A047849 -id:A047849 - cp's OEIS frontend

A196791 a(n) = A047848(9, n).

1, 2, 14, 158, 1886, 22622, 271454, 3257438, 39089246, 469070942, 5628851294, 67546215518, 810554586206, 9726655034462, 116719860413534, 1400638324962398, 16807659899548766, 201691918794585182, 2420303025535022174, 29043636306420266078, 348523635677043192926

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856.

Cf. A001021 (first differences).

Magma
```
[(12^n+10)/11: n in [0..20]];
```

LinearRecurrence[{13,-12},{1,2},30] (* Harvey P. Dale, Sep 07 2015 *)
(12^Range[0,40] +10)/11 (* G. C. Greubel, Jan 17 2025 *)

def A196791(n): return (pow(12, n) + 10)//11
print([A196791(n) for n in range(41)]) # G. C. Greubel, Jan 17 2025

a(n) = (12^n + 10)/11.
a(n) = 12*a(n-1) - 10, with a(0) = 1.
G.f.: (1-11*x)/((1-x)*(1-12*x)). - Bruno Berselli, Oct 11 2011
From Elmo R. Oliveira, Aug 30 2024: (Start)
E.g.f.: exp(x)*(exp(11*x) + 10)/11.
a(n) = 13*a(n-1) - 12*a(n-2) for n > 1. (End)

A196792 a(n) = A047848(10, n).

Original entry on oeis.org

1, 2, 15, 184, 2381, 30942, 402235, 5229044, 67977561, 883708282, 11488207655, 149346699504, 1941507093541, 25239592216022, 328114698808275, 4265491084507564, 55451384098598321, 720867993281778162, 9371283912663116095, 121826690864620509224, 1583746981240066619901

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (14,-13).

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856.

Cf. A001022 (first differences).

Magma
```
[(13^n+11)/12: n in [0..20]];
```

(13^Range[0,40] +11)/12 (* G. C. Greubel, Jan 17 2025 *)

def A196792(n): return (pow(13, n) + 11)//12
print([A196792(n) for n in range(41)]) # G. C. Greubel, Jan 17 2025

a(n) = (13^n + 11)/12.
a(n) = 13*a(n-1) - 11, with a(0) = 1.
G.f.: (1-12*x)/((1-x)*(1-13*x)). - Bruno Berselli, Oct 11 2011
From Elmo R. Oliveira, Aug 30 2024: (Start)
E.g.f.: exp(x)*(exp(12*x) + 11)/12.
a(n) = 14*a(n-1) - 13*a(n-2) for n > 1. (End)

A196793 a(n) = A047848(n, n).

Original entry on oeis.org

1, 2, 7, 44, 401, 4682, 66431, 1111112, 21435889, 469070942, 11488207655, 311505013052, 9267595563617, 300239975158034, 10523614159962559, 396861212733968144, 16024522975978953761, 689852631578947368422, 31544039619835776489479

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856

Magma
```
[((n+3)^n+n+1)/(n+2): n in [0..20]]
```

A196793:=n->((n+3)^n+n+1)/(n+2); seq(A196793(k), k=0..50); # Wesley Ivan Hurt, Nov 08 2013

Table[((n+3)^n+n+1)/(n+2), {n,0,50}] (* Wesley Ivan Hurt, Nov 08 2013 *)

def A196793(n): return (pow(n+3, n) +n+1)//(n+2)
print([A196793(n) for n in range(51)]) # G. C. Greubel, Jan 17 2025

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

A343138 Array A(k, n) read by descending antidiagonals: A(k, n) = Sum_{m=0..n} F(k, m)^2, where F are the k-generalized Fibonacci numbers A092921.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 6, 2, 1, 0, 1, 5, 15, 6, 2, 1, 0, 1, 6, 40, 22, 6, 2, 1, 0, 1, 7, 104, 71, 22, 6, 2, 1, 0, 1, 8, 273, 240, 86, 22, 6, 2, 1, 0, 1, 9, 714, 816, 311, 86, 22, 6, 2, 1, 0, 1, 10, 1870, 2752, 1152, 342, 86, 22, 6, 2, 1, 0

