A048483 -id:A048483 - cp's OEIS frontend

A048487 a(n) = T(4,n), array T given by A048483.

1, 6, 16, 36, 76, 156, 316, 636, 1276, 2556, 5116, 10236, 20476, 40956, 81916, 163836, 327676, 655356, 1310716, 2621436, 5242876, 10485756, 20971516, 41943036, 83886076, 167772156, 335544316, 671088636, 1342177276, 2684354556, 5368709116, 10737418236, 21474836476

Offset: 0

Clark Kimberling

Row sums of triangle A131113. - Gary W. Adamson, Jun 15 2007
a(n) = sum of (n+1)-th row terms of triangle A134636. This sequence is the binomial transform of 1, 5, 5, (5 continued). - Gary W. Adamson, Nov 04 2007
Row sums of triangle A135856. - Gary W. Adamson, Dec 01 2007

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

Cf. A010716 (n-th difference of a(n), a(n-1), ..., a(0)).
Diagonal of A062001.
A column of A119726.
Cf. A048483, A123208, A131113, A134636, A135856.

[5*2^n-4: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

a=1; lst={a}; k=5; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 15 2008 *)
a=6; lst={1, a}; k=10; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)

a(n) = 5*2^n - 4. - Henry Bottomley, May 29 2001
a(n) = 2*a(n-1) + 4 for n > 0 with a(0) = 1. - Paul Barry, Aug 25 2004
From Colin Barker, Sep 13 2012: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n >= 2.
G.f.: (1 + 3*x)/((1 - x)*(1 - 2*x)). (End)
a(n) = A123208(2*n). - Philippe Deléham, Apr 15 2013
E.g.f.: exp(x)*(5*exp(x) - 4). -  Stefano Spezia, Oct 03 2023

A048490 a(n) = T(7,n), array T given by A048483.

Original entry on oeis.org

1, 9, 25, 57, 121, 249, 505, 1017, 2041, 4089, 8185, 16377, 32761, 65529, 131065, 262137, 524281, 1048569, 2097145, 4194297, 8388601, 16777209, 33554425, 67108857, 134217721, 268435449, 536870905, 1073741817, 2147483641, 4294967289, 8589934585

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is (8, 8, 8, ...).

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

a=1; lst={a}; k=8; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *)

Vec((6*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Nov 26 2014

a(n) = 8 * 2^n - 7. - Ralf Stephan, Jan 09 2009
Equals binomial transform of [1, 8, 8, 8, ...]. - Gary W. Adamson, Apr 29 2008
a(n) = 2*a(n-1) + 7 for n > 0, a(0)=1. - Vincenzo Librandi, Aug 06 2010
For n>=1, a(n) = 6<+>(n+3), where the operation <+> is defined in A206853. - Vladimir Shevelev, Feb 17 2012
From Colin Barker, Nov 26 2014: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: (6*x+1) / ((x-1)*(2*x-1)). (End)

More terms from Colin Barker, Nov 26 2014

A048491 a(n) = T(8,n), array T given by A048483.

Original entry on oeis.org

1, 10, 28, 64, 136, 280, 568, 1144, 2296, 4600, 9208, 18424, 36856, 73720, 147448, 294904, 589816, 1179640, 2359288, 4718584, 9437176, 18874360, 37748728, 75497464, 150994936, 301989880, 603979768, 1207959544, 2415919096

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is (9, 9, 9, ...).

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

a=1; lst={a}; k=9; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *)
9*2^Range[0,30]-8 (* Harvey P. Dale, May 14 2021 *)

a(n) = 9 * 2^n - 8. - Ralf Stephan
Equals binomial transform of [1, 9, 9, 9, ...]. - Gary W. Adamson, Apr 29 2008
a(n) = 2*a(n-1) + 8, with a(0)=1. - Vincenzo Librandi, Aug 06 2010
a(n) = 2*A053209(n), n>0. - Philippe Deléham, Apr 15 2013
a(n) = 3*a(n-1) - 2*a(n-2) with a(0)=1, a(1)=10. - Philippe Deléham, Apr 15 2013
G.f.: (1+7*x)/((1-x)*(1-2*x)). - Philippe Deléham, Apr 15 2013

A048488 a(n) = 6*2^n - 5.

