A073371 -id:A073371 - cp's OEIS frontend

A073370 Convolution triangle of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0.

1, 1, 1, 3, 2, 1, 5, 7, 3, 1, 11, 16, 12, 4, 1, 21, 41, 34, 18, 5, 1, 43, 94, 99, 60, 25, 6, 1, 85, 219, 261, 195, 95, 33, 7, 1, 171, 492, 678, 576, 340, 140, 42, 8, 1, 341, 1101, 1692, 1644, 1106, 546, 196, 52, 9, 1

Offset: 0

Wolfdieter Lang, Aug 02 2002

The g.f. for the row polynomials P(n,x) = Sum_{m=0..n} T(n,m)*x^m is 1/(1-(1+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.
Riordan array (1/(1-x-2*x^2), x/(1-x-2*x^2)). - Paul Barry, Mar 15 2005
Subtriangle (obtained by dropping the first column) of the triangle given by (0, 1, 2, -2, 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, Feb 19 2013
The number of ternary words of length n having k letters equal 2 and 0,1 avoid runs of odd lengths. - Milan Janjic, Jan 14 2017

			Triangle begins as:
    1;
    1,   1;
    3,   2,   1;
    5,   7,   3,   1;
   11,  16,  12,   4,   1;
   21,  41,  34,  18,   5,   1;
   43,  94,  99,  60,  25,   6,   1;
   85, 219, 261, 195,  95,  33,   7,   1;
  171, 492, 678, 576, 340, 140,  42,   8,   1;
The triangle (0, 1, 2, -2, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  3,  2,  1;
  0,  5,  7,  3,  1;
  0, 11, 16, 12,  4,  1;
  0, 21, 41, 34, 18,  5,  1; - _Philippe Deléham_, Feb 19 2013

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
W. Lang, First 10 rows.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Columns: A001045 (k=0), A073371 (k=1), A073372 (k=2), A073373 (k=3), A073374 (k=4), A073375 (k=5), A073376 (k=6), A073377 (k=7), A073378 (k=8), A073379 (k=9).
Cf. A002605 (row sums), A006130 (diagonal sums), A073399, A073400.
Cf. A000045, A077957.

A073370:= func< n,k | (&+[Binomial(n-j,k)*Binomial(n-k-j,j)*2^j: j in [0..Floor((n-k)/2)]]) >;
[A073370(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 01 2022

# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> (2^n - (-1)^n) / 3); # Peter Luschny, Oct 07 2022

T[n_, k_]:= T[n, k]= Sum[Binomial[n-j,k]*Binomial[n-k-j,j]*2^j, {j,0,Floor[(n- k)/2]}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 01 2022 *)

def A073370(n,k): return binomial(n,k)*sum( 2^j * binomial(2*j,j) * binomial(n-k,2*j)/binomial(n,j) for j in range(1+(n-k)//2))
flatten([[A073370(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Oct 01 2022

T(n, m) = Sum_{k=0..floor((n-m)/2)} binomial(n-k, m)*binomial(n-m-k, k)*2^k, if n > m, else 0.
Sum_{k=0..n} T(n, k) = A002605(n+1).
T(n, m) = (1*(n-m+1)*T(n, m-1) + 2*2*(n+m)*T(n-1, m-1))/((1^2 + 4*2)*m), n >= m >= 1, T(n, 0) = A001045(n+1), n >= 0, else 0.
T(n, m) = (p(m-1, n-m)*1*(n-m+1)*T(n-m+1) + q(m-1, n-m)*2*(n-m+2)*T(n-m))/(m!*9^m), n >= m >= 1, with T(n) = T(n, m=0) = A001045(n+1), else 0; p(k, n) = Sum_{j=0..k} (A(k, j)*n^(k-j) and q(k, n) = Sum_{j=0..k} B(k, j)*n^(k-j), with the number triangles A(k, m) = A073399(k, m) and B(k, m) = A073400(k, m).
G.f.: 1/(1-(1+2*x)*x)^(m+1) = 1/((1+x)*(1-2*x))^(m+1), m >= 0, for column m (without leading zeros).
T(n, 0) = A001045(n), T(1, 1) = 1, T(n, k) = 0 if k>n, T(n, k) = T(n-1, k-1) + 2*T(n-2, k) + T(n-1, k) otherwise. - Paul Barry, Mar 15 2005
G.f.: (1+x)*(1-2*x)/(1-x-2*x^2-x*y) for the triangle including the 1, 0, 0, 0, 0, ... column. - R. J. Mathar, Aug 11 2015
From Peter Bala, Oct 07 2019: (Start)
Recurrence for row polynomials: R(n,x) = (1 + x)*R(n-1,x) + 2*R(n-2,x) with R(0,x) = 1 and R(1,x) = 1 + x.
The row reverse polynomial x^n*R(n,1/x) is equal to the numerator polynomial of the finite continued fraction 1 + x/(1 - 2*x/(1 + ... + x/(1 - 2*x/(1)))) (with 2*n partial numerators). Cf. A110441. (End)
From G. C. Greubel, Oct 01 2022: (Start)
T(n, k) = binomial(n,k)*Sum_{j=0..floor((n-k)/2)} 2^j*binomial(2*j, j)*binomial(n-k, 2*j)/binomial(n, j).
T(n, k) = binomial(n, k)*Hypergeometric2F1([(k-n)/2, (k-n+1)/2], [-2*n], -8).
Sum_{k=0..n} (-1)^k * T(n, k) = A077957(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A006130(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A000045(n+1). (End)