Offset: 0

Peter Luschny, Apr 06 2021

			Array starts:
  n = 0  1  2  3   4   5    6     7     8     9      10
------------------------------------------------------------
[k=0] 0, 1, 1, 1,  1,  1,   1,    1,    1,     1,     1, ...  [A057427]
[k=1] 0, 1, 2, 3,  4,  5,   6,    7,    8,     9,    10, ...  [A001477]
[k=2] 0, 1, 2, 6, 15, 40, 104,  273,  714,  1870,  4895, ...  [A001654]
[k=3] 0, 1, 2, 6, 22, 71, 240,  816, 2752,  9313, 31514, ...  [A107239]
[k=4] 0, 1, 2, 6, 22, 86, 311, 1152, 4288, 15952, 59216, ...
[k=5] 0, 1, 2, 6, 22, 86, 342, 1303, 5024, 19424, 75120, ...
[k=6] 0, 1, 2, 6, 22, 86, 342, 1366, 5335, 20960, 82464, ...
[k=7] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21591, 85600, ...
[k=8] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 86871, ...
[k=9] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, ...
[...]
[ oo] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, ...  [A047849]
Note that the first parameter in A(k, n) refers to rows, and the second parameter refers to columns, as always. The usual naming convention for the indices is not adhered to because the row sequences are the sums of the squares of the k-bonacci numbers.

Greg Dresden and Yichen Wang, Sums and convolutions of k-bonacci and k-Lucas numbers, draft 2021.

Russell Jay Hendel, Sums of Squares: Methods for Proving Identity Families, arXiv:2103.16756 [math.NT], 2021

Cf. A092921, A057427, A001477, A001654, A107239, A047849, A343125.

F := (k, n) -> (F(k, n) := `if`(n<2, n, add(F(k, n-j), j = 1..min(k, n)))):
A := (k, n) -> add(F(k, m)^2, m = 0..n):
seq(seq(A(k, n-k), k=0..n), n = 0..11);
# The following two functions implement Russell Jay Hendel's formula for k >= 2:
T := (k, n) -> (n + 3)*(k - n) - 4:
H := (k, n) -> (2*add(j*add((m-k+1)*F(k, n+j)*F(k, n+m), m = j+1..k), j = 1..k-1)
- add(T(k, j-1)*F(k, n+j)^2, j = 1..k) + (k - 2))/(2*k - 2):
seq(lprint([k], seq(H(k, n), n = 0..11)), k=2..9); # Peter Luschny, Apr 07 2021

A343138[k_, len_] := Take[Accumulate[LinearRecurrence[PadLeft[{1}, k, 1], PadLeft[{1}, k], len + k]^2], -len - 2];
A343138[0, len_] := Table[Boole[n != 0], {n, 0, len}];
A343138[1, len_] := Table[n, {n, 0, len}];
(* Table: *) Table[A343138[k, 12], {k, 0, 9}]
(* Sequence / descending antidiagonals: *)
Table[Table[Take[A343138[j, 12], {k + 1 - j, k + 1 - j}], {j, 0, k}], {k, 0, 10}] // Flatten (* Georg Fischer, Apr 08 2021 *)

Russell Jay Hendel gives the following representation, valid for k >= 2:

A(n, k) = Sum_{m=0..n} F(k, m)^2 = (1/(2*k-2)) * (2*Sum_{j=1..k-1}(j*Sum_{m=j+1..k} (m-k+1) * F(k, n+j) * F(k, n+m)) - Sum_{j=1..k}(A343125(k, j-1) * F(k, n+j)^2) + (k - 2)). - Peter Luschny, Apr 07 2021

A377885 Cogrowth sequence of the 16-element quasihedral group SD16 = .

Original entry on oeis.org

1, 1, 1, 4, 28, 136, 544, 2080, 8128, 32512, 130816, 524800, 2099200, 8390656, 33550336, 134201344, 536854528, 2147516416, 8590065664, 34359869440, 137438691328, 549754765312, 2199022206976, 8796095119360, 35184380477440, 140737496743936, 562949936644096

Offset: 0

Sean A. Irvine, Nov 10 2024

Gives the even terms, all the odd terms are 0.

Also called QD16, Q8:C2. Gap identifier 16,8.

Index entries for linear recurrences with constant coefficients, signature (6, -12, 16).

Cf. A047849 (D4), A007582 (D8), A071930 (Q8), A377840 (C8 X C2), A377883 (M4(2)).

