A009001 -id:A009001 - cp's OEIS frontend

A093178 If n is even then 1, otherwise n.

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

Offset: 0

Michael Somos, Mar 27 2004

Continued fraction expansion for tan(1).
1 followed by run lengths of A062557 = 2n-1 1's followed by a 2. - Jeremy Gardiner, Aug 12 2012
Greatest common divisor of n and (n+1) mod 2. - Bruno Berselli, Mar 07 2017

			1.557407724654902230506974807... = 1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + ...))))
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 + ...

Harry J. Smith, Table of n, a(n) for n = 0..20000
D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
Simon Plouffe, A Search for a mathematical expression for mass ratios using a large database. page 3.
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Equals |A009001(n)|.

Cf. A133080, A049471 (decimal expansion), A009001, A161738, A062557, A124625.

A093178:=n->(n+1+(1-n)*(-1)^n)/2; seq(A093178(k), k=0..100); # Wesley Ivan Hurt, Oct 19 2013

Join[{1},Riffle[Range[1,85,2],1]] (* or *) Array[If[EvenQ[#],1,#]&,87,0] (* Harvey P. Dale, Nov 23 2011 *)

PARI
```
{a(n) = if( n%2, n, 1)};
```

G.f.: (1+x-x^2+x^3)/(1-x^2)^2.
a(n) = (-1)^n * a(-n) for all n in Z.
a(n) = (1/2) * [ 1 + n + (1-n)*(-1)^n ]. - Ralf Stephan, Dec 02 2004
a(n) = n^n mod (n+1) for n > 0. - Amarnath Murthy, Apr 18 2004
Satisfies a(0) = 1, a(n+1) = a(n) + n if a(n) < n else a(n+1) = a(n)/n. - Amarnath Murthy, Oct 29 2002
a(n) = ((n+1)+(1-n)(-1)^n)/2 and have e.g.f. (1+x)cosh(x). - Paul Barry, Apr 09 2003
a(n) = binomial(n, 2*floor(n/2)). - Paul Barry, Dec 28 2006
Starting (1, 1, 3, 1, 5, 1, 7, ...) = A133080^(-1) * [1,2,3,...]. - Gary W. Adamson, Sep 08 2007
a(n) = denom(b(n+2)/b(n+1)) with b(n) = product((2*n-3-2*k), k=0..floor(n/2-1)). - Johannes W. Meijer, Jun 18 2009
a(n) = 2*floor(n/2) - n*(n-1 mod 2) + 1. - Wesley Ivan Hurt, Oct 19 2013
a(n) = n^(n mod 2). - Wesley Ivan Hurt, Apr 16 2014

A049471 Decimal expansion of tan(1).

Original entry on oeis.org

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

Offset: 1

Albert du Toit (dutwa(AT)intekom.co.za), N. J. A. Sloane

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			1.5574077246549022305...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Mohammad K. Azarian, Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 36, No. 5, November 2005, pp. 413-414.
Mohammad K. Azarian, Solution of Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395.
Index entries for transcendental numbers

Cf. A093178 (continued fraction), A009001, A073449.

RealDigits[Tan[1], 10, 100][[1]] (* Amiram Eldar, May 15 2021 *)

default(realprecision, 20080); x=tan(1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b049471.txt", n, " ", d)); \\

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

A009531 Expansion of the e.g.f. sin(x)*(1+x).

Original entry on oeis.org

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

Offset: 0

R. H. Hardin

Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Cf. A009001, A029578, A049240.

[(((2*n+3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n) div 4)+((2*n-1-(-1)^n) div 2)*(-1)^((6*n+5-(-1)^n) div 4))/2: n in [0..90]]; // Vincenzo Librandi, Jul 19 2015

CoefficientList[Series[x*(1+x)^2/(1+x^2)^2, {x, 0, 100}], x] (* Vaclav Kotesovec, Oct 03 2014 *)

concat(0, Vec(x*(1+x)^2/(1+x^2)^2 + O(x^80))) \\ Michel Marcus, Oct 03 2014

A009531(n) = (((n^(n+1)) % (n+1)) * ((-1)^((n-1)\2))); \\ Antti Karttunen, Nov 02 2017, after Henry Bottomley's formula.