A062111 Upper-right triangle resulting from binomial transform calculation for nonnegative integers.

Original entry on oeis.org

0, 1, 1, 4, 3, 2, 12, 8, 5, 3, 32, 20, 12, 7, 4, 80, 48, 28, 16, 9, 5, 192, 112, 64, 36, 20, 11, 6, 448, 256, 144, 80, 44, 24, 13, 7, 1024, 576, 320, 176, 96, 52, 28, 15, 8, 2304, 1280, 704, 384, 208, 112, 60, 32, 17, 9, 5120, 2816, 1536, 832, 448, 240, 128, 68, 36, 19, 10

Offset: 0

Henry Bottomley, May 30 2001

From Philippe Deléham, Apr 15 2007: (Start)
This triangle can be found in the Laisant reference in the following form:
  .......................5...11..
  ...................4...9...20..
  ...............3...7..16...36..
  ...........2...5..12..28.......
  .......1...3...8..20..48.......
  ...0...1...4..12..32..80....... (End)
Triangle A152920 reversed. - Philippe Deléham, Apr 21 2009

			As a lower triangle (T(n, k)):
    0;
    1,   1;
    4,   3,   2;
   12,   8,   5,  3;
   32,  20,  12,  7,  4;
   80,  48,  28, 16,  9,  5;
  192, 112,  64, 36, 20, 11,  6;
  448, 256, 144, 80, 44, 24, 13, 7;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
F. Ellermann, Illustration of binomial transforms
C.-A. Laisant, Sur les tableaux de sommes - Nouvelles applications, Compt. Rendus de l'Association Francaise pour l'Avancement des Sciences, Aout 04 1893, pp. 206-216 (table given on p. 212).

Rows include (essentially) A001787, A001792, A034007, A045623, A045891.
Diagonals include (essentially) A001477, A005408, A008586, A008598, A017113.
Column sums are A058877.
Cf. A111297, A159694, A159695, A159696, A159697. - Philippe Deléham, Apr 21 2009
Cf. A058877, A073371, A130129, A152920.

[2^(n-k-1)*(n+k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 28 2022

Table[2^(n-k-1)*(n+k), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 28 2022 *)

def A062111(n,k): return 2^(n-k-1)*(n+k)
flatten([[A062111(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Sep 28 2022

A(n, k) = A(n, k-1) + A(n+1, k) if k > n with A(n, n) = n.
A(n, k) = (k+n)*2^(k-n-1) if k >= n.
T(2*n, n) = 3*n*2^(n-1) = 3*A001787(n). - Philippe Deléham, Apr 21 2009
From G. C. Greubel, Sep 28 2022: (Start)
T(n, k) = 2^(n-k-1)*(n+k) for 0 <= k <= n, n >= 0.
T(m*n, n) = 2^((m-1)*n-1)*(m+1)*A001477(n), m >= 1.
T(2*n-1, n-1) = A130129(n-1).
T(2*n+1, n-1) = 12*A001787(n).
Sum_{k=0..n} T(n, k) = A058877(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = 3*A073371(n-2), n >= 2.
T(n, k) = A152920(n, n-k). (End)

A127978 a(n) = ((15n + 34)/54)2^(n-1) - (-1)^(n-1)(6n + 5)/27.

