A113405 -id:A113405 - cp's OEIS frontend

A275788 a(0) = 0, a(n+1) = 2*a(n) + (-1)^floor(n/3).

0, 1, 3, 7, 13, 25, 49, 99, 199, 399, 797, 1593, 3185, 6371, 12743, 25487, 50973, 101945, 203889, 407779, 815559, 1631119, 3262237, 6524473, 13048945, 26097891, 52195783, 104391567, 208783133, 417566265, 835132529, 1670265059, 3340530119, 6681060239

Offset: 0

Paul Curtz, Aug 09 2016

nonn

a(n) and its successive differences:
0,  1,  3,  7, 13, 25, 49, ...
1,  2,  4,  6, 12, 24, 50, 100, ...
1,  2,  2,  6, 12, 26, 50, 100, 198, ...
1,  0,  4,  6, 14, 24, 50,  98, 200, 398, ...
-1, 4,  2,  8, 10, 26, 48, 102, 198, 400, 794, ...
5, -2,  6,  2, 16, 22, 54,  96, 202, 394, 800, 1590, ...
-7, 8, -4, 14,  6, 32, 42, 106, 192, 406, 790, 1600, 3178, ...
... .
Each row has the recurrence a(n) + a(n+3) = 7*2^n.
Main diagonal: 2*A001045(n).
Upper diagonals: A084214(n+1), 3*2^n, ... .
Subdiagonals: 2^n, A078008(n), A084214(n+1), -2^n, ... .
a(-n) = 0, 1/2, 3/4, 7/8, -1/16, -17/32, -49/64, 15/128, ... .
b(n), numerators of a(-n), and first differences:
0, 1, 3,  7,  -1, -17, -49,  15, 143,  399,  -113, -1137, ...
1, 2, 4, -8, -16, -32,  64, 128, 256, -512, -1024, ... = A000079(n)*A130151(n), not in the OEIS.

			a(1)=2*0+1=1, a(2)=2*1+1=3, a(2)=2*3+1=7, a(3)=2*7-1=13, a(4)=2*13-1=25, ... .

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Cf. A000079, A001045, A002264, A005009, A007283, A078008, A084214, A113405, A130151, A274817.

CoefficientList[Series[x (1 + x + x^2)/((1 + x) (1 - 2 x) (1 - x + x^2)), {x, 0, 33}], x] (* Michael De Vlieger, Aug 11 2016 *)
LinearRecurrence[{2,0,-1,2}, {0, 1, 3, 7}, 25] (* G. C. Greubel, Aug 16 2016 *)

concat(0, Vec(x*(1+x+x^2)/((1+x)*(1-2*x)*(1-x+x^2)) + O(x^40))) \\ Colin Barker, Aug 10 2016

From Colin Barker, Aug 09 2016: (Start)
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3.
G.f.: x*(1 + x + x^2) / ((1+x)*(1-2*x)*(1-x+x^2)).
(End)
a(n+3) = 7*2^n - a(n), a(0)=0, a(1)=1, a(2)=3.

More terms from Colin Barker, Aug 10 2016

A328881 a(n+3) = 2^n - a(n), a(0)=a(2)=1, a(1)=0 for n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 8, 14, 29, 56, 114, 227, 456, 910, 1821, 3640, 7282, 14563, 29128, 58254, 116509, 233016, 466034, 932067, 1864136, 3728270, 7456541, 14913080, 29826162, 59652323, 119304648, 238609294, 477218589, 954437176, 1908874354, 3817748707

