A079583 -id:A079583 - cp's OEIS frontend

A280193 a(2n) = 2, a(2n + 1) = -1, a(0) = 1.

1, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2

Offset: 0

Michael Somos, Dec 28 2016

			G.f. = 1 - x + 2*x^2 - x^3 + 2*x^4 - x^5 + 2*x^6 - x^7 + 2*x^8 - x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000225, A000325, A040001, A028242, A079583, A083329, A117575.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 - x+x^2)/(1-x^2))); // G. C. Greubel, Jul 29 2018

a[ n_] := Which[ n < 1, Boole[n == 0], OddQ[n], -1, True, 2];
a[ n_] := SeriesCoefficient[ (1 - x + x^2) / (1 - x^2), {x, 0, n}];
LinearRecurrence[{0,1},{1,-1,2},80] (* Harvey P. Dale, Aug 06 2025 *)

PARI
```
{a(n) = if( n<1, n==0, 2 - 3*(n%2))};
```

{a(n) = if( n<1, n==0, [2, -1][n%2 + 1])};

{a(n) = if( n<0, 0, polcoeff( (1 - x + x^2) / (1 - x^2) + x * O(x^n), n))};

Euler transform of length 6 sequence [-1, 2, 1, 0, 0, -1].
Moebius transform is length 2 sequence [-1, 3].
a(n) = -b(n) where b() is multiplicative with b(2^e) = -2 if e>0, b(p^e) = 1 otherwise.
G.f.: (1 - x + x^2) / (1 - x^2).
G.f.: (1 - x) * (1 - x^6) / ((1 - x^3) * (1 -x^2)^2).
G.f.: 1 / (1 + x / (1 + x / (1 - 3*x / (1 + x)))).
a(n) = (-1)^n * A040001(n).
A028242(n) = Sum_{k=0..n} a(k).
A117575(n+1) = Product_{k=0..n} a(k).
A000225(n-1) = Sum_{k=0..n} binomial(n, k) * a(k) if n>0.
A000325(n) = Sum_{k=0..n} binomial(n, k+1) * a(k) if n>0.
a(n) = Sum_{k=0..n} binomial(n, k) * (-1)^k * A083329(k).
A079583(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n.
a(n) = A168361(n+1), n>0. - R. J. Mathar, Jan 04 2017

A097810 a(n) = 72^n - 3n - 6.

Original entry on oeis.org

1, 5, 16, 41, 94, 203, 424, 869, 1762, 3551, 7132, 14297, 28630, 57299, 114640, 229325, 458698, 917447, 1834948, 3669953, 7339966, 14679995, 29360056, 58720181, 117440434, 234880943, 469761964, 939524009, 1879048102, 3758096291

Offset: 0

Paul Barry, Aug 25 2004

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A079583, A097809, A131068.

s=1;lst={s};Do[s+=(s+=n)+n++;AppendTo[lst, s], {n, 1, 5!, 1}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
Table[7*2^n-3n-6,{n,0,30}] (* or *) LinearRecurrence[{4,-5,2},{1,5,16},30] (* Harvey P. Dale, Nov 15 2011 *)

G.f.: (1 + x + x^2)/((1-x)^2*(1-2*x)).
a(n) = 2*a(n-1) + 3*n, n > 0, a(0)=1.
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
From Elmo R. Oliveira, Mar 06 2025: (Start)
E.g.f.: exp(x)*(7*exp(x) - 3*(x + 2)).
a(n) = A131068(n+1)/2. (End)

A132110 A007318 + A059268 - A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 6, 8, 1, 5, 9, 11, 16, 1, 6, 13, 17, 20, 32, 1, 7, 18, 27, 30, 37, 64, 1, 8, 24, 42, 50, 52, 70, 128, 1, 9, 31, 63, 85, 87, 91, 135, 256, 1, 10, 39, 91, 141, 157, 147, 163, 264, 512

Offset: 0

Gary W. Adamson, Aug 09 2007

Row sums = A079583: (1, 3, 8, 19, 42, 89, 184, ...).

			First few rows of the triangle:
  1;
  1, 2;
  1, 3,  4;
  1, 4,  6,  8;
  1, 5,  9, 11, 16;
  1, 6, 13, 17, 20, 32;
  1, 7, 18, 27, 30, 37, 64;
  1, 8, 24, 42, 50, 52, 70, 128;
  ...

Cf. A007318, A059268, A000012, A079583.

A227712 a(n) = 92^n - 3n - 5.

Original entry on oeis.org

4, 10, 25, 58, 127, 268, 553, 1126, 2275, 4576, 9181, 18394, 36823, 73684, 147409, 294862, 589771, 1179592, 2359237, 4718530, 9437119, 18874300, 37748665, 75497398, 150994867, 301989808, 603979693, 1207959466, 2415919015, 4831838116, 9663676321, 19327352734

Offset: 0

Emeric Deutsch, Aug 06 2013

Denoting by P[n] the path on n vertices, a(n) is the number of vertices of the tree obtained by identifying the roots of 3 identical rooted trees g[n], where g[n] is obtained recursively in the following manner: g[0]=P[2] and g[n] (n>=1) is obtained by identifying the roots of 2 copies of g[n-1] and one of the extremities of P[n+1]; the root of g[n] is defined to be the other extremity of P[n+1]. Most references contain pictures of these trees; however, the small circles have to be viewed as vertices rather than hexagons.

			a(1) = 10 because g[1] is the rooted tree in the shape of Y (4 vertices) and a "bouquet" of three Y's has 3*4 - 2 = 10 vertices.