Original entry on oeis.org

3, 5, 15, 31, 75, 163, 367, 799, 1747, 3771, 8119, 17367, 37019, 78579, 166271, 350735, 737891, 1548587, 3242823, 6776903, 14136363, 29437795, 61205775, 127071871, 263464435, 545570203, 1128423127, 2331411639, 4811954107

Offset: 2

Artur Jasinski, Feb 09 2007

In the Bosma's paper there is an error (see table of the first few values at p. 37): for n=1 ((15*n+34)/54)*2^(n-1)-(-1)^(n-1)*(6*n+5)/27 is 1/2 and not 1.

G. C. Greubel, Table of n, a(n) for n = 2..1000
W. Bosma, Signed bits and fast exponentiation, Journal de Théorie des Nombres de Bordeaux, Vol. 13, Fasc. 1 (2001), p. 38 (Proposition 7).
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4)

Cf. A073371, A127976.

I:=[3, 5, 15, 31]; [n le 4 select I[n] else 2*Self(n-1) + 3*Self(n-2) -4*Self(n-3) -4*Self(n-4): n in [1..30]]; // G. C. Greubel, May 07 2018

Table[((15n+34)/54)2^(n-1) -((-1)^(n-1))(6n+5)/27, {n, 2, 50}]
LinearRecurrence[{2,3,-4,-4}, {3, 5, 15, 31}, 50] (* G. C. Greubel, May 07 2018 *)

x='x+O('x^30); Vec(x^2*(3-x-4*x^2-2*x^3)/((1+x)^2*(1-2*x)^2)) \\ G. C. Greubel, May 07 2018

G.f.: x^2*(3-x-4*x^2-2*x^3)/((1+x)^2*(1-2*x)^2). - Colin Barker, Apr 02 2012

Offset changed from 1 to 2 (according to Bosma's Proposition 5) from Bruno Berselli, Apr 02 2012

A127979 a(n) = (5n/18 + 19/54)2^n - (-1)^(n-1)(3n + 4)/27.

Original entry on oeis.org

1, 4, 9, 24, 55, 130, 293, 660, 1459, 3206, 6977, 15096, 32463, 69482, 148061, 314332, 665067, 1402958, 2951545, 6194368, 12971271, 27107634, 56545429, 117751204, 244823075, 508287510, 1053857713, 2182280840, 4513692479, 9325646586

Offset: 1

Artur Jasinski, Feb 09 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Bosma, Signed bits and fast exponentiation, Journal de Théorie des Nombres de Bordeaux, Vol. 13, Fasc. 1 (2001), p. 37 (Proposition 5).
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).

Cf. A073371, A127976, A127978, A073371.

[(5*n/18 +19/54)*2^n -(-1)^(n-1)*(3*n+4)/27: n in [1..50]]; // G. C. Greubel, May 08 2018

Table[(5n/18 + 19/54)2^(n) - ((-1)^(n - 1) )(3n + 4)/27, {n, 1, 100}]
LinearRecurrence[{2,3,-4,-4}, {1,4,9,24}, 50] (* G. C. Greubel, May 08 2018 *)

for(n=1, 50, print1((5*n/18 +19/54)*2^n -(-1)^(n-1)*(3*n+4)/27, ", ")) \\ G. C. Greubel, May 08 2018

G.f.: x*(1+2*x-2*x^2-2*x^3)/((1+x)^2*(1-2x)^2). - Bruno Berselli, Apr 02 2012

a(n) = 2*a(n-1)+3*a(n-2)-4*a(n-3)-4*a(n-4). - Wesley Ivan Hurt, May 07 2021

A073372 Second convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 3, 12, 34, 99, 261, 678, 1692, 4149, 9959, 23568, 55014, 127031, 290457, 658602, 1482240, 3314025, 7365915, 16285300, 35832810, 78500811, 171293293, 372412782, 806963364, 1743173469, 3754782351, 8066319768, 17285917742, 36957928479, 78847115649

Offset: 0

Wolfdieter Lang, Aug 02 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,3,-11,-6,12,8).

Third (m=2) column of triangle A073370.

Cf. A001045, A073371.