Offset: 0

Paul Curtz, Oct 29 2019

The array of a(n) and its repeated differences:
    1,   0,   1,   0,   2,   3,   8,  14, ...
   -1,   1,  -1,   2,   1,   5,   6,  15, ...
    2,  -2,   3,  -1,   4,   1,   9,  12, ...
   -4,   5,  -4,   5,  -3,   8,   3,  19, ...
    9,  -9,   9,  -8,  11,  -5,  16,   5, ...
  -18,  18, -17,  19, -16,  21, -11,  32, ...
   36, -35,  36, -35,  37, -32,  43, -21, ...
  -71,  71, -71,  72, -69,  75, -64,  85, ...
  ...
The recurrence is the same for every row.
From Jean-François Alcover, Nov 28 2019: (Start)
It appears that, when odd, a(n) is never a multiple of 5.
Main and 3rd upper diagonals of the difference array are A001045 (Jacobsthal numbers); first upper diagonal is negated A001045; second upper diagonal is A000079 (powers of 2); 4th upper diagonal is A062092.
(End)

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Cf. A024495, A062092, A131714.

Cf. A015565, A092220, A113405.

a[0] = a[2] = 1; a[1] = 0; a[n_] := a[n] = 2^(n - 3) - a[n - 3]; Array[a, 36, 0] (* Amiram Eldar, Nov 06 2019 *)

Vec((1 - 2*x + x^2 - x^3) / ((1 + x)*(1 - 2*x)*(1 - x + x^2)) + O(x^40)) \\ Colin Barker, Oct 29 2019

a(n+1) - 2*a(n) = period 6: repeat [-2, 1, -2, 2, -1, 2].
a(n+12) - a(n) = 455*2^n.
From Colin Barker, Oct 29 2019: (Start)
G.f.: (1 - 2*x + x^2 - x^3) / ((1 + x)*(1 - 2*x)*(1 - x + x^2)).
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3.
(End)
a(n+2) - a(n) = A024495(n).
a(n+6) - a(n) = 7*2^n.
a(n+9) + a(n) = 57*2^n.
a(n) = A113405(n) + A092220(n+5).
9*a(n) = 2^n + 5*(-1)^n + 3*A010892(n). - R. J. Mathar, Nov 28 2019

A167280 Period length 12: 0,0,1,2,4,7,4,8,7,4,8,5 (and repeat).

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Nov 01 2009

The sum of the terms in the period is 50, so the partial sums of the sequence are also 12-periodic if reduced modulo 50 or modulo 10.
The weighted partial sums b(n) = sum_{i=0..n} a(i)*2^i obey b(n) = b(n+12) (mod 10).
Third column is A000689. (Which table or array is this referring to? R. J. Mathar, Nov 01 2009)
The set of digits in the period is the same as in A141425.
A derived sequence with terms a(n)+a(n+6) has period length 6: 4, 8, 8, 6, 12, 12 (repeat).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

a(n) = A113405(n+1) mod 10.

G.f.: x^2*(1+2*x+4*x^2+7*x^3+4*x^4+8*x^5+7*x^6+4*x^7+8*x^8+5*x^9)/( (1-x)*(1+x+x^2)*(1+x)*(1-x+x^2)*(1+x^2)*(x^4-x^2+1)) [R. J. Mathar, Nov 03 2009]

Edited by R. J. Mathar, Nov 05 2009

A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = xZ(n-1,x) - 2Z(n-2,x), n >= 2.

Original entry on oeis.org

1, 1, 0, 1, 0, -2, 1, 0, -4, 0, 1, 0, -6, 0, 4, 1, 0, -8, 0, 12, 0, 1, 0, -10, 0, 24, 0, -8, 1, 0, -12, 0, 40, 0, -32, 0, 1, 0, -14, 0, 60, 0, -80, 0, 16, 1, 0, -16, 0, 84, 0, -160, 0, 80, 0, 1, 0, -18, 0, 112, 0, -280, 0, 240, 0, -32

Offset: 0

Paul Curtz, Jun 19 2011

The coefficients of the Z(n,x) polynomials by decreasing exponents, see the formulas, define this triangle.

