A139634 -id:A139634 - cp's OEIS frontend

A173803 Primes in A139634.

11, 31, 71, 151, 311, 631, 2551, 655351, 2621431, 10485751, 5497558138871, 10995116277751, 175921860444151, 737869762948382064631, 396140812571321687967719751671

Offset: 1

Vincenzo Librandi, Mar 05 2010

nonn

Cf. A139634

Select[10*2^Range[0,100]-9,PrimeQ] (* Harvey P. Dale, Jun 01 2019 *)

Edited by Jon E. Schoenfield, Jun 23 2010

A139697 Binomial transform of [1, 12, 12, 12, ...].

Original entry on oeis.org

1, 13, 37, 85, 181, 373, 757, 1525, 3061, 6133, 12277, 24565, 49141, 98293, 196597, 393205, 786421, 1572853, 3145717, 6291445, 12582901, 25165813, 50331637, 100663285, 201326581, 402653173, 805306357, 1610612725, 3221225461, 6442450933, 12884901877

Offset: 1

Gary W. Adamson, Apr 29 2008

The binomial transform of [1, c, c, c, ...] has the terms a(n) = 1 - c + c*2^(n-1) if the offset 1 is chosen. The o.g.f. of the a(n) is x*(1+(c-2)*x)/((2x-1)*(x-1)). This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

			a(4) = 85 = (1, 3, 3, 1) dot (1, 12, 12, 12) = (1 + 36 + 36 + 12).

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A139634, A139635.

seq(12*2^(n-1)-11,n=1..25); # Emeric Deutsch, May 05 2008

a=1; lst={a}; k=12; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)

A007318 * [1, 12, 12, 12, ...].
a(n) = 12*2^(n-1) - 11. - Emeric Deutsch, May 05 2008
a(n) = 2*a(n-1) + 11 (with a(1)=1). - Vincenzo Librandi, Nov 24 2010
From Colin Barker, Oct 10 2013: (Start)
a(n) = 3*2^(n+1) - 11.
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(10*x+1) / ((x-1)*(2*x-1)). (End)

More terms from Emeric Deutsch, May 05 2008

More terms from Colin Barker, Oct 10 2013

A099003 Number of 4 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (11;0).

Original entry on oeis.org

1, 16, 46, 106, 226, 466, 946, 1906, 3826, 7666, 15346, 30706, 61426, 122866, 245746, 491506, 983026, 1966066, 3932146, 7864306, 15728626, 31457266, 62914546, 125829106, 251658226, 503316466, 1006632946, 2013265906, 4026531826

Offset: 0

Sergey Kitaev, Nov 13 2004

An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by 2^(m+n) - 2^m - 2^n + 2.
Binomial transform of 1,15,15,... (15 infinitely repeated). - Gary W. Adamson, Apr 29 2008
The binomial transform of [1, c, c, c, ...] has the terms a(n) = 1 - c + c*2^(n-1) if the offset 1 is chosen. The o.g.f. of the a(n) is x*(1+(c-2)x)/((2x-1)*(x-1)). This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A048489 (m=3).

LinearRecurrence[{3,-2},{1,16},40] (* Harvey P. Dale, May 20 2018 *)

a(n) = 15*2^n - 14.

O.g.f.: (1+13x)/((x-1)(2x-1)). - R. J. Mathar, May 06 2008

A139635 Binomial transform of [1, 11, 11, 11, ...].

Original entry on oeis.org

1, 12, 34, 78, 166, 342, 694, 1398, 2806, 5622, 11254, 22518, 45046, 90102, 180214, 360438, 720886, 1441782, 2883574, 5767158, 11534326, 23068662, 46137334, 92274678, 184549366, 369098742, 738197494, 1476394998, 2952790006, 5905580022, 11811160054

Offset: 1

Gary W. Adamson, Apr 29 2008

A007318 * [1, 11, 11, 11, ...].