[(2^(n+3)*(16+15*n+3*n^2) +(-1)^n*(34+21*n+3*n^2))/162: n in [0..40]]; // G. C. Greubel, Sep 28 2022

CoefficientList[Series[-(-1+x+2x^2)^(-3),{x,0,78}],x] (* or *) Table[(-3*(-1)^n*n^2+3*2^(n+2)*n^2-15*(-1)^n*n+9*2^(n+2)*n-16*(-1)^n+2^(n+4))/162,{n,42}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

def A073372(n): return (2^(n+3)*(16+15*n+3*n^2) +(-1)^n*(34+21*n+3*n^2))/162
[A073372(n) for n in range(40)] # G. C. Greubel, Sep 28 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073371(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, 2) * binomial(n-k, k) * 2^k.
a(n) = ((30+9*n)*(n+1)*U(n+1) + 2*(33+9*n)*(n+2)*U(n))/162 with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1 - (1+2*x)*x)^3.
E.g.f.: (1/162)*(32*(4 + 9*x + 3*x^2)*exp(2*x) + (34 - 24*x + 3*x^2)*exp(-x)). - G. C. Greubel, Sep 28 2022

A127980 a(n) = (n + 2/3)2^(n-1) - 1/2 - (-1)^(n-1)(1/6).

Original entry on oeis.org

1, 5, 14, 37, 90, 213, 490, 1109, 2474, 5461, 11946, 25941, 55978, 120149, 256682, 546133, 1157802, 2446677, 5155498, 10835285, 22719146, 47535445, 99265194, 206918997, 430615210, 894784853, 1856678570, 3847574869, 7963585194

Offset: 1

Artur Jasinski, Feb 09 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Bosma, Signed bits and fast exponentiation, Journal de Théorie des Nombres de Bordeaux, Vol. 13, Fasc. 1 (2001), p. 38 (Proposition 7).
Index entries for linear recurrences with constant coefficients, signature (4,-3,-4,4).

Cf. A073371, A127976, A127978, A127979, A073371.

I:=[1,5,14,37]; [n le 4 select I[n] else 4*Self(n-1)-3*Self(n-2)-4*Self(n-3)+4*Self(n-4): n in [1..30]]; // G. C. Greubel, May 08 2018

Table[(n+2/3)2^(n-1) - 1/2 -(-1)^(n-1)*(1/6), {n, 1, 50}]
LinearRecurrence[{4,-3,-4,4}, {1,5,14,37}, 50] (* G. C. Greubel, May 08 2018 *)

x='x+O('x^30); Vec(x*(1+x-3*x^2)/((1-x)*(1+x)*(1-2*x)^2)) \\ G. C. Greubel, May 08 2018

G.f.: x*(1+x-3*x^2)/((1-x)*(1+x)*(1-2*x)^2). - Colin Barker, Apr 02 2012

E.g.f.: ((1 + 3*x)*cosh(2*x) - 2*sinh(x) + cosh(x)*((2 + 6*x)*sinh(x) - 1))/3. - Stefano Spezia, May 25 2023

A127984 a(n) = (n/3 + 7/9)*2^(n - 1) + (-1)^n/9.

Original entry on oeis.org

1, 3, 7, 17, 39, 89, 199, 441, 967, 2105, 4551, 9785, 20935, 44601, 94663, 200249, 422343, 888377, 1864135, 3903033, 8155591, 17010233, 35418567, 73633337, 152859079, 316902969, 656175559, 1357090361, 2803659207, 5786275385, 11930464711, 24576757305

Offset: 1

Artur Jasinski, Feb 09 2007

a(n) is the number of runs of strictly increasing parts in all compositions of n. a(3) = 7: (1)(1)(1), (12), (2)(1), (3). - Alois P. Heinz, Apr 30 2017
From Hugo Pfoertner, Feb 19 2020: (Start)
a(n)/2^(n-2) apparently is the expected number of flips of a fair coin to completion of a game where the player advances by 1 for heads and by 2 for tails, starting at position 0 and repeating to flip until the target n+1 is exactly reached. If the position n (1 below the target) is reached, the player stays at this position and continues to flip the coin and count the flips until he can advance by 1.
The expected number of flips for targets 1, 2, 3,... , found by inversion of the corresponding Markov matrices, is 2, 2, 3, 7/2, 17/4, 39/8, 89/16, 199/32, 441/64, ...
Target 1 needs an expected number of 2 flips and would require a(0) = 1/2.
  n=1, target n+1 = 2: 1 / 2^(1-2) = 2;
  n=2, target n+1 = 3: 3 / 2^(2-2) = 3;
  n=3, target n+1 = 4: 7 / 2^(3-2) = 7/2.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
W. Bosma, Signed bits and fast exponentiation, J. Th. des Nombres de Bordeaux Vol.13, Fasc. 1, 2001.
Aruna Gabhe, Problem 11623, Am. Math. Monthly 119 (2012) 161.
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A059570, A073371, A127976, A127978, A127979, A127980, A127981, A127982, A127983, A073371, A000337, A172481.