			The first few rows of the coefficients of the Z(n,x) are
  1;
  1,    0;
  1,    0,   -2;
  1,    0,   -4,    0;
  1,    0,   -6,    0,    4;
  1,    0,   -8,    0,   12,    0;
  1,    0,  -10,    0,   24,    0,   -8;
  1,    0,  -12,    0,   40,    0,  -32,    0;
  1,    0,  -14,    0,   60,    0,  -80,    0,   16;
  1,    0,  -16,    0,   84,    0, -160,    0,   80,    0;

Row sums: A107920(n+1). Main diagonal: A077966(n).
Z(n,x=1) = A107920(n+1), Z(n,x=2)  = A009545(n+1),
Z(n,x=3) = A000225(n+1), Z(n,x=4)  = A007070(n),
Z(n,x=5) = A107839(n),   Z(n,x=6)  = A154244(n),
Z(n,x=7) = A186446(n),   Z(n,x=8)  = A190975(n+1),
Z(n,x=9) = A190979(n+1), Z(n,x=10) = A190869(n+1).
Row sum without sign: A113405(n+1).
Cf. A128099, A013609.

nmax:=10: Z(0, x):=1 : Z(1, x):=x: for n from 2 to nmax do Z(n, x) := x*Z(n-1, x) - 2*Z(n-2, x) od: for n from 0 to nmax do for k from 0 to n do T(n, k) := coeff(Z(n, x), x, n-k) od: od: seq(seq(T(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 27 2011, revised Nov 29 2012

a[n_, k_] := If[OddQ[k], 0, 2^(k/2)*Coefficient[ ChebyshevU[n, x/2], x, n-k]]; Flatten[ Table[ a[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Aug 02 2012, from 2nd formula *)

Z(0,x) = 1, Z(1,x) = x and Z(n,x) = x*Z(n-1,x) - 2*Z(n-2,x), n >= 2.

a(n,k) = A077957(k) * A053119(n,k). - Paul Curtz, Sep 30 2011

Edited and information added by Johannes W. Meijer, Jun 27 2011

A275771 a(n+3) = A008578(n+1) -a(n), a(0) = a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 5, 8, 9, 12, 11, 14, 17, 20, 23, 24, 23, 24, 29, 36, 37, 38, 35, 36, 41, 48, 53, 56, 53, 50, 51, 56, 63, 76, 75, 74, 63, 74, 77, 94, 89, 90, 79, 90, 91, 112, 103, 106, 87, 108, 117, 140

Offset: 0

Paul Curtz, Aug 08 2016

nonn

A008578 gives the noncomposite numbers, the prime numbers at the beginning of the 20th century which included 1.
a(2n) = 0, 0, 2, 4, 8, 12, 14, 20, 24, 24, ... always even?
a(2n+3) =  1, 3, 5, 9, 11, 17, 23, 23, 29, ... always odd?
First differences: 0, 0, 1, 1, 1, 1, 1, 3, 1, 3, -1, 3, 3, 3, 3, 1, -1, 1, 5, 7, ... .

			a(3) = 1-0 = 1, a(4) = 2-0 = 2, a(5) = 3-0 = 3, a(6) = 5-1 = 4, a(6) = 7-2 = 5, ... .

Cf. A000040, A008578, A113405, A274817.

RecurrenceTable[{a[n + 3] == If[n == 0, 1, Prime[n]] - a[n], a[0] == 0, a[1] == 0, a[2] == 0}, a, {n, 0, 52}] (* Michael De Vlieger, Aug 08 2016 *)

cp's OEIS Frontend

A275788 a(0) = 0, a(n+1) = 2*a(n) + (-1)^floor(n/3).

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A328881 a(n+3) = 2^n - a(n), a(0)=a(2)=1, a(1)=0 for n >= 0.

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A167280 Period length 12: 0,0,1,2,4,7,4,8,7,4,8,5 (and repeat).

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A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = x*Z(n-1,x) - 2*Z(n-2,x), n >= 2.

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A275771 a(n+3) = A008578(n+1) -a(n), a(0) = a(1) = a(2) = 0.

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A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = xZ(n-1,x) - 2Z(n-2,x), n >= 2.