A099036 -id:A099036 - cp's OEIS frontend

A212323 a(n) = 3^n - Fibonacci(n).

1, 2, 8, 25, 78, 238, 721, 2174, 6540, 19649, 58994, 177058, 531297, 1594090, 4782592, 14348297, 43045734, 129138566, 387417905, 1162257286, 3486777636, 10460342257, 31381041898, 94143150170, 282429490113, 847288534418, 2541865706936, 7625597288569

Offset: 0

Bruno Berselli, May 09 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-3).

Cf. A000045, A000244, A099036, A212262.

Magma
```
[3^n-Fibonacci(n): n in [0..27]];
```

Table[3^n - Fibonacci[n], {n, 0, 27}]
LinearRecurrence[{4,-2,-3},{1,2,8},30] (* Harvey P. Dale, Dec 08 2023 *)

for(n=0, 27, print1(3^n-fibonacci(n)", "));

G.f.: (1 - 2*x + 2*x^2)/((1 - 3*x)*(1 - x - x^2)).

A335712 The sum of the sizes of the minimal fixed points over all compositions of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 27, 54, 115, 237, 486, 997, 2030, 4122, 8350, 16881, 34054, 68609, 138052, 277500, 557328, 1118546, 2243589, 4498004, 9014053, 18058159, 36166338, 72415886, 144970116, 290170091, 580721926, 1162077483, 2325206168, 4652155420, 9307199819

Offset: 1

Margaret Archibald, Jun 18 2020

nonn

			Example: For n=3 the a(3)=2 values are the first 1s in 111 and 12 (the other compositions 21 and 3 do not have any fixed points).

M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

Alois P. Heinz, Table of n, a(n) for n = 1..500
M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

Cf. A099036, A335713, A335714.

my(N=44,x='x+O('x^N)); Vec( sum(j=1, N, prod(i=1, j-1, (x/(1-x)-x^i) ) *j*x^j * (1-x)/(1-2*x) ) ) \\ Joerg Arndt, Jun 18 2020

G.f.: Sum_{j>=1} (Product_{i=1..j-1} (x/(1-x)-x^i)) j x^j (1-x)/(1-2x).

a(21)-a(34) from Alois P. Heinz, Jun 18 2020

A228078 a(n) = 2^n - Fibonacci(n) - 1.

Original entry on oeis.org

0, 0, 2, 5, 12, 26, 55, 114, 234, 477, 968, 1958, 3951, 7958, 16006, 32157, 64548, 129474, 259559, 520106, 1041810, 2086205, 4176592, 8359950, 16730847, 33479406, 66987470, 134021309, 268117644, 536356682, 1072909783, 2146137378, 4292788986, 8586410013

Offset: 0

Reinhard Zumkeller, Aug 15 2013

a(n+1) = sum of n-th row of the triangle in A228074.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 30.
Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,2).

Haskell
```
a228078 = subtract 1 . a099036
```

[2^n - Fibonacci(n) - 1: n in [0..40]]; // Vincenzo Librandi, Aug 16 2013

Table[(2^n - Fibonacci[n] - 1), {n, 0, 40}] (* Vincenzo Librandi, Aug 16 2013 *)

concat([0,0], Vec(x^2*(3*x-2)/((x-1)*(2*x-1)*(x^2+x-1)) + O(x^100))) \\ Colin Barker, Mar 20 2015

a(n) = A000079(n) - A000045(n) - 1 = A000225(n) - A000045(n) = A000079(n) - A001611(n) = A099036(n) - 1.
From Colin Barker, Mar 20 2015: (Start)
a(n) = 4*a(n-1)-4*a(n-2)-a(n-3)+2*a(n-4) for n>3.
G.f.: x^2*(3*x-2) / ((x-1)*(2*x-1)*(x^2+x-1)). (End)
a(n) = (-1+2^n+(((1-sqrt(5))/2)^n-((1+sqrt(5))/2)^n)/sqrt(5)). - Colin Barker, Nov 02 2016

A335714 The sum of the sizes (positions) of fixed points over all compositions of n.

Original entry on oeis.org

1, 1, 4, 8, 19, 41, 89, 189, 398, 830, 1719, 3539, 7251, 14797, 30096, 61044, 123531, 249501, 503117, 1013165, 2037986, 4095546, 8223919, 16502823, 33097639, 66349021, 132954724, 266337584, 533388643, 1067965265, 2137907009, 4279099869, 8563658486, 17136379382

Offset: 1

Margaret Archibald, Jun 18 2020

			For n=3 the a(3)=4 values are the first 1 in the composition 111 and both values in the composition 12 (the compositions 21 and 3 have no fixed points).