[(n/3+7/9)*2^(n-1)+(-1)^n/9: n in [1..35]]; // Vincenzo Librandi, Jun 15 2017

A127984:=n->(n/3 + 7/9)*2^(n - 1) + (-1)^n/9; seq(A127984(n), n=1..50); # Wesley Ivan Hurt, Mar 14 2014

Table[(n/3 + 7/9)2^(n - 1) + (-1)^n/9, {n, 50}] (* Artur Jasinski *)
CoefficientList[Series[(1 - 2 x^2) / ((-1 + 2 x)^2 (1 + x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 15 2017 *)

a(n) = (n/3 + 7/9)*2^(n - 1) + (-1)^n/9.
From R. J. Mathar, Apr 04 2008: (Start)
O.g.f.: -x*(-1+2x^2)/((-1+2x)^2*(1+x)).
a(n) = 3*a(n-1) - 4*a(n-3). (End)
a(n) + a(n+1) = A087447(n+1). - R. J. Mathar, Feb 21 2009
A172481(n) = a(n) + 2^(n-1). Application: Problem 11623, AMM 119 (2012) 161. - Stephen J. Herschkorn, Feb 11 2012
From Wolfdieter Lang, Jun 14 2017: (Start)
a(n) = f(n+1)*2^(n-1), where f(n) is a rational Fibonacci type sequence based on fuse(a,b) = (a+b+1)/2 with f(0) = 0, f(1) = 1 and f(n) = fuse(f(n-1),f(n-2)), for n >= 2. For fuse(a,b) see the Jeff Erickson link under A188545. Proof: f(n) = (3*n+4 - (-1)^n/2^(n-2))/9, n >= 0, by induction.
a(n) = a(n-1) + a(n-2) + 2^(n-2), n >= 1, with inputs a(-1) = 0, a(0) = 1/2.
(End)
E.g.f.: (2*exp(-x) + exp(2*x)*(7 + 6*x) - 9)/18. - Stefano Spezia, Feb 19 2020

A127981 a(n) = (n + 1/3)2^(n-1) - 1/2 + (-1)^(n-1)(1/6).

Original entry on oeis.org

1, 4, 13, 34, 85, 202, 469, 1066, 2389, 5290, 11605, 25258, 54613, 117418, 251221, 535210, 1135957, 2402986, 5068117, 10660522, 22369621, 46836394, 97867093, 204122794, 425022805, 883600042, 1834308949, 3802835626, 7874106709

Offset: 1

Artur Jasinski, Feb 09 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Bosma, Signed bits and fast exponentiation, Journal de Théorie des Nombres de Bordeaux, Vol. 13, Fasc. 1 (2001), p. 38 (Proposition 7).
Index entries for linear recurrences with constant coefficients, signature (4,-3,-4,4).

Cf. A073371, A127976, A127978, A127979, A127980, A073371.

[(n+1/3)*2^(n-1) -1/2 +(-1)^(n-1)*(1/6): n in [1..50]]; // G. C. Greubel, May 08 2018

Table[(n+1/3)*2^(n-1) -1/2 +(-1)^(n-1)*(1/6), {n, 1, 50}]
LinearRecurrence[{4,-3,-4,4}, {1,4,13,34}, 50] (* G. C. Greubel, May 08 2018 *)

for(n=1,50, print1((n+1/3)*2^(n-1) -1/2 +(-1)^(n-1)*(1/6), ", ")) \\ G. C. Greubel, May 08 2018

G.f.: x*(1-2*x^3)/((1-x)*(1+x)*(1-2*x)^2). - Colin Barker, Apr 02 2012

A127982 a(n) = (n - 1/3)*2^n - n/2 + 1/4 + (-1)^n/12.