The binomial transform of [1, c, c, c, ...] has the terms a(n) = 1 - c + c*2^(n-1) if the offset 1 is chosen. The o.g.f. of the a(n) is x*(1+(c-2)*x)/((2x-1)*(x-1)). This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

			a(4) = 78 = (1, 3, 3, 1) dot (1, 11, 11, 11) = (1 + 33 + 33 + 11).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A139634.

seq(11*2^(n-1)-10,n=1.. 25); # Emeric Deutsch, May 03 2008

a=1; lst={a}; k=11; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)
CoefficientList[Series[(9 x + 1)/((x - 1) (2 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 13 2014 *)
LinearRecurrence[{3,-2},{1,12},40] (* Harvey P. Dale, Oct 26 2015 *)

Vec(x*(9*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 11 2014

a(n) = 11*2^(n-1) - 10. - Emeric Deutsch, May 03 2008
a(n) = 2*a(n-1) + 10, with n > 1, a(1)=1. - Vincenzo Librandi, Nov 24 2010
From Colin Barker, Mar 11 2014: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(9*x+1) / ((x-1)*(2*x-1)). (End)

More terms from Emeric Deutsch, May 03 2008

More terms from Colin Barker, Mar 11 2014

A139698 Binomial transform of [1, 25, 25, 25, ...].

Original entry on oeis.org

1, 26, 76, 176, 376, 776, 1576, 3176, 6376, 12776, 25576, 51176, 102376, 204776, 409576, 819176, 1638376, 3276776, 6553576, 13107176, 26214376, 52428776, 104857576, 209715176, 419430376, 838860776, 1677721576, 3355443176, 6710886376, 13421772776, 26843545576

Offset: 1

Gary W. Adamson, Apr 29 2008

The binomial transform of [1, c, c, c, ...] has the terms a(n)=1-c+c*2^(n-1) if the offset 1 is chosen. The o.g.f. of the a(n) is x{1+(c-2)x}/{(2x-1)(x-1)}. This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

			a(3) = 76 = (1, 2, 1) dot (1, 25, 25) = (1 + 50 + 25).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A139634, A139635, A139697.

[25*2^(n-1)-24 : n in [1..40]]; // Wesley Ivan Hurt, Jan 17 2017

seq(25*2^(n-1)-24,n=1..25); # Emeric Deutsch, May 03 2008

LinearRecurrence[{3,-2},{1,26},40] (* Harvey P. Dale, Jul 25 2021 *)

Vec(x*(23*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 11 2014

A007318 * [1, 25, 25, 25, ...].
a(n) = 25*2^(n-1)-24. - Emeric Deutsch, May 03 2008
a(n) = 2*a(n-1) + 24 (with a(1)=1). - Vincenzo Librandi, Nov 24 2010
a(n) = 3*a(n-1)-2*a(n-2). G.f.: x*(23*x+1) / ((x-1)*(2*x-1)). - Colin Barker, Mar 11 2014

More terms from Emeric Deutsch, May 03 2008

More terms from Colin Barker, Mar 11 2014

A139700 Binomial transform of [1, 30, 30, 30, ...].

Original entry on oeis.org

1, 31, 91, 211, 451, 931, 1891, 3811, 7651, 15331, 30691, 61411, 122851, 245731, 491491, 983011, 1966051, 3932131, 7864291, 15728611, 31457251, 62914531, 125829091, 251658211, 503316451, 1006632931, 2013265891, 4026531811, 8053063651, 16106127331

Offset: 1

Gary W. Adamson, Apr 29 2008

The binomial transform of [1, c, c, c, ...] has the terms a(n) = 1 - c + c*2^(n-1) if the offset 1 is chosen. The o.g.f. of the a(n) is x{1+(c-2)x}/{(2x-1)(x-1)}. This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

			a(3) = 91 = (1, 2, 1) dot (1, 30, 30) = (1 + 60 + 30).

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A139699, A139698, A139697, A139635, A139634.

seq(30*2^(n-1)-29,n=1..27); # Emeric Deutsch, May 07 2008

LinearRecurrence[{3,-2},{1,31},30] (* Harvey P. Dale, Apr 18 2018 *)

Vec(x*(28*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 11 2014

A007318 * [1, 30, 30, 30, ...].
a(n) = 30*2^(n-1) - 29. - Emeric Deutsch, May 07 2008
a(n) = 2*a(n-1) + 29 (with a(1)=1). - Vincenzo Librandi, Nov 24 2010
From Colin Barker, Mar 11 2014: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(28*x+1) / ((x-1)*(2*x-1)). (End)

More terms from Emeric Deutsch, May 07 2008

More terms from Colin Barker, Mar 11 2014

A139701 Binomial transform of [1, 100, 100, 100, ...].