M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

Cf. A099036, A335712, A335713.

Vec((x*(1-x)^3)/((1-2*x)*(1-x-x^2)^2) + O(x^40)) \\ Michel Marcus, Jun 18 2020

G.f.: x*(1-x)^3/((1-2*x)*(1-x-x^2)^2).

More terms from Michel Marcus, Jun 18 2020

A125104 Triangle read by rows counting compositions (ordered partitions) by minimal part size.

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 0, 1, 6, 1, 0, 0, 2, 13, 1, 0, 0, 1, 3, 27, 1, 0, 0, 0, 2, 5, 56, 1, 0, 0, 0, 1, 2, 9, 115, 1, 0, 0, 0, 0, 2, 3, 15, 235, 1, 0, 0, 0, 0, 1, 2, 5, 25, 478, 1, 0, 0, 0, 0, 0, 2, 2, 8, 42, 969, 1, 0, 0, 0, 0, 0, 1, 2, 3, 12, 70, 1959, 1, 0, 0, 0, 0, 0, 0, 2, 2, 5, 18, 116, 3952, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 8, 27, 192, 7959, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 11, 41, 317, 16007

Offset: 0

Alford Arnold, Nov 28 2006, corrected Nov 28 2006

The diagonals of this array can be generated from Table A099238 as follows: A000079 - A000045 = [1, 2, 4, 8, 16, 32, ...] - [0, 1, 1, 2, 3, 5, ...] = [1, 1, 3, 6, 13, 27, ...] = A099036, A000045 - A000930, A000930 - A003269, A003269 - A003520, etc.

			Row 4 of the array is (1, 0, 1, 6) because there are six compositions with minimum part of size one: 1111, 31, 13, 211, 121, 112; one of size two: 22; none of size three; and 1 of size four: 4.
Triangle (after 45-degree counterclockwise rotation) begins:
1 1 3 6 13 27 56 115 235 478 969 1959 3952 7959
.1 0 1 2 3 5 9 15 25 42 70 116 192
..1 0 0 1 2 2 3 5 8 12 18 27
...1 0 0 0 1 2 2 2 3 5 8
....1 0 0 0 0 1 2 2 2 2
.....1 0 0 0 0 0 1 2 2
......1 0 0 0 0 0 0 1
.......1 0 0 0 0 0 0
........1 0 0 0 0 0

Cf. A000079, A000045, A000930, A003269, A003520, A099036, A099238.

Cf. A105147.

Edited by N. J. A. Sloane, Dec 21 2006

More terms from Vladeta Jovovic, Jul 10 2007

A131350 2*A007318 - A049310 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 1, 4, 1, 2, 4, 6, 1, 1, 8, 9, 8, 1, 2, 7, 20, 16, 10, 1, 1, 12, 24, 40, 25, 12, 1, 2, 10, 42, 60, 70, 36, 14, 1, 1, 16, 46, 112, 125, 112, 49, 16, 1, 2, 13, 72, 148, 252, 231, 168, 64, 18, 1

Offset: 0

Gary W. Adamson, Jul 02 2007

Row sums = A099036 starting (1, 3, 6, 13, 27, 56, 115,...).

			First few rows of the triangle are:
1;
2, 1;
1, 4, 1;
2, 4, 6, 1;
1, 8, 9, 8, 1;
2, 7, 20, 16, 10, 1;
1, 12, 24, 40, 25, 12, 1;
...

Cf. A007318, A049310, A131350.

A354267 A Fibonacci-Pascal triangle read by rows: T(n, n) = 1, T(n, n-1) = n - 1, T(n, 0) = T(n-1, 1) and T(n, k) = T(n-1, k-1) + T(n-1, k) for 0 < k < n-1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 4, 3, 1, 3, 5, 7, 7, 4, 1, 5, 8, 12, 14, 11, 5, 1, 8, 13, 20, 26, 25, 16, 6, 1, 13, 21, 33, 46, 51, 41, 22, 7, 1, 21, 34, 54, 79, 97, 92, 63, 29, 8, 1, 34, 55, 88, 133, 176, 189, 155, 92, 37, 9, 1, 55, 89, 143, 221, 309, 365, 344, 247, 129, 46, 10, 1

Offset: 0

Peter Luschny, May 31 2022

			[0]  1;
[1]  0,  1;
[2]  1,  1,  1;
[3]  1,  2,  2,  1;
[4]  2,  3,  4,  3,  1;
[5]  3,  5,  7,  7,  4,  1;
[6]  5,  8, 12, 14, 11,  5,  1;
[7]  8, 13, 20, 26, 25, 16,  6,  1;
[8] 13, 21, 33, 46, 51, 41, 22,  7, 1;
[9] 21, 34, 54, 79, 97, 92, 63, 29, 8, 1;

Index entries for triangles and arrays related to Pascal's triangle

Cf. A212804 (first column, which is also row 0 of A352744), A099036 (row sums), A228074 (subtriangle), A000045 (Fibonacci), A371870 (central terms).

