A123208 -id:A123208 - 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

A051633 a(n) = 5*2^n - 2.

Original entry on oeis.org

3, 8, 18, 38, 78, 158, 318, 638, 1278, 2558, 5118, 10238, 20478, 40958, 81918, 163838, 327678, 655358, 1310718, 2621438, 5242878, 10485758, 20971518, 41943038, 83886078, 167772158, 335544318, 671088638, 1342177278, 2684354558, 5368709118, 10737418238, 21474836478

Offset: 0

Asher Auel

			a(5) = 5*2^4 - 2 = 80 - 2 = 78.

Stefano Spezia, Table of n, a(n) for n = 0..3300
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000079, A020714, A118654.

Cf. A123208, A048487, A131051, A153894.

LinearRecurrence[{3, -2},{3, 8},30] (* Ray Chandler, Jul 18 2020 *)

a(n) = A118654(n, 5).
a(n) = A000079(n)*5 - 2 = A020714(n) - 2. - Omar E. Pol, Dec 23 2008
a(n) = 2*(a(n-1)+1) with a(0)=3. - Vincenzo Librandi, Aug 06 2010
a(n) = A123208(2*n+1) = A048487(n)+2 = A131051(n+2) = A153894(n)-1. - Philippe Deléham, Apr 15 2013
G.f.: ( 3-x ) / ( (2*x-1)*(x-1) ). - R. J. Mathar, Mar 23 2023
E.g.f.: exp(x)*(5*exp(x) - 2). - Stefano Spezia, Oct 03 2023

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

A219605 Square array T(n,k), read by antidiagonals: T(n,2k) = T(n,2k-1)n, T(n,2k+1) = T(n,2*k)+n, T(n,0) = 1.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 2, 3, 1, 0, 3, 6, 4, 1, 0, 3, 8, 12, 5, 1, 0, 4, 16, 15, 20, 6, 1, 0, 4, 18, 45, 24, 30, 7, 1, 0, 5, 36, 48, 96, 35, 42, 8, 1, 0, 5, 38, 144, 100, 175, 48, 56, 9, 1, 0, 6, 76, 147, 400, 180, 288, 63, 72, 10, 1, 0, 6, 78, 441, 404, 900, 294, 441

Offset: 0

Philippe Deléham, Apr 12 2013

			Square array begins:
1..1....0....0....0....0....0....0.....0.....0...
1..2....2....3....3....4....4....5.....5.....5...
1..3....6....8...16...18...36...38....76....78...
1..4...12...15...45...48..144..147...441...444...
1..5...20...24...96..100..400..404..1616..1620...
1..6...30...35..175..180..900..905..4525..4530...
1..7...42...48..288..294.1764.1770.10620.10626...
1..8...56...63..441..448.3136.3143.22001.22008...
1..9...72...80..640..648.5184.5192.41536.41544...
1.10...90...99..891..900.8100.8109.72971.72980...
...

t[n_, k_] /; n < 0 || k < 0 = 0; t[n_, 0] = 1; t[n_, 1] = n+1; t[0, k_ /; k > 1] = 0; t[n_?Positive, k_?EvenQ] := t[n, k] = t[n, k-1]*n; t[n_?Positive, k_?OddQ] := t[n, k] = t[n, k-1] + n; Table[t[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 19 2013 *)

T(n,0) = A000012(n).
T(n,1) = A000027(n).
T(n,2) = A002378(n+1).
T(n,3) = A005563(n).
T(n,4) = A152618(n+1).
T(n,5) = A045991(n+1).
T(n,6) = A035287(n+1).
T(0,k) = A019590(k+1).
T(1,k) = A008619(k+1).
T(2,k) = A123208(k).

A220354 Irregular triangle T(n,k), read by rows; row n gives coefficients in expansion of P_n(x), which is defined by: P_0(x) = 1, P_n(x) = P_(n-1)(x)+x if n odd, P_n(x) = P_(n-1)(x)*x if n even.

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Apr 13 2013

Row lengths are: 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ...

			Irregular triangle begins:
1
1, 1
0, 1, 1
0, 2, 1
0, 0, 2, 1
0, 1, 2, 1
0, 0, 1, 2, 1
0, 1, 1, 2, 1
0, 0, 1, 1, 2, 1
0, 1, 1, 1, 2, 1
0, 0, 1, 1, 1, 2, 1
0, 1, 1, 1, 1, 2, 1
0, 0, 1, 1, 1, 1, 2, 1
0, 1, 1, 1, 1, 1, 2, 1
...

Cf. A219605.

Sum_(T(n,k)*x^k, k>=0) = A019590(n+1), A008619(n+1), A123208(n) for x = 0, 1, 2 respectively.

cp's OEIS Frontend

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

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A051633 a(n) = 5*2^n - 2.

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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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A219605 Square array T(n,k), read by antidiagonals: T(n,2*k) = T(n,2*k-1)*n, T(n,2*k+1) = T(n,2*k)+n, T(n,0) = 1.

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A220354 Irregular triangle T(n,k), read by rows; row n gives coefficients in expansion of P_n(x), which is defined by: P_0(x) = 1, P_n(x) = P_(n-1)(x)+x if n odd, P_n(x) = P_(n-1)(x)*x if n even.

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Formula

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).

A219605 Square array T(n,k), read by antidiagonals: T(n,2k) = T(n,2k-1)n, T(n,2k+1) = T(n,2*k)+n, T(n,0) = 1.