Original entry on oeis.org

1, 101, 301, 701, 1501, 3101, 6301, 12701, 25501, 51101, 102301, 204701, 409501, 819101, 1638301, 3276701, 6553501, 13107101, 26214301, 52428701, 104857501, 209715101, 419430301, 838860701, 1677721501, 3355443101, 6710886301, 13421772701, 26843545501

Offset: 1

Gary W. Adamson, Apr 29 2008

The binomial transform of [1, c, c, c, ...] has the terms a(n)=1-c+c*2^(n-1) if the offset 1 is chosen. The o.g.f. of a(n) is x{1+(c-2)x}/{(2x-1)(x-1)}. This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

			a(3) = 301 = (1, 2, 1) dot (1, 100, 100) = (1 + 200 + 100).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A007318, A099003, A139634, A139635, A139697, A139698, A139700, A139701.

[100*2^(n-1)-99 : n in [1..30]]; // Wesley Ivan Hurt, Aug 16 2016

a:=proc(n) options operator, arrow: 100*2^(n-1)-99 end proc: seq(a(n), n=1.. 30); # Emeric Deutsch, May 03 2008

100*2^(Range[30] - 1) - 99 (* Wesley Ivan Hurt, Aug 16 2016 *)
LinearRecurrence[{3, -2}, {1, 101}, 40] (* Vincenzo Librandi, Aug 17 2016 *)

Vec(x*(98*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 11 2014

A007318 * [1, 100, 100, 100, ...].
a(n) = 100*2^(n-1)-99. - Emeric Deutsch, May 03 2008
a(n) = 2*a(n-1)+99 for n > 1. [Vincenzo Librandi, Nov 24 2010]
a(n) = 3*a(n-1) - 2*a(n-2) for n > 2. G.f.: x*(98*x+1) / ((x-1)*(2*x-1)). - Colin Barker, Mar 11 2014

More terms from Emeric Deutsch, May 03 2008

More terms from Colin Barker, Mar 11 2014

A172171 (1, 9) Pascal Triangle read by horizontal rows. Same as A093644, but mirrored and without the additional row/column (1, 9, 9, 9, 9, ...).

Original entry on oeis.org

1, 1, 10, 1, 11, 19, 1, 12, 30, 28, 1, 13, 42, 58, 37, 1, 14, 55, 100, 95, 46, 1, 15, 69, 155, 195, 141, 55, 1, 16, 84, 224, 350, 336, 196, 64, 1, 17, 100, 308, 574, 686, 532, 260, 73, 1, 18, 117, 408, 882, 1260, 1218, 792, 333, 82

Offset: 1

Mark Dols, Jan 28 2010

Binomial transform of A017173.

			Triangle begins:
  1;
  1, 10;
  1, 11,  19;
  1, 12,  30,  28;
  1, 13,  42,  58,   37;
  1, 14,  55, 100,   95,   46;
  1, 15,  69, 155,  195,  141,   55;
  1, 16,  84, 224,  350,  336,  196,   64;
  1, 17, 100, 308,  574,  686,  532,  260,   73;
  1, 18, 117, 408,  882, 1260, 1218,  792,  333,   82;
  1, 19, 135, 525, 1290, 2142, 2478, 2010, 1125,  415,  91;
  1, 20, 154, 660, 1815, 3432, 4620, 4488, 3135, 1540, 506, 100;

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

Cf. A007318, A017173, A050489 (central terms), A093644, A139634 (row sums).

T[n_, k_]:= T[n, k]= If[k<1 || k>n, 0, If[k==1, 1, If[n==2 && k==2, 10, T[n-1, k] + 2*T[n-1, k-1] - T[n-2, k-1] - T[n-2, k-2]]]];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 24 2022 *)

@CachedFunction
def T(n,k):
    if (k<1 or k>n): return 0
    elif (k==1): return 1
    elif (n==2 and k==2): return 10
    else: return T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2)
flatten([[T(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 24 2022

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(n,1) = 1, T(2,2) = 10, T(n,k) = 0 if k < 1 or if k > n.
Sum_{k=0..n} T(n, k) = A139634(n).
T(2*n-1, n) = A050489(n).

