A083318 -id:A083318 - cp's OEIS frontend

A135224 Triangle A103451 * A007318 * A000012, read by rows. T(n, k) for 0 <= k <= n.

1, 3, 1, 5, 3, 1, 9, 7, 4, 1, 17, 15, 11, 5, 1, 33, 31, 26, 16, 6, 1, 65, 63, 57, 42, 22, 7, 1, 129, 127, 120, 99, 64, 29, 8, 1, 257, 255, 247, 219, 163, 93, 37, 9, 1, 513, 511, 502, 466, 382, 256, 130, 46, 10, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = A132750: (1, 4, 9, 21, 49, 113, ...).

Left border = A083318: (1, 3, 5, 9, 17, 33, ...).

			First few rows of the triangle:
   1;
   3,  1;
   5,  3,  1;
   9,  7,  4,  1;
  17, 15, 11,  5,  1;
  33, 31, 26, 16,  6,  1;
  65, 63, 57, 42, 22,  7,  1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A083318, A103451, A132750.

function T(n,k)
  if k eq 0 and n eq 0 then return 1;
  elif k eq 0 then return 2^n +1;
  else return (&+[Binomial(n, k+j): j in [0..n]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=0 and n=0 then 1
    elif k=0 then 2^n +1
    else add(binomial(n, k+j), j=0..n)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k] = If[k==n==0, 1, If[k==0, 2^n +1, Sum[Binomial[n, k + j], {j, 0, n}]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==0 && n==0, 1, if(k==0, 2^n +1, sum(j=0, n, binomial(n, k+j)) )); \\ G. C. Greubel, Nov 20 2019

def T(n, k):
    if (k==0 and n==0): return 1
    elif (k==0): return 2^n + 1
    else: return sum(binomial(n, k+j) for j in (0..n))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 20 2019

T(n, k) = A103451(n,k) * A007318(n,k) * A000012(n,k) as infinite lower triangular matrices.
T(n, k) = Sum_{j=0..n} binomial(n, k+j), with T(0,0) = 1 and T(n,0) = 2^n + 1. - G. C. Greubel, Nov 20 2019
T(n, k) = binomial(n, k)*hypergeom([1, k-n], [k+1], -1) - binomial(n, k+n+1)* hypergeom([1, k+1], [k+n+2], -1) + 0^k - 0^n. - Peter Luschny, Nov 20 2019

A143096 a(n) = 2*a(n-1)-1, with a(1)=1, a(2)=4, a(3)=5.

Original entry on oeis.org

1, 4, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297, 8589934593

Offset: 1

Gary W. Adamson and Roger L. Bagula, Jul 23 2008

			a(4) = 9 = 2*a(3) - 1 = 2*5 - 1.
a(4) = 9 = (1, 3, 3, 1) dot (1, 3, -2, 5) = (1 + 9 - 6 + 5).

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A065190.

Essentially the same as A083318, A048578 and A000051.

Join[{1,4},NestList[2#-1&,5,40]] (* or *) Join[{1,4},LinearRecurrence[ {3,-2},{5,9},40]] (* Harvey P. Dale, Feb 18 2014 *)

Binomial transform of 0, 1, 2, -4, 9, -13, 20, -26, 35, ... (offset 0).
O.g.f.: x*(1+x-5*x^2+2*x^3)/((1-x)*(1-2*x)). a(n) = 1+2^(n-1), n>2. - R. J. Mathar, Jul 31 2008
a(n) = A048578(n-2), n>=3. - R. J. Mathar, Aug 10 2008

A267089 T(n,k) is decimal conversion of 1's in an n X n table that lie on its principal diagonals.

Original entry on oeis.org

1, 3, 3, 5, 2, 5, 9, 6, 6, 9, 17, 10, 4, 10, 17, 33, 18, 12, 12, 18, 33, 65, 34, 20, 8, 20, 34, 65, 129, 66, 36, 24, 24, 36, 66, 129, 257, 130, 68, 40, 16, 40, 68, 130, 257, 513, 258, 132, 72, 48, 48, 72, 132, 258, 513, 1025, 514, 260, 136, 80, 32, 80, 136, 260, 514

Offset: 0

Kival Ngaokrajang, Jan 10 2016

Inspired by A137932 and A042948.
Conjectures:
(i) The first column is A083318.
(ii) T(n,k) = A086066(m) where m >= 10, n = m - 9*k, k = floor(m/10).

			See the "Illustration of initial terms" link for explicit examples.
Triangle begins:
n\k 0   1  2  3  4  5  6   7   8 ...
0   1
1   3   3
2   5   2  5
3   9   6  6  9
4  17  10  4 10 17
5  33  18 12 12 18 33
6  65  34 20  8 20 34 65
7 129  66 36 24 24 36 66 129
8 257 130 68 40 16 40 68 130 257
...

Kival Ngaokrajang, Illustration of initial terms, T(n,k) for n = 0..15, k = 0..10

Cf. A042948, A083318, A086066, A137932.

A162779 Rows of A162777 when written as a triangle converge to this sequence.