T := proc(n, k) option remember;
if n = k then 1 elif k = n-1 then n-1 elif k = 0 then T(n-1, 1) else
T(n-1, k) + T(n-1, k-1) fi end: seq(seq(T(n, k), k = 0..n), n = 0..11);

T[n_, k_] := Which[n == k, 1, k == n-1, n-1, k == 0, T[n-1, 1], True, T[n-1, k] + T[n-1, k-1]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023 *)

from functools import cache
@cache
def A354267row(n):
    if n == 0: return [1]
    if n == 1: return [0, 1]
    row = A354267row(n - 1) + [1]
    s = row[1]
    for k in range(n-1, 0, -1):
        row[k] += row[k - 1]
    row[0] = s
    return row
for n in range(10): print(A354267row(n))

T(n, 0) = Fibonacci(n - 1).

A357459 The total number of fixed points among all partitions of n, when parts are written in nondecreasing order.

Original entry on oeis.org

0, 1, 1, 3, 4, 7, 10, 17, 22, 34, 46, 66, 88, 123, 160, 218, 283, 375, 482, 630, 799, 1030, 1299, 1651, 2066, 2602, 3230, 4032, 4976, 6157, 7554, 9288, 11326, 13837, 16793, 20393, 24632, 29763, 35783, 43031, 51527, 61683, 73577, 87729, 104252, 123834, 146664

Offset: 0

Jeremy Lovejoy, Sep 29 2022

nonn

For instance, the partition (1,3,3,3,5) = (y(1),y(2),y(3),y(4),y(5)) has 3 fixed points, since y(i) = i for i=1,3,5.

			The 7 partitions of 5 are (1,1,1,1,1), (1,1,1,2), (1,2,2), (1,1,3), (1,4), (2,3), and (5), containing 1, 1, 2, 2, 1, 0, and 0 fixed points, respectively, and so a(5) = 1+1+2+2+1+0+0=7.

A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.

Cf. A001522 (parts decreasing), A099036.

G.f.: (Product_{k>=1}(1/(1-q^k)))*Sum_{n>=1}q^(2*n-1)*Product_{k=n..2*n-2}(1-q^k).

A130711 Number of compositions of n such that the smallest part divides every part.

Original entry on oeis.org

1, 2, 4, 8, 14, 32, 57, 123, 239, 493, 970, 1997, 3953, 8017, 16024, 32281, 64550, 129742, 259561, 520606, 1041871, 2087177, 4176594, 8362063, 16730862, 33483361, 66987710, 134029333, 268117646, 536373213, 1072909785, 2146169660

Offset: 1

Vladeta Jovovic, Jul 01 2007

			a(5)=14 because among the 16 compositions of 5 only 2+3 and 3+2 do not qualify; the others, except for the composition 5, have at least one component equal to 1.

Cf. A083710.

G:=sum(x^n*(1-x^n)^2/((1-2*x^n)*(1-x^n-x^(2*n))), n=1..50); Gser:=series(G, x =0,40): seq(coeff(Gser,x,n),n=1..33); # Emeric Deutsch, Sep 08 2007

Inverse Moebius transform of A099036.

G.f.: Sum_{n>0} x^n*(1-x^n)^2/((1-2*x^n)*(1-x^n-x^(2*n))).

More terms from Emeric Deutsch, Sep 08 2007

A317023 Square array A(n,k), n >= 0, k >= 0, read by ascending antidiagonals, where the sequence of row n is the expansion of (1-x^(n+1))/((1-x)^(n+1)).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 6, 2, 0, 1, 5, 10, 9, 2, 0, 1, 6, 15, 20, 12, 2, 0, 1, 7, 21, 35, 34, 15, 2, 0, 1, 8, 28, 56, 70, 52, 18, 2, 0, 1, 9, 36, 84, 126, 125, 74, 21, 2, 0, 1, 10, 45, 120, 210, 252, 205, 100, 24, 2, 0, 1, 11, 55, 165, 330, 462, 461, 315, 130, 27, 2, 0, 1, 12, 66

Offset: 0

Werner Schulte, Jul 19 2018

Conjecture: alternating row sums of the triangle give A106510 for n >= 0.