G.f.: (6*x^3-7*x^2+5*x-1) / ((4*x-1) * (4*x^2-2*x+1)).

A120462 Expansion of -2x(-3-2x+4x^2) / ((x-1)(2x+1)(2x-1)*(1+x)).

Original entry on oeis.org

0, 6, 4, 22, 20, 86, 84, 342, 340, 1366, 1364, 5462, 5460, 21846, 21844, 87382, 87380, 349526, 349524, 1398102, 1398100, 5592406, 5592404, 22369622, 22369620, 89478486, 89478484, 357913942, 357913940, 1431655766, 1431655764, 5726623062

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jun 28 2006

Top element of the vector obtained by multiplying the n-th power of the 6 X 6 matrix [[0, 1, 0, 0, 0, 1], [1, 0, 1, 0, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 0, 1], [1, 0, 0, 0, 1, 0]] by the column vector [0, 1, 1, 2, 3, 5].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

M = {{0, 1, 0, 0, 0, 1}, {1, 0, 1, 0, 0, 0}, {0, 1, 0, 1, 0, 0}, {0, 0, 1, 0, 1, 0}, {0, 0, 0, 1, 0, 1}, {1, 0, 0, 0, 1, 0}} v[1] = {0, 1, 1, 2, 3, 5} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
LinearRecurrence[{0,5,0,-4},{0,6,4,22},40] (* Harvey P. Dale, Jul 28 2024 *)

concat(0, Vec(2*x*(3+2*x-4*x^2)/((1-x)*(1+x)*(1-2*x)*(1+2*x)) + O(x^40))) \\ Colin Barker, Sep 09 2016

a(2*n+1) = A047849(n+2). a(2*n)= 2*A020988(n). - R. J. Mathar, Nov 07 2011
From Colin Barker, Sep 09 2016: (Start)
a(n) = -2*(1/6 + (-2)^n/3 + (-1)^n/2 - 2^n).
a(n) = 5*a(n-2)-4*a(n-4) for n>3.
(End)

A142595 Triangle T(n,k) = 2T(n-1, k-1) + 2T(n-1, k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 22, 40, 22, 1, 1, 46, 124, 124, 46, 1, 1, 94, 340, 496, 340, 94, 1, 1, 190, 868, 1672, 1672, 868, 190, 1, 1, 382, 2116, 5080, 6688, 5080, 2116, 382, 1, 1, 766, 4996, 14392, 23536, 23536, 14392, 4996, 766, 1

Offset: 1

Roger L. Bagula, Sep 22 2008

This triangle is dominated by the Eulerian numbers A008292.

			Triangle begins as:
  1;
  1,   1;
  1,   4,    1;
  1,  10,   10,     1;
  1,  22,   40,    22,     1;
  1,  46,  124,   124,    46,     1;
  1,  94,  340,   496,   340,    94,     1;
  1, 190,  868,  1672,  1672,   868,   190,    1;
  1, 382, 2116,  5080,  6688,  5080,  2116,  382,   1;
  1, 766, 4996, 14392, 23536, 23536, 14392, 4996, 766, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A008292, A047849 (row sums), A119258.

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return 2*(T(n-1, k-1) + T(n-1, k));
  end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 13 2021

T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, 2*(T[n-1, k-1] +T[n-1, k])];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 13 2021 *)
a[0] = {1}; a[1] = {1, 1};
a[n_]:= a[n]= 2*Join[a[n-1], {-1/2}] + 2*Join[{-1/2}, a[n-1]];
Table[a[n], {n,0,10}]//Flatten (* Roger L. Bagula, Dec 09 2008 *)

@CachedFunction
def T(n,k): return 1 if k==1 or k==n else 2*(T(n-1, k-1) + T(n-1, k))
flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 13 2021

Sum_{k=0..n} T(n, k) = (4^(n-1) + 2)/3 = A047849(n-1).

Edited by N. J. A. Sloane, Dec 11 2008

A308124 a(n) = (2 + 7*4^n)/3.

