Peter Bala, Sep 30 2023
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LINKS
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FORMULA
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G.f.: x*(2+x+3*x^2-x^3+x^4)/(1-x^2)^2.
a(n) = 2*a(n-2)-a(n-4) with n>4, a(0)=0, a(1)=2, a(2)=1, a(3)=7, a(4)=1.
a(n) = 1+3*(1-(-1)^n)*(n-1)/2 with n>1, a(0)=0, a(1)=2.
For k>0: a(2k) = 1, a(4k+1) = 2*a(2k+1)-1 and a(4k+3) = 2*a(2k+1)+5, with a(0)=0, a(1)=2. (End)
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EXAMPLE
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0.346253549510575491038543565... = 0 + 1/(2 + 1/(1 + 1/(7 + 1/(1 + ...)))). - Harry J. Smith, Jun 13 2009
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MATHEMATICA
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PROG
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(PARI) { allocatemem(932245000); default(realprecision, 88000); x=contfrac(tan(1/3)); for (n=0, 20000, write("b019426.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 13 2009
(Magma) [n le 1 select 2*n else 1+3*(1-(-1)^n)*(n-1)/2: n in [0..80]]; // Bruno Berselli, Sep 21 2012
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CROSSREFS
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Cf. A161012 (decimal expansion of tan(1/3)).
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KEYWORD
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nonn,easy,cofr
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AUTHOR
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STATUS
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approved
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