A009531(n) = (lift(Mod(n, n+1)^(n+1)) * ((-1)^((n-1)\2))); \\ (like above, but quicker) - Antti Karttunen, Nov 02 2017

There's an obvious formula for the n-th term!
G.f.: x*(1+x)^2/(1+x^2)^2.
abs(a(n)) = Sum_{k=0..floor((n-1)/2)} (C(n-k-1, k) mod 2)*(-1)^k*2^A000120(n-2k-1). - Paul Barry, Jan 06 2005
a(n) = (n^(n+1) mod (n+1)) * (-1)^[(n-1)/2] = a(n-1)-a(n-2)+(-1)^n*a(n-1) = -2a(n-2)-a(n-4). - Henry Bottomley, May 07 2005
a(n+2) is the Hankel transform of A086622(n+1). - Paul Barry, Nov 06 2007
E.g.f.: sin(x)*(1+x)=x*Q(0); Q(k)=1+x/(1-x/(x-2*(k+1)*(2k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2011
a(n) = sin(Pi*n/2)-n*cos(Pi*n/2). - Vaclav Kotesovec, Oct 03 2014
a(n) = (((2*n+3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4)+((2*n-1-(-1)^n)/2)*(-1)^((6*n+5-(-1)^n)/4))/2. - Luce ETIENNE, Jul 18 2015

A062172 Table T(n,k) by antidiagonals of n^(k-1) mod k [n,k > 0].

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 3, 1, 1, 0, 1, 2, 1, 0, 1, 0, 0, 1, 1, 3, 1, 1, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 0, 0, 1, 4, 3, 1, 5, 1, 3, 1, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 3, 7, 5, 1, 1, 1, 1, 1, 1, 0, 1, 8, 1, 4, 7, 0, 0, 2, 1, 0, 1, 0, 0, 1, 1, 3, 1, 5, 0, 7, 1, 3, 0, 3, 0, 1, 0

Offset: 1

Henry Bottomley, Jun 12 2001

			T(5,3)=5^(3-1) mod 3=25 mod 3=1. Rows start (0,1,1,1,1,...), (0,0,1,0,1,...), (0,1,0,3,1...), (0,0,1,0,1,...), (0,1,1,1,0,...), ...

Cf. A002997, A060154. Rows include A057427, A062173, A062174, A062175, A062176. Columns include A000004, A000035, A011655, A010684 with interleaved 0's, A011558, A010875. Diagonals include all the rows again and A000004 and A009001 unsigned.

A344935 a(0)=1; for n > 0, a(n) = n(a(n-1) + i^(n-1)) if n is odd, na(n-1) + i^n otherwise, where i = sqrt(-1).

Original entry on oeis.org

1, 2, 3, 6, 25, 130, 779, 5446, 43569, 392130, 3921299, 43134278, 517611337, 6728947394, 94205263515, 1413078952710, 22609263243361, 384357475137154, 6918434552468771, 131450256496906630, 2629005129938132601, 55209107728700784642, 1214600370031417262123

Offset: 0

Amrit Awasthi, Jun 03 2021

nonn

			a(0) = 1;
a(1) = 1*(a(0) + i^(1-1)) =  2;
a(2) = 2*a(1)  + i^2      =  3;
a(3) = 3*(a(2) + i^2)     =  6;
a(4) = 4*a(3)  + i^4      = 25.

Cf. A344262, A344419, A049470, A009001.

A344935 := proc(n)
    option remember ;
    if n = 0 then
        1;
    elif type(n,'odd') then
        n*(procname(n-1)+I^(n-1)) ;
    else
        n*procname(n-1)+I^n ;
    end if;
    simplify(%) ;
end proc:
seq(A344935(n),n=0..40) ; # R. J. Mathar, Aug 19 2022

a[0] = 1; a[n_] := a[n] = If[OddQ[n], n*(a[n - 1] + I^(n - 1)), n*a[n - 1] + I^n]; Array[a, 30, 0] (* Amiram Eldar, Jun 03 2021 *)

a(n) = if (n==0, 1, if (n%2, n*(a(n-1) + I^(n-1)), n*a(n-1) + I^n)); \\ Michel Marcus, Jun 05 2021

E.g.f.: (1+x)*cos(x)/(1-x).
Lim_{n->infinity} a(n)/n! = 2*cos(1) = 2*A049470.
D-finite with recurrence a(n) -n*a(n-1) +2*a(n-2) +2*(-n+2)*a(n-3) +a(n-4) +(-n+4)*a(n-5)=0. - R. J. Mathar, Aug 19 2022

cp's OEIS Frontend

A093178 If n is even then 1, otherwise n.

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A049471 Decimal expansion of tan(1).

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Mathematica

PARI

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A009531 Expansion of the e.g.f. sin(x)*(1+x).

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Magma

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PARI

PARI

PARI

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A062172 Table T(n,k) by antidiagonals of n^(k-1) mod k [n,k > 0].

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A344935 a(0)=1; for n > 0, a(n) = n*(a(n-1) + i^(n-1)) if n is odd, n*a(n-1) + i^n otherwise, where i = sqrt(-1).

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Programs

Maple

Mathematica

PARI

Formula

A344935 a(0)=1; for n > 0, a(n) = n(a(n-1) + i^(n-1)) if n is odd, na(n-1) + i^n otherwise, where i = sqrt(-1).