Original entry on oeis.org

1, 3, 5, 5, 7, 13, 15, 9, 7, 13, 17, 19, 29, 43, 39, 17, 7, 13, 17, 19, 29, 43, 41, 27, 29, 45, 55, 69, 103, 127, 95, 33, 7, 13, 17, 19, 29, 43, 41, 27, 29, 45, 55, 69, 103, 127, 97, 43, 29, 45, 55, 69, 103, 129, 111, 85, 105, 147, 181, 243, 335, 351, 223, 65, 7

Offset: 0

Omar E. Pol, Jul 23 2009

It appears that the right border of triangle gives A083318. - Omar E. Pol, Mar 15 2020

			From _Omar E. Pol_, Mar 15 2020: (Start)
Written as an irregular triangle in which row lengths give A011782 the sequence begins:
1;
3;
5,  5;
7, 13, 15,  9;
7, 13, 17, 19, 29, 43, 39, 17;
7, 13, 17, 19, 29, 43, 41, 27, 29, 45, 55, 69, 103, 127, 95, 33;
7, 13, 17, 19, 29, 43, 41, 27, 29, 45, 55, 69, 103, 127, 97, 43, 29, 45, 55, ...
(End)

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A083318, A139250, A153003, A153006, A162777.

More terms from Jinyuan Wang, Mar 15 2020

A176665 Triangle of polynomial coefficients of p(x,n) = Sum_{k=0..n} (k + 1)^n * k! * binomial(x, k), read by rows.

Original entry on oeis.org

1, 1, 2, 1, -5, 9, 1, 109, -165, 64, 1, -3303, 6188, -3494, 625, 1, 169711, -357254, 254434, -74635, 7776, 1, -13084359, 30063342, -24927719, 9549230, -1718079, 117649, 1, 1417404703, -3486909736, 3229823067, -1474126800, 354928391, -43216649, 2097152

Offset: 0

Roger L. Bagula, Apr 23 2010

Row sums are: A083318 = {1, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, ...}.

			Triangle begins as:
  1;
  1,         2;
  1,        -5,        9;
  1,       109,     -165,        64;
  1,     -3303,     6188,     -3494,     625;
  1,    169711,  -357254,    254434,  -74635,     7776;
  1, -13084359, 30063342, -24927719, 9549230, -1718079, 117649;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A083318.

(* First program *)
p[x_, n_]:= Sum[(k+1)^n*k!*Binomial[x, k], {k, 0, n}];
Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}]//Flatten
(* Second program *)
f[n_]:= CoefficientList[Sum[(k+1)^n*Product[x-j, {j,0,k-1}], {k,0,n}], x];
Table[f[n], {n, 0, 10}] (* G. C. Greubel, Feb 07 2021 *)

def p(n, x): return sum( (k+1)^n*factorial(k)*binomial(x, k) for k in (0..n))
flatten([[( p(n, x) ).series(x, n+1).list()[k] for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 07 2021

Let p(x,n) = Sum_{k=0..n} (k + 1)^n * k! * binomial(x, k) then the number triangle is given by T(n, m) = coefficients( p(x,n) ).

Edited by G. C. Greubel, Feb 07 2021

A265852 n such that A261807(n) = n^3 - n.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 15, 17, 31, 33, 35, 39, 41, 63, 65, 67, 79, 81, 103, 105, 127, 129, 131, 133, 135, 143, 145, 159, 161, 163, 169, 231, 255, 257, 259, 261, 265, 287, 289, 319, 321, 323, 359, 391, 399, 401, 419, 425, 511, 513, 515, 517, 519, 527, 543, 545

Offset: 1

Robert Israel, Dec 16 2015

n such that the base-2 representation of n^3 has a 1 whenever the representation of n has a 1.
All terms after the first are odd.
Contains A083318 and A000225.

			5 is in the sequence because A261807(5) = 120 = 5^3 - 5.  The base-2 representations of 5 and 5^3 are 101 and 1111101, and every 1 in 101 corresponds to a 1 in 1111101.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A261807.

select(t -> Bits[Xor](t,t^3) = t^3 - t, [$0..10000]);

for(n=0, 1e3, if(bitxor(n, n^3) == n^3-n, print1(n, ", "))) \\ Altug Alkan, Dec 16 2015

A287811 Number of septenary sequences of length n such that no two consecutive terms have distance 5.

Original entry on oeis.org

1, 7, 45, 291, 1881, 12159, 78597, 508059, 3284145, 21229047, 137226717, 887047443, 5733964809, 37064931183, 239591481525, 1548743682699, 10011236540769, 64713650292711, 418315611378573, 2704034619149571, 17479154549033145, 112987031151647583

Offset: 0

David Nacin, Jun 01 2017

			For n=2 the a(2) = 49-4 = 45 sequences contain every combination except these four: 05, 50, 16, 61.

Index entries for linear recurrences with constant coefficients, signature (6, 3).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473.

Cf. A287804-A287819.