More terms from Philippe Deléham, Dec 25 2013

A062001 Table by antidiagonals of n-Stohr sequences: T(n,k) is least positive integer not the sum of at most n distinct terms in the n-th row from T(n,1) through to T(n,k-1).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 7, 4, 2, 1, 6, 10, 8, 4, 2, 1, 7, 13, 15, 8, 4, 2, 1, 8, 16, 22, 16, 8, 4, 2, 1, 9, 19, 29, 31, 16, 8, 4, 2, 1, 10, 22, 36, 46, 32, 16, 8, 4, 2, 1, 11, 25, 43, 61, 63, 32, 16, 8, 4, 2, 1, 12, 28, 50, 76, 94, 64, 32, 16, 8, 4, 2, 1, 13, 31, 57, 91, 125, 127, 64, 32, 16, 8, 4, 2, 1

Offset: 1

Henry Bottomley, May 29 2001

			Array begins as:
  1, 2, 3, 4,  5,  6,  7,   8,   9, ... A000027;
  1, 2, 4, 7, 10, 13, 16,  19,  22, ... A033627;
  1, 2, 4, 8, 15, 22, 29,  36,  43, ... A026474;
  1, 2, 4, 8, 16, 31, 46,  61,  76, ... A051039;
  1, 2, 4, 8, 16, 32, 63,  94, 125, ... A051040;
  1, 2, 4, 8, 16, 32, 64, 127, 190, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 255, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;
Antidiagonal triangle begins as:
   1;
   2,  1;
   3,  2,  1;
   4,  4,  2,  1;
   5,  7,  4,  2,   1;
   6, 10,  8,  4,   2,   1;
   7, 13, 15,  8,   4,   2,  1;
   8, 16, 22, 16,   8,   4,  2,  1;
   9, 19, 29, 31,  16,   8,  4,  2,  1;
  10, 22, 36, 46,  32,  16,  8,  4,  2, 1;
  11, 25, 43, 61,  63,  32, 16,  8,  4, 2, 1;
  12, 28, 50, 76,  94,  64, 32, 16,  8, 4, 2, 1;
  13, 31, 57, 91, 125, 127, 64, 32, 16, 8, 4, 2, 1;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Rows include A000027, A033627, A026474, A051039, A051040.
Diagonals include A000079, A000225, A033484, A036563, A048487.
Cf. A048488, A048489, A048490, A048491, A139634, A139635, A139697.
A048483 can be seen as half this table.

T[n_, k_]:= If[kG. C. Greubel, May 03 2022 *)

def A062001(n,k):
    if (kA062001(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, May 03 2022

If k <= n+1 then A(n, k) = 2^(k-1), while if k > n+1, A(n, k) = (2^n - 1)*(k - n) + 1 (array).
T(n, k) = A(k, n-k+1) (antidiagonals).
T(2*n-1, n) = A000079(n-1), n >= 1.
T(2*n, n) = A000079(n), n >= 1.
T(2*n+1, n) = A000225(n+1), n >= 1.
T(2*n+2, n) = A033484(n), n >= 1.
T(2*n+3, n) = A036563(n+3), n >= 1.
T(2*n+4, n) = A048487(n), n >= 1.
From G. C. Greubel, May 03 2022: (Start)
T(n, k) = (2^k - 1)*(n-2*k+1) + 1 for k < n/2, otherwise 2^(n-k).
T(2*n+5, n) = A048488(n), n >= 1.
T(2*n+6, n) = A048489(n), n >= 1.
T(2*n+7, n) = A048490(n), n >= 1.
T(2*n+8, n) = A048491(n), n >= 1.
T(2*n+9, n) = A139634(n), n >= 1.
T(2*n+10, n) = A139635(n), n >= 1.
T(2*n+11, n) = A139697(n), n >= 1. (End)

cp's OEIS Frontend

A173803 Primes in A139634.

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A139697 Binomial transform of [1, 12, 12, 12, ...].

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A099003 Number of 4 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (11;0).

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A139635 Binomial transform of [1, 11, 11, 11, ...].

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A139698 Binomial transform of [1, 25, 25, 25, ...].

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Magma

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A139700 Binomial transform of [1, 30, 30, 30, ...].

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A139701 Binomial transform of [1, 100, 100, 100, ...].

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