Original entry on oeis.org

1, 7, 19, 43, 91, 187, 379, 763, 1531, 3067, 6139, 12283, 24571, 49147, 98299, 196603, 393211, 786427, 1572859, 3145723, 6291451, 12582907, 25165819, 50331643, 100663291, 201326587, 402653179, 805306363, 1610612731

Offset: 0

Clark Kimberling, Dec 11 1999

a(n) = T(5, n), array T given by A048483.

Sequence is generated by the Northwest (NW) direction of circles put around circle(s). See illustration. - Odimar Fabeny, Aug 09 2008

			a(2) = 6 * 2^2 - 5 = 6 * 4 - 5 = 24 - 5 = 19.
a(3) = 6 * 2^3 - 5 = 6 * 8 - 5 = 48 - 5 = 43.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Odimar Fabeny, Illustration for this sequence
Index entries for linear recurrences with constant coefficients, signature (3,-2).

n-th difference of a(n), a(n-1), ..., a(0) is (6, 6, 6, ...).

Cf. A000079, A007283. - Omar E. Pol, Dec 21 2008

[6*2^n - 5: n in [0..30]]; // Vincenzo Librandi, May 18 2011

A048488:=n->6*2^n - 5; seq(A048488(n), n=0..30); # Wesley Ivan Hurt, May 09 2014

6(2^Range[0, 35]) - 5 (* Alonso del Arte, May 03 2014 *)

a(n)=6*2^n-5 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*a(n-1) + 5, n > 0, a(0) = 1. - Paul Barry, Aug 25 2004
Equals binomial transform of [1, 6, 6, 6, ...]. - Gary W. Adamson, Apr 29 2008
a(n) = A000079(n)*6 - 5 = A007283(n)*2 - 5. - Omar E. Pol, Dec 21 2008
From Colin Barker, Sep 17 2012: (Start)
a(n) = 3*2^(1+n) - 5. a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: (1+4*x)/((1-x)*(1-2*x)). (End)
a(n + 1) = 3 * 2^n - 5 = 1 + 2 * (Sum_{i=0..n-1} 3i) for n > 0. - Gerasimov Sergey and Alonso del Arte, May 03 2014
a(n) = A000225(n+1)+4*A000225(n). - R. J. Mathar, Feb 27 2019

Simpler definition from Ralf Stephan

A048489 a(n) = 7 * 2^n - 6.

Original entry on oeis.org

1, 8, 22, 50, 106, 218, 442, 890, 1786, 3578, 7162, 14330, 28666, 57338, 114682, 229370, 458746, 917498, 1835002, 3670010, 7340026, 14680058, 29360122, 58720250, 117440506, 234881018, 469762042, 939524090, 1879048186

Offset: 0

Clark Kimberling

Number of 3 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (11;0). 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 i1Sergey Kitev, Nov 13 2004
Row sums of triangle A131115. - N. J. A. Sloane, Nov 10 2007
Equals binomial transform of [1, 7, 7, 7, ...]. - Gary W. Adamson, Apr 28 2008
Number of variations of a Componium barrel which produces n phrases. This sequence describes the variations produced by the Componium, a historical mechanical organ. Another way of describing it is: Number of base 8 n-digit numbers produced by repeating or advancing along this 14-step cycle: (0,1,2,3,4,5,6,7,6,5,4,3,2,1). Subset of A126362. - Jim Bumgardner, Dec 10 2013
a(n) = the sum of the terms in row(n) in a triangle with first column T(n,0)=
1+2*n and diagonal T(n,n)=1+4*n with T(i,j)=T(i-1,j-1) + T(i-1,j). - J. M. Bergot, May 11 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jim Bumgardner, Variations of the Componium, 2013
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).

a(n)=T(6, n), array T given by A048483.
n-th difference of a(n), a(n-1), ..., a(0) is (7, 7, 7, ...).
Cf. A131115.