Mathematica
```
LinearRecurrence[{6, 3}, {1,7}, 40]
```

def a(n):
 if n in [0, 1]:
  return [1, 7][n]
 return 6*a(n-1)-3*a(n-2)

a(n) = 6*a(n-1) + 3*a(n-2), a(0)=1, a(1)=7.
G.f.: (1 + x)/(1 - 6*x - 3*x^2).
a(n) = A090018(n-1)+A090018(n). - R. J. Mathar, Oct 20 2019

A287838 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 8.

Original entry on oeis.org

1, 11, 115, 1205, 12625, 132275, 1385875, 14520125, 152130625, 1593906875, 16699721875, 174966753125, 1833166140625, 19206495171875, 201230782421875, 2108340300078125, 22089556912890625, 231437270629296875, 2424820490857421875, 25405391261720703125

Offset: 0

David Nacin, Jun 07 2017

In general, the number of sequences on {0,1,...,10} such that no two consecutive terms have distance 6+k for k in {0,1,2,3,4} has generating function (-1 - x)/(-1 + 10*x + (2*k+1)*x^2).

Colin Barker, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (10,5).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287839.

LinearRecurrence[{10, 5}, {1, 11, 115}, 20]

Vec((1 + x) / (1 - 10*x - 5*x^2) + O(x^40)) \\ Colin Barker, Nov 25 2017

def a(n):
 if n in [0,1,2]:
  return [1, 11, 115][n]
 return 10*a(n-1) + 5*a(n-2)

For n > 2, a(n) = 10*a(n-1) + 5*a(n-2), a(0)=1, a(1)=11, a(2)=115.
G.f.: (-1 - x)/(-1 + 10*x + 5*x^2).
a(n) = (((5-sqrt(30))^n*(-6+sqrt(30)) + (5+sqrt(30))^n*(6+sqrt(30)))) / (2*sqrt(30)). - Colin Barker, Nov 25 2017

A295165 Numbers n such that !n and n!! (A000166(n) and A006882(n)) are coprime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 17, 18, 20, 24, 30, 32, 33, 44, 48, 54, 62, 65, 68, 72, 74, 80, 84, 98, 102, 110, 114, 128, 140, 150, 158, 168, 180, 182, 198, 200, 212, 224, 228, 230, 234, 252, 257, 264, 270, 272, 278, 282, 308, 312, 314, 318, 332, 348, 354, 374, 380, 384, 402, 410, 420, 422, 432

Offset: 1

Robert Israel, Nov 16 2017

nonn

Odd n is in the sequence iff !n is not divisible by any odd primes < n.
Even n is in the sequence iff !n is not divisible by any odd primes < n/2.
All odd terms are in A083318, all even terms > 2 are in A008864, but both of these are strict inclusions.
Odd terms include 1,3,5,9,17,33,65,257,513,32769.

			!5 = 44 and 5!! = 15 are coprime so 5 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000166, A006882, A008864, A083318.

sf:= proc(n) option remember; n*procname(n-1)+(-1)^n end proc:
sf(0):= 1:
select(n -> igcd(sf(n),doublefactorial(n))=1, [$0..1000]);

Select[Range[0, 1000], CoprimeQ[Subfactorial[#], #!!]&] (* Jean-François Alcover, Oct 16 2020 *)

A287805 Number of quinary sequences of length n such that no two consecutive terms have distance 2.

Original entry on oeis.org

1, 5, 19, 73, 281, 1083, 4175, 16097, 62065, 239307, 922711, 3557761, 13717913, 52893147, 203943935, 786361409, 3032030689, 11690820555, 45077144455, 173807214241, 670161078089, 2583988659867, 9963272432111, 38416111919777, 148123788152017, 571131629935179

Offset: 0

David Nacin, Jun 01 2017

			For n=2 the a(2)=19=25-6 sequences contain every combination except these six: 02,20,13,31,24,42.

Index entries for linear recurrences with constant coefficients, signature (4, 1, -6).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819.

LinearRecurrence[{4, 1, -6}, {1, 5, 19, 73}, 40]

def a(n):
 if n in [0,1,2,3]:
  return [1,5,19,73][n]
 return 4*a(n-1)+a(n-2)-6*a(n-3)

For n>0, a(n) = 4*a(n-1) + a(n-2) - 6*a(n-3), a(1)=5, a(2)=19, a(3)=73.

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

cp's OEIS Frontend

A135224 Triangle A103451 * A007318 * A000012, read by rows. T(n, k) for 0 <= k <= n.

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A143096 a(n) = 2*a(n-1)-1, with a(1)=1, a(2)=4, a(3)=5.

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A267089 T(n,k) is decimal conversion of 1's in an n X n table that lie on its principal diagonals.

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A162779 Rows of A162777 when written as a triangle converge to this sequence.

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A176665 Triangle of polynomial coefficients of p(x,n) = Sum_{k=0..n} (k + 1)^n * k! * binomial(x, k), read by rows.

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A265852 n such that A261807(n) = n^3 - n.

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Maple

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A287811 Number of septenary sequences of length n such that no two consecutive terms have distance 5.

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A287838 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 8.

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