Original entry on oeis.org

3, 10, 38, 150, 598, 2390, 9558, 38230, 152918, 611670, 2446678, 9786710, 39146838, 156587350, 626349398, 2505397590, 10021590358, 40086361430, 160345445718, 641381782870, 2565527131478, 10262108525910, 41048434103638, 164193736414550, 656774945658198, 2627099782632790

Offset: 0

Paul Curtz, Jul 23 2019

Consider A092808 and its differences:
1,  0,  3,  1, 11,  5,  43,  21, 171, ...
-1, 3, -2, 10, -6, 38, -22, 150, ... = b(n).
a(n) is the second bisection of b(n). The first is A047849.
a(n) mod 9 is the period 9 sequence: repeat [3, 1, 2, 6, 4, 5, 0, 7, 8].
b(n) + b(n+1) = A135520(n).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000302, A001045, A002042, A092808, A047849, A135520, A141495.

LinearRecurrence[{5,-4},{3,10},30] (* Paolo Xausa, Nov 13 2023 *)
(2+7*4^Range[0,30])/3 (* Harvey P. Dale, Aug 15 2025 *)

a(n) = (2 + 7*4^n)/3; \\ Stefano Spezia, Jul 23 2019

Vec((3 - 5*x) / ((1 - x)*(1 - 4*x)) + O(x^40)) \\ Colin Barker, Jul 23 2019

a(n) = 4*a(n-1) - 2 for n=1,2,... , a(0) = 3.
a(n+1) = a(n) + A002042(n).
Binomial transform of A141495(n+1) = 3, 7, 21, ....
From Colin Barker, Jul 23 2019: (Start)
G.f.: (3 - 5*x) / ((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n>1.
(End)
a(n+2) = a(n) + 35*A000302(n) for n=0,1,2, ... .

a(14)-a(25) from Stefano Spezia, Jul 23 2019

A354602 a(n) is the number of trivial braids on 3 strands with 2*n crossings.

Original entry on oeis.org

1, 4, 28, 244, 2412, 25804, 290932, 3403404, 40914508

Offset: 0

Alexei Vernitski, Jul 08 2022

In other words, a(n) is the number of products of 2*n generators in the braid group B_3 which are equal to the identity element of the group.
Only braids with an even number of crossings are considered because a braid with an odd number of crossings cannot be trivial.
If we do include the 0s corresponding to the odd values of the number of crossings, a group-theoretical name for this sequence is the cogrowth sequence of B_3.

Cf. A000984 (number of trivial braids on 2 strands with 2*n crossings), A047849 (number of trivial permutations of 3 elements after 2*n adjacent transpositions).

A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Philippe Deléham, Oct 07 2009

			Triangle begins :
1 ;
0,2 ;
0,1,2 ;
0,1,1,2 ;
0,1,1,1,2 ;
0,1,1,1,1,2 ;
0,1,1,1,1,1,2 ; ...

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k)= A166122(n), A166114(n), A084222(n), A084247(n), A000034(n), A040000(n), A000027(n+1), A000079(n), A007051(n), A047849(n), A047850(n), A047851(n), A047852(n), A047853(n), A047854(n), A047855(n), A047856(n) for x= -5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^k= A000007(n), A000027(n+1), A033484(n), A134931(n), A083597(n) for x= 0,1,2,3,4 respectively.
T(n,k)= A166065(n,k)/2^(n-k).
G.f.: (1-x+x*y)/(1-x-x*y+x^2*y). - Philippe Deléham, Nov 09 2013
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013

cp's OEIS Frontend

A196791 a(n) = A047848(9, n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Python

Formula

A196792 a(n) = A047848(10, n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Python

Formula

A196793 a(n) = A047848(n, n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

Formula

A343138 Array A(k, n) read by descending antidiagonals: A(k, n) = Sum_{m=0..n} F(k, m)^2, where F are the k-generalized Fibonacci numbers A092921.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A377885 Cogrowth sequence of the 16-element quasihedral group SD16 = .

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A120462 Expansion of -2*x*(-3-2*x+4*x^2) / ((x-1)*(2*x+1)*(2*x-1)*(1+x)).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

A142595 Triangle T(n,k) = 2*T(n-1, k-1) + 2*T(n-1, k), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A308124 a(n) = (2 + 7*4^n)/3.

A120462 Expansion of -2x(-3-2x+4x^2) / ((x-1)(2x+1)(2x-1)*(1+x)).

A142595 Triangle T(n,k) = 2T(n-1, k-1) + 2T(n-1, k), read by rows.