A048489:=n->7*2^n-6: seq(A048489(n), n=0..40); # Wesley Ivan Hurt, Apr 18 2017

CoefficientList[Series[(1 + 5 x)/((2 x - 1) (x - 1)), {x, 0, 28}], x] (* Michael De Vlieger, May 22 2018 *)
7*2^Range[0,30]-6 (* or *) LinearRecurrence[{3,-2},{1,8},30] (* Harvey P. Dale, May 19 2019 *)

a(n)=7<Charles R Greathouse IV, Dec 10 2013

a(n) = A000079(n)*7-6 = A005009(n)-6. - Omar E. Pol, Dec 21 2008
a(n) = 2*a(n-1)+6 with n>0, a(0)=1. - Vincenzo Librandi, Aug 06 2010
G.f.: ( 1+5*x ) / ( (2*x-1)*(x-1) ). - R. J. Mathar, Oct 21 2012
a(n) = A063757(2*n). - Philippe Deléham, Apr 13 2013

A048493 a(n) = (n+1)*2^n - n.

Original entry on oeis.org

1, 3, 10, 29, 76, 187, 442, 1017, 2296, 5111, 11254, 24565, 53236, 114675, 245746, 524273, 1114096, 2359279, 4980718, 10485741, 22020076, 46137323, 96468970, 201326569, 419430376, 872415207, 1811939302, 3758096357, 7784628196, 16106127331, 33285996514

Offset: 0

Clark Kimberling

Old definition was: "a(n) = T(n,n), array T given by A048483".

Also the number of connected induced subgraphs in the n-sunlet graph. - Eric W. Weisstein, May 25 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Sunlet Graph
Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A058877.

[(n+1)*2^n-n: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011

Table[(n + 1) 2^n - n, {n, 20}] (* Eric W. Weisstein, May 25 2017 *)
Table[2^n + (2^n - 1) n, {n, 20}] (* Eric W. Weisstein, May 25 2017 *)
LinearRecurrence[{6, -13, 12, -4}, {3, 10, 29, 76}, 20] (* Eric W. Weisstein, May 25 2017 *)

Vec(-(4*x^3-5*x^2+3*x-1)/((x-1)^2*(2*x-1)^2) + O(x^100)) \\ Colin Barker, Nov 26 2014

a(n) = (n+1)*2^n-n. - Vladeta Jovovic, Feb 28 2003
a(n) = 5*a(n-1)-7*a(n-2)-a(n-3)+8*a(n-4)-4*a(n-5). - Colin Barker, Nov 26 2014
G.f.: -(4*x^3-5*x^2+3*x-1) / ((x-1)^2*(2*x-1)^2). - Colin Barker, Nov 26 2014

Description changed to more explicit formula by Eric W. Weisstein, May 25 2017

A119726 Triangular array read by rows: T(n,1) = T(n,n) = 1, T(n,k) = 4T(n-1, k-1) + 2T(n-1, k).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 16, 26, 1, 1, 36, 116, 106, 1, 1, 76, 376, 676, 426, 1, 1, 156, 1056, 2856, 3556, 1706, 1, 1, 316, 2736, 9936, 18536, 17636, 6826, 1, 1, 636, 6736, 30816, 76816, 109416, 84196, 27306, 1, 1, 1276, 16016, 88576, 276896, 526096, 606056, 391396, 109226, 1

Offset: 1

Zerinvary Lajos, Jun 14 2006

Second column is A048487.

Second diagonal is A020989.

