A104537 -id:A104537 - cp's OEIS frontend

A130481 a(n) = Sum_{k=0..n} (k mod 3) (i.e., partial sums of A010872).

0, 1, 3, 3, 4, 6, 6, 7, 9, 9, 10, 12, 12, 13, 15, 15, 16, 18, 18, 19, 21, 21, 22, 24, 24, 25, 27, 27, 28, 30, 30, 31, 33, 33, 34, 36, 36, 37, 39, 39, 40, 42, 42, 43, 45, 45, 46, 48, 48, 49, 51, 51, 52, 54, 54, 55, 57, 57, 58, 60, 60, 61, 63, 63, 64, 66, 66, 67, 69, 69, 70, 72, 72

Offset: 0

Hieronymus Fischer, May 29 2007

Essentially the same as A092200. - R. J. Mathar, Jun 13 2008
Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 3, A[i,i]:=1, A[i,i-1]=-1. Then, for n>=1, a(n)=det(A). - Milan Janjic, Jan 24 2010
2-adic valuation of A104537(n+1). - Gerry Martens, Jul 14 2015
Conjecture: a(n) is the exponent of the largest power of 2 that divides all the entries of the matrix {{3,1},{1,-1}}^n. - Greg Dresden, Sep 09 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A130482, A130483, A130484.

Cf. A092200, A104537.

List([0..80], n-> Int((n+1)/3) + Int(2*(n+1)/3)); # G. C. Greubel, Aug 31 2019

[Floor((n+1)/3) + Floor(2*(n+1)/3): n in [0..80]]; // G. C. Greubel, Aug 31 2019

seq(coeff(series(x*(1+2*x)/((1-x^3)*(1-x)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Aug 31 2019

a[n_]:= Floor[(n+1)/3] + Floor[2(n+1)/3]; Table[a[n], {n, 0, 80}] (* Clark Kimberling, May 28 2012 *)
a[n_]:= IntegerExponent[A104537[n + 1], 2];
Table[a[n], {n, 0, 80}]  (* Gerry Martens, Jul 14 2015 *)
CoefficientList[Series[x(1+2x)/((1-x^3)(1-x)), {x, 0, 80}], x] (* Stefano Spezia, Sep 09 2018 *)
LinearRecurrence[{1,0,1,-1},{0,1,3,3},100] (* Harvey P. Dale, Jun 14 2021 *)

main(size)=my(n,k);vector(size,n,sum(k=0,n,k%3)) \\ Anders Hellström, Jul 14 2015

first(n)=my(s); concat(0, vector(n,k,s+=k%3)) \\ Charles R Greathouse IV, Jul 14 2015

a(n)=n\3*3+[0,1,3][n%3+1] \\ Charles R Greathouse IV, Jul 14 2015

def A130481_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1+2*x)/((1-x^3)*(1-x))).list()
A130481_list(80) # G. C. Greubel, Aug 31 2019

a(n) = 3*floor(n/3) + A010872(n)*(A010872(n) + 1)/2.
G.f.: x*(1 + 2*x)/((1-x^3)*(1-x)).
a(n) = n + 1 - (Fibonacci(n+1) mod 2). - Gary Detlefs, Mar 13 2011
a(n) = floor((n+1)/3) + floor(2*(n+1)/3). - Clark Kimberling, May 28 2010
a(n) = n when n+1 is not a multiple of 3, and a(n) = n+1 when n+1 is a multiple of 3. - Dennis P. Walsh, Aug 06 2012
a(n) = n + 1 - sign((n+1) mod 3). - Wesley Ivan Hurt, Sep 25 2017
a(n) = n + (1-cos(2*(n+2)*Pi/3))/3 + sin(2*(n+2)*Pi/3)/sqrt(3). - Wesley Ivan Hurt, Sep 27 2017
a(n) = n + 1 - (n+1)^2 mod 3. - Ammar Khatab, Aug 14 2020
E.g.f.: ((1 + 3*x)*cosh(x) - (cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))*(cosh(x/2) - sinh(x/2)) + (1 + 3*x)*sinh(x))/3. - Stefano Spezia, May 28 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) + log(2)/3. - Amiram Eldar, Sep 17 2022

A138230 Expansion of (1-x)/(1 - 2x + 4x^2).

Original entry on oeis.org

1, 1, -2, -8, -8, 16, 64, 64, -128, -512, -512, 1024, 4096, 4096, -8192, -32768, -32768, 65536, 262144, 262144, -524288, -2097152, -2097152, 4194304, 16777216, 16777216, -33554432, -134217728, -134217728, 268435456, 1073741824, 1073741824, -2147483648, -8589934592

Offset: 0

Paul Barry, Mar 06 2008

In general, the expansion of (1-x)/(1 - 2*x + (m+1)*x^2) has general term given by a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*(-m)^k = ((1+sqrt(-m))^n + (1-sqrt(-m))^n)/2.