R. Kopelman, M. Shortreed, Z. Y. Shi, W. Tan, Z. F. Xu, J. S. Moore, A. Bar-Haim, J. Klafter, Spectroscopic evidence for excitonic localization in fractal antenna supermolecules, Phys. Rev. Letters, 78, 1997, 1239-1242.
M. A. Martín-Delgado, J. Rodriguez-Laguna, G. Sierra, A density matrix renormalization group study of excitons in dendrimers, Phys. Rev. B 65, 2002, 155116(1-11).
S. Raychaudhuri, Y. Shapir, S. Mukamel, Disorder and funneling effect on exciton migration in treelike dendrimers, Phys. Rev. E, 65, 2002, 021803(1-12).
S. Tretiak, V. Chernyak, S. Mukamel, Localized electronic excitations in phenylacetylene dendrimers, J. Phys. Chem. B, 102, 1998, 3310-3315.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A079583.

[9*2^n-3*n-5: n in [0..40]]; // Vincenzo Librandi, Feb 19 2016

a := proc (n) options operator, arrow: 9*2^n-3*n-5 end proc: seq(a(n), n = 0 .. 35);

Table[9*2^n-3n-5,{n,0,40}] (* or *) LinearRecurrence[{4,-5,2},{4,10,25},40] (* Harvey P. Dale, Apr 15 2015 *)

Vec((4-6*x+5*x^2)/((1-2*x)*(1-x)^2) + O(x^100)) \\ Altug Alkan, Oct 17 2015

G.f.: (4-6*x+5*x^2)/((1-2*x)*(1-x)^2).
a(0)=4, a(1)=10, a(2)=25, a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3). - Harvey P. Dale, Apr 15 2015
a(n)= 3*A079583(n) + 1. - Emeric Deutsch, Feb 18 2016

A271590 First differences of column n of A271589 for arbitrarily large n.

Original entry on oeis.org

3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 3, 4, 12, 26, 3, 4, 2, 1, 2, 4, 11, 1, 2, 4, 2, 3, 4, 12, 36, 71, 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 1, 2, 4, 12, 26, 3, 4, 2, 1, 2, 4, 11, 1, 2, 4, 2, 3, 4, 12, 36, 108, 206, 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 3, 4, 12, 26, 3, 4, 2, 1

Offset: 1

Max Barrentine, Apr 10 2016

nonn

Max Barrentine, Table of n, a(n) for n = 1..16368

Cf. A271589.

Cf. A000295, A079583.

For all k>2:
a(A000295(k)) = a(A000295(k+1)) = 3;
For A000295(k)For n = 2^k-2, a(n) = a(n+2^(k+1)-1), a(2^(k+1)-2) = a(n)+5*3^(k-2);
a(2^(k+1)-3) = 4*3^(k-2);
For 2^k-2A079583(k-1)-1, a(n) = a(n+2^k)=a(n+2^(k+1)-1);
For n = A079583(k-1)-1, a(n) = a(n+2^(k+1)-1) = 1;
For n = A079583(k-1), a(n) = a(n+2^(k+1)-1) = 2;
a(A079583(k-1)+2^k-1) = 3;
For A079583(k-1)A000295(k+1), a(n) = a(n+2^k-1) = a(n+2^(k+1)-1)

A368826 Square array T(n,k) = 3*2^k - n read by ascending antidiagonals.

Original entry on oeis.org

3, 2, 6, 1, 5, 12, 0, 4, 11, 24, -1, 3, 10, 23, 48, -2, 2, 9, 22, 47, 96, -3, 1, 8, 21, 46, 95, 192, -4, 0, 7, 20, 45, 94, 191, 384, -5, -1, 6, 19, 44, 93, 190, 383, 768, -6, -2, 5, 18, 43, 92, 189, 382, 767, 1536, -7, -3, 4, 17, 42, 91, 188, 381, 766, 1535, 3072

Offset: 0

Paul Curtz, Jan 07 2024

Similar to A367559.

			Table begins:
   3  6 12 24 48 96 ...
   2  5 11 23 47 95 ...
   1  4 10 22 46 94 ...
   0  3  9 21 45 93 ...
  -1  2  8 20 44 92 ...
  -2  1  7 19 43 91 ...
  ...

Cf. A000225, A000918, A007283, A033484, A048488, A083329, A165751, A367559.

Cf. diagonals: A123720, A079583, A095151, A101945 (after -1).

T[n_, k_] := 3*2^k - n; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jan 15 2024 *)

T(0,k) = 3*2^k     =   A007283(k).
T(1,k) = 3*2^k - 1 =   A083329(k+1).
T(2,k) = 3*2^k - 2 =   A033484(k).
T(3,k) = 3*2^k - 3 = 3*A000225(k).
T(4,k) = 3*2^k - 4 =  -A165751(k).
T(5,k) = 3*2^k - 5 =   A048488(k-1)
T(6,k) = 3*2^k - 6 = 3*A000918(k).

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cp's OEIS Frontend

A280193 a(2*n) = 2, a(2*n + 1) = -1, a(0) = 1.

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A097810 a(n) = 7*2^n - 3*n - 6.

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A132110 A007318 + A059268 - A000012 as infinite lower triangular matrices.

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A227712 a(n) = 9*2^n - 3*n - 5.

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A271590 First differences of column n of A271589 for arbitrarily large n.

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A368826 Square array T(n,k) = 3*2^k - n read by ascending antidiagonals.

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Comments

Examples

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Programs

Mathematica

Formula

A280193 a(2n) = 2, a(2n + 1) = -1, a(0) = 1.

A097810 a(n) = 72^n - 3n - 6.

A227712 a(n) = 92^n - 3n - 5.