			Triangle begins as:
  1;
  1,    1;
  1,    6,     1;
  1,   16,    26,     1;
  1,   36,   116,   106,      1;
  1,   76,   376,   676,    426,      1;
  1,  156,  1056,  2856,   3556,   1706,      1;
  1,  316,  2736,  9936,  18536,  17636,   6826,      1;
  1,  636,  6736, 30816,  76816, 109416,  84196,  27306,      1;
  1, 1276, 16016, 88576, 276896, 526096, 606056, 391396, 109226, 1;

TERMESZET VILAGA XI.TERMESZET-TUDOMANY DIAKPALYAZAT 133.EVF. 6.SZ. jun. 2002. Vegh Lea (and Vegh Erika): "Pascal-tipusu haromszogek" http://www.kfki.hu/chemonet/TermVil/tv2002/tv0206/tartalom.html

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A007318, A020989, A048483, A048487, A119725, A119727, A123208.

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return 4*T(n-1,k-1) + 2*T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 18 2019

T:= proc(n, k) option remember;
      if k=1 and k=n then 1
    else 4*T(n-1, k-1) + 2*T(n-1, k)
      fi
end: seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Nov 18 2019

T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, 4*T[n-1, k-1] + 2*T[n-1, k]]; Table[T[n,k], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==1 || k==n, 1, 4*T(n-1,k-1) + 2*T(n-1,k));

@CachedFunction
def T(n, k):
    if (k==1 or k==n): return 1
    else: return 4*T(n-1, k-1) + 2*T(n-1, k)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 18 2019

Edited by Don Reble, Jul 24 2006

A048492 a(n) = ( 8(2^n) - n^2 - 3n - 6 )/2.

Original entry on oeis.org

1, 3, 8, 20, 47, 105, 226, 474, 977, 1991, 4028, 8112, 16291, 32661, 65414, 130934, 261989, 524115, 1048384, 2096940, 4194071, 8388353, 16776938, 33554130, 67108537, 134217375, 268435076, 536870504, 1073741387, 2147483181

Offset: 0

Clark Kimberling

Partial sums of A000325, starting at n=1. - Klaus Brockhaus, Oct 13 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

a(n)=T(0, n)+T(1, n-1)+...+T(n, 0), array T given by A048483.

Cf. A000325 (2^n - n), A145070. - Klaus Brockhaus, Oct 13 2008

a:=0; for n:=1 to 30 do a:=a+2**n-n; write(a, ","); end; # Klaus Brockhaus, Oct 13 2008

[( 8*(2^n) -n^2 -3*n -6 )/2: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

lst={};s=0;Do[s+=2^n-n;AppendTo[lst, s], {n, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 30 2008 *)
Table[(8*2^n-n^2-3n-6)/2,{n,0,30}]
LinearRecurrence[{5,-9,7,-2},{1,3,8,20},40] (* Harvey P. Dale, Aug 28 2019 *)

Vec((2*x^2-2*x+1) / ((x-1)^3*(2*x-1)) + O(x^100)) \\ Colin Barker, Oct 27 2014

a(0) = 1; a(n) = a(n-1) + 2^(n+1) - (n+1) for n > 0. - Klaus Brockhaus, Oct 13 2008
From Colin Barker, Oct 27 2014: (Start)
a(n) = (-2+2^(2+n)-1/2*(1+n)*(2+n)).
a(n) = 5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4).
G.f.: (2*x^2-2*x+1) / ((x-1)^3*(2*x-1)).
(End)

Better description from Frank Ellermann, Mar 16 2002

Corrected by T. D. Noe, Nov 08 2006

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

A048487 a(n) = T(4,n), array T given by A048483.

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A048490 a(n) = T(7,n), array T given by A048483.

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A048491 a(n) = T(8,n), array T given by A048483.

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A048488 a(n) = 6*2^n - 5.

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A048489 a(n) = 7 * 2^n - 6.

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Maple

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A048493 a(n) = (n+1)*2^n - n.

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Magma

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A119726 Triangular array read by rows: T(n,1) = T(n,n) = 1, T(n,k) = 4*T(n-1, k-1) + 2*T(n-1, k).

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A119726 Triangular array read by rows: T(n,1) = T(n,n) = 1, T(n,k) = 4T(n-1, k-1) + 2T(n-1, k).

A048492 a(n) = ( 8(2^n) - n^2 - 3n - 6 )/2.