Binomial transform of [1, 0, -3, 0, 9, 0, -27, 0, 81, 0, ...] = powers of -3 with interpolated zeros. - Philippe Deléham, Dec 02 2008

Michael De Vlieger, Table of n, a(n) for n = 0..3322
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras, Vol. 29 (2019), Article 54.
Index entries for linear recurrences with constant coefficients, signature (2,-4).

Cf. A087204, A088138, A098158, A104537, A124182, A128018.

[2^n*Evaluate(ChebyshevFirst(n), 1/2): n in [0..30]]; // G. C. Greubel, Feb 11 2023

CoefficientList[Series[(1-x)/(1-2x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,-4},{1,1},30] (* Harvey P. Dale, Nov 11 2014 *)

[2^n*chebyshev_T(n,1/2) for n in range(31)] # G. C. Greubel, Feb 11 2023

From Philippe Deléham, Nov 14 2008: (Start)
a(n) = 2*a(n-1) - 4*a(n-2), a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A098158(n,k)*(-3)^(n-k). (End)
a(n) = Sum_{k=0..n} A124182(n,k)*(-4)^(n-k). - Philippe Deléham, Nov 15 2008
a(n) = 2^n*cos(Pi*n/3). - Richard Choulet, Nov 19 2008
a(n) = -8*a(n-3). - Paul Curtz, Apr 22 2011
From Sergei N. Gladkovskii, Jul 27 2012: (Start)
G.f.: G(0) where G(k) = 1 + x/(1 + 2*x/(1 - 2*x - 4*x/(4*x + 1/G(k+1)))); (continued fraction).
E.g.f.: exp(x)*cos(sqrt(3)*x) = G(0) where G(k) = 1 + x/(3*k+1 + 2*x*(3*k+1)/(3*k+2 - 2*x - 4*x*(3*k+2)/(4*x + 3*(k+1)/G(k+1)))); (continued fraction). (End)
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = A088138(n+1) - A088138(n). - R. J. Mathar, Mar 04 2018
a(n) = (-1)^n*A104537(n). - R. J. Mathar, May 21 2019
a(n) = 2^(n-1)*A087204(n). - G. C. Greubel, Feb 11 2023
Sum_{n>=0} 1/a(n) = 4/3. - Amiram Eldar, Feb 14 2023

A116412 Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

Original entry on oeis.org

1, 3, 1, 6, 6, 1, 12, 21, 9, 1, 24, 60, 45, 12, 1, 48, 156, 171, 78, 15, 1, 96, 384, 558, 372, 120, 18, 1, 192, 912, 1656, 1473, 690, 171, 21, 1, 384, 2112, 4608, 5160, 3225, 1152, 231, 24, 1, 768, 4800, 12240, 16584, 13083, 6219, 1785, 300, 27, 1, 1536, 10752

Offset: 0

Paul Barry, Feb 13 2006

Row sums are A003688. Diagonal sums are A116413. Product of A007318 and A116413 is A116414. Product of A007318 and A105475.

Subtriangle of triangle given by (0, 3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012

			Triangle begins
1,
3, 1,
6, 6, 1,
12, 21, 9, 1,
24, 60, 45, 12, 1,
48, 156, 171, 78, 15, 1
Triangle T(n,k), 0<=k<=n, given by (0, 3, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins :
1
0, 1
0, 3, 1
0, 6, 6, 1
0, 12, 21, 9, 1
0, 24, 60, 45, 12, 1
0, 48, 156, 171, 78, 15, 1
... - _Philippe Deléham_, Jan 18 2012

Michael De Vlieger, Table of n, a(n) for n = 0..11475
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A003688, A003945.

With[{n = 10}, DeleteCases[#, 0] & /@ CoefficientList[Series[(1 + x)/(1 - (y + 2) x - y x^2), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Number triangle T(n,k)=sum{j=0..n, C(k+1,j)*C(n-j,k)2^(n-k-j)}
From Vladimir Kruchinin, Mar 17 2011: (Start)
T((m+1)*n+r-1, m*n+r-1) * r/(m*n+r) = sum(k=1..n, k/n * T((m+1)*n-k-1, m*n-1) * T(r+k-1,r-1)), n>=m>1.
T(n-1,m-1) = m/n * sum(k=1..n-m+1, k*A003945(k-1)*T(n-k-1,m-2)), n>=m>1. (End)
G.f.: (1+x)/(1-(y+2)*x -y*x^2). - Philippe Deléham, Jan 18 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A104537(n), A110523(n), (-2)^floor(n/2), A057079(n), A003945(n), A003688(n+1), A123347(n), A180035(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Jan 18 2012
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(2,0) = T(2,1) = 6, T(2,2) = 1, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Oct 31 2013

cp's OEIS Frontend

A130481 a(n) = Sum_{k=0..n} (k mod 3) (i.e., partial sums of A010872).

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A138230 Expansion of (1-x)/(1 - 2*x + 4*x^2).

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A116412 Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

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A138230 Expansion of (1-x)/(1 - 2x + 4x^2).