Original entry on oeis.org

1, 6, 20, 57, 147, 360, 850, 1959, 4433, 9894, 21840, 47781, 103759, 223908, 480590, 1026723, 2184525, 4631202, 9786700, 20621985, 43341131, 90876576, 190141770, 397060767, 827675977, 1722460830, 3579139400, 7426714269, 15390299463

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Bosma, Signed bits and fast exponentiation, J. Th. des Nombres de Bordeaux Vol.13, Fasc. 1, 2001.
Index entries for linear recurrences with constant coefficients, signature (5,-7,-1,8,-4).

Cf. A073371, A127976, A127978, A127979, A127980, A073371, A000337.

[(n-1/3)*2^n -n/2 +1/4 +(-1)^n/12: n in [1..50]]; // G. C. Greubel, May 08 2018

Table[(n-1/3)*2^n -n/2 +1/4 +(-1)^n/12, {n, 1, 50}]
LinearRecurrence[{5,-7,-1,8,-4}, {1,6,20,57,147}, 50] (* G. C. Greubel, May 08 2018 *)

a(n) = (n-1/3)*2^n -n/2 +1/4 +(-1)^n/12 \\ G. C. Greubel, May 08 2018

a(n) = (n - 1/3)*2^n - n/2 + 1/4 + (-1)^n/12.

G.f.: -x*(3*x^2-x-1)/((1+x)*(2*x-1)^2*(x-1)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009 [checked and corrected by R. J. Mathar, Sep 16 2009]

A127983 a(n) = (n - 2/3)*2^n - n/2 + 3/4 - (-1)^n/12.

Original entry on oeis.org

1, 5, 18, 52, 137, 339, 808, 1874, 4263, 9553, 21158, 46416, 101029, 218447, 469668, 1004878, 2140835, 4543821, 9611938, 20272460, 42642081, 89478475, 187345568, 391468362, 816491167, 1700091209, 3534400158, 7337235784, 15211342493

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Bosma, Signed bits and fast exponentiation, J. Th. des Nombres de Bordeaux Vol.13, Fasc. 1, 2001.
Index entries for linear recurrences with constant coefficients, signature (5,-7,-1,8,-4).

Cf. A073371, A127976, A127978, A127979, A127980, A127981, A127982, A073371, A000337.

[(n-2/3)*2^n -n/2 +3/4 -(-1)^n/12: n in [1..50]]; // G. C. Greubel, May 08 2018

Table[(n-2/3)*2^n -n/2 +3/4 -(-1)^n/12, {n, 1, 50}]
LinearRecurrence[{5,-7,-1,8,-4}, {1,5,18,52,137}, 50] (* G. C. Greubel, May 08 2018 *)

a(n) = (n-2/3)*2^n -n/2 +3/4 -(-1)^n/12 \\ G. C. Greubel, May 08 2018

a(n) = (n - 2/3)*2^n - n/2 + 3/4 - (-1)^n/12.

G.f.: x*(1-2*x^3)/(1+x)/((2*x-1)^2*(x-1)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009 [checked and corrected by R. J. Mathar, Sep 16 2009]

cp's OEIS Frontend

A073370 Convolution triangle of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A062111 Upper-right triangle resulting from binomial transform calculation for nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A127978 a(n) = ((15*n + 34)/54)*2^(n-1) - (-1)^(n-1)*(6*n + 5)/27.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A127979 a(n) = (5*n/18 + 19/54)*2^n - (-1)^(n-1)*(3*n + 4)/27.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A073372 Second convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A127980 a(n) = (n + 2/3)*2^(n-1) - 1/2 - (-1)^(n-1)*(1/6).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A127984 a(n) = (n/3 + 7/9)*2^(n - 1) + (-1)^n/9.

Views

Author

Keywords

Comments

Links

A127978 a(n) = ((15n + 34)/54)2^(n-1) - (-1)^(n-1)(6n + 5)/27.

A127979 a(n) = (5n/18 + 19/54)2^n - (-1)^(n-1)(3n + 4)/27.

A127980 a(n) = (n + 2/3)2^(n-1) - 1/2 - (-1)^(n-1)(1/6).

A127981 a(n) = (n + 1/3)2^(n-1) - 1/2 + (-1)^(n-1)(1/6).