			The square array A(n,k) begins:
  n\k |  0  1  2   3    4    5    6     7     8     9     10
  ====+=====================================================
   0  |  1  0  0   0    0    0    0     0     0     0      0
   1  |  1  2  2   2    2    2    2     2     2     2      2
   2  |  1  3  6   9   12   15   18    21    24    27     30
   3  |  1  4 10  20   34   52   74   100   130   164    202
   4  |  1  5 15  35   70  125  205   315   460   645    875
   5  |  1  6 21  56  126  252  461   786  1266  1946   2877
   6  |  1  7 28  84  210  462  924  1715  2996  4977   7924
   7  |  1  8 36 120  330  792 1716  3432  6434 11432  19412
   8  |  1  9 45 165  495 1287 3003  6435 12870 24309  43749
   9  |  1 10 55 220  715 2002 5005 11440 24310 48620  92377
  10  |  1 11 66 286 1001 3003 8008 19448 43758 92378 184756
  etc.
The triangle T(n,k) begins:
  n\k |  0  1  2   3   4   5   6   7   8   9 10 11 12
  ====+==============================================
   0  |  1
   1  |  1  0
   2  |  1  2  0
   3  |  1  3  2   0
   4  |  1  4  6   2   0
   5  |  1  5 10   9   2   0
   6  |  1  6 15  20  12   2   0
   7  |  1  7 21  35  34  15   2   0
   8  |  1  8 28  56  70  52  18   2   0
   9  |  1  9 36  84 126 125  74  21   2   0
  10  |  1 10 45 120 210 252 205 100  24   2  0
  11  |  1 11 55 165 330 462 461 315 130  27  2  0
  12  |  1 12 66 220 495 792 924 786 460 164 30  2  0
  etc.

Row sums of the triangle give A099036 for n >= 0.
Cf. A000984 (main diagonal), A000012 (column 0), A087156 (column 1).
Cf. A099036, A106510.
In the square array; row 0..12 are: A000007, A040000, A008486, A005893, A008487, A008488, A008489, A008490, A008491, A008492, A008493, A008494, A008495.
A173265 is based on the same square array, but is read by descending antidiagonals with special treatment of column 0.

nmax:=15;; A:=List([0..nmax],n->List([0..nmax],k->Binomial(n+k,k)-Binomial(k-1,k-1-n)));;   b:=List([2..nmax],n->OrderedPartitions(n,2));;
a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->A[b[i][j][2]][b[i][j][1]]))); # Muniru A Asiru, Jul 20 2018

Table[SeriesCoefficient[(1 - x^(# + 1))/((1 - x)^(# + 1)), {x, 0, k}] &[n - k], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 20 2018 *)

T(n,k) = binomial(n+k,k) - binomial(k-1,k-1-n); \\ Michel Marcus, Aug 07 2018

A(n,k) = binomial(n+k,k) - binomial(k-1,k-1-n) for n >= 0 and k >= 0 with binomial(i,j) = 0 if i < j or j < 0.
G.f.: Sum_{k>=0,n>=0} A(n,k)*x^k*y^n = ((1-x)^2)/((1-x-y)*(1-x-x*y)).
Seen as a triangle T(n,k) = A(n-k,k) = binomial(n,k)-binomial(k-1,2*k-1-n) for 0 <= k <= n with binomial(i,j) = 0 if i < j or j < 0.
Mirror image of the triangle equals A173265 except column 0.

cp's OEIS Frontend

A212323 a(n) = 3^n - Fibonacci(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A335712 The sum of the sizes of the minimal fixed points over all compositions of n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A228078 a(n) = 2^n - Fibonacci(n) - 1.

Views

Author

Keywords

Comments

Links

Programs

Haskell

Magma

Mathematica

PARI

Formula

A335714 The sum of the sizes (positions) of fixed points over all compositions of n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A125104 Triangle read by rows counting compositions (ordered partitions) by minimal part size.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A131350 2*A007318 - A049310 as infinite lower triangular matrices.

Views

Author

Keywords

Comments

Examples

Crossrefs

A354267 A Fibonacci-Pascal triangle read by rows: T(n, n) = 1, T(n, n-1) = n - 1, T(n, 0) = T(n-1, 1) and T(n, k) = T(n-1, k-1) + T(n-1, k) for 0 < k < n-1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A357459 The total number of fixed points among all partitions of n, when parts are written in nondecreasing order.

Views

Author

Keywords

Comments