A060816 -id:A060816 - cp's OEIS frontend

A084214 Inverse binomial transform of a math magic problem.

1, 1, 4, 6, 14, 26, 54, 106, 214, 426, 854, 1706, 3414, 6826, 13654, 27306, 54614, 109226, 218454, 436906, 873814, 1747626, 3495254, 6990506, 13981014, 27962026, 55924054, 111848106, 223696214, 447392426, 894784854, 1789569706, 3579139414, 7158278826, 14316557654

Offset: 0

Paul Barry, May 19 2003

Inverse binomial transform of A060816.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000975, A001045, A020989, A048654, A060816, A081254.

Cf. A048573.

a084214 n = a084214_list !! n
a084214_list = 1 : xs where
   xs = 1 : 4 : zipWith (+) (map (* 2) xs) (tail xs)
-- Reinhard Zumkeller, Aug 01 2011

[(5*2^n-3*0^n+4*(-1)^n)/6: n in [0..35]]; // Vincenzo Librandi, Jun 15 2011

A084214 := proc(n)
    (5*2^n - 3*0^n + 4*(-1)^n)/6 ;
end proc:
seq(A084214(n),n=0..60) ; # R. J. Mathar, Aug 18 2024

f[n_]:=2/(n+1);x=3;Table[x=f[x];Numerator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
LinearRecurrence[{1,2},{1,1,4},50] (* Harvey P. Dale, Mar 05 2021 *)

a(n) = 5<<(n-1)\3 + bitnegimply(1,n); \\ Kevin Ryde, Dec 20 2023

a(n) = (5*2^n - 3*0^n + 4*(-1)^n)/6.
G.f.: (1+x^2)/((1+x)*(1-2*x)).
E.g.f.: (5*exp(2*x) - 3*exp(0) + 4*exp(-x))/6.
From Paul Barry, May 04 2004: (Start)
The binomial transform of a(n+1) is A020989(n).
a(n) = A001045(n-1) + A001045(n+1) - 0^n/2. (End)
a(n) = Sum_{k=0..n} A001045(n+1)*C(1, k/2)*(1+(-1)^k)/2. - Paul Barry, Oct 15 2004
a(n) = a(n-1) + 2*a(n-2) for n > 2. - Klaus Brockhaus, Dec 01 2009
From Yuchun Ji, Mar 18 2019: (Start)
a(n+1) = Sum_{i=0..n} a(i) + 1 - (-1)^n, a(0)=1.
a(n) = A000975(n-3)*10 + 5 + (-1)^(n-3), a(0)=1, a(1)=1, a(2)=4. (End)
a(n) = A081254(n) + (n-1 mod 2). - Kevin Ryde, Dec 20 2023
a(n) = 2*A048573(n-2) for n>=2. - Alois P. Heinz, May 20 2025

A134931 a(n) = (5*3^n-3)/2.

Original entry on oeis.org

1, 6, 21, 66, 201, 606, 1821, 5466, 16401, 49206, 147621, 442866, 1328601, 3985806, 11957421, 35872266, 107616801, 322850406, 968551221, 2905653666, 8716961001, 26150883006, 78452649021, 235357947066, 706073841201, 2118221523606

Offset: 0

Rolf Pleisch, Jan 29 2008

Numbers n where the recurrence s(0)=1, if s(n-1) >= n then s(n) = s(n-1) - n else s(n) = s(n-1) + n produces s(n)=0. - Hugo Pfoertner, Jan 05 2012
A046901(a(n)) = 1. - Reinhard Zumkeller, Jan 31 2013
Binomial transform of A146523: (1, 5, 10, 20, 40, ...) and double binomial transform of A010685: (1, 4, 1, 4, 1, 4, ...). - Gary W. Adamson, Aug 25 2016
Also the number of maximal cliques in the (n+1)-Hanoi graph. - Eric W. Weisstein, Dec 01 2017
a(n) is the least k such that f(a(n-1)+1) + ... + f(k) > f(a(n-2)+1) + ... + f(a(n-1)) for n > 1, where f(n) = 1/(n+1). Because Sum_{k=1..5*3^(n-1)} 1/(a(n)+3*k-1) + 1/(a(n)+3*k) + 1/(a(n)+3*k+1) - 1/((a(n)+1+5*3^n)*5*3^(n-1)) < Sum_{k=1..5*3^(n-1)} 1/(a(n-1)+k+1) < Sum_{k=1..5*3^(n-1)} 1/(a(n)+3*k-1) + 1/(a(n)+3*k) + 1/(a(n)+3*k+1), we have 1 < 1/3 + 1/4 + ... + 1/7 < 1/8 + 1/9 + ... + 1/22 < ... . - Jinyuan Wang, Jun 15 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Hanoi Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A003462, A007051, A034472, A024023, A067771, A029858.

Cf. A005030, A010685, A046901, A060816, A116952, A146523, A225918.

[(5*3^n-3)/2: n in [0..30]]; // Vincenzo Librandi, Jun 05 2011

seq((5*3^n-3)/2, n= 0..25); # Gary Detlefs, Jun 22 2010

a=1; lst={a}; Do[a=a*3+3; AppendTo[lst,a], {n,0,100}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)
Table[(5 3^n - 9)/6, {n, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
(5 3^Range[20] - 9)/6 (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{4, -3}, {1, 6}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[(1 + 2 x)/(1 - 4 x + 3 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

a(n) = (5*3^n-3)/2; /* Joerg Arndt, Apr 14 2013 */

a(n) = 3*(a(n-1) + 1), with a(0)=1.
From R. J. Mathar, Jan 31 2008: (Start)
O.g.f.: (5/2)/(1-3*x) - (3/2)/(1-x).
a(n) = (A005030(n) - 3)/2. (End)
a(n) = A060816(n+1) - 1. - Philippe Deléham, Apr 14 2013
E.g.f.: exp(x)*(5*exp(2*x) - 3)/2. - Stefano Spezia, Aug 28 2023

More terms from Vladimir Joseph Stephan Orlovsky, Dec 25 2008

A057198 a(n) = (5*3^(n-1)+1)/2.

Original entry on oeis.org

3, 8, 23, 68, 203, 608, 1823, 5468, 16403, 49208, 147623, 442868, 1328603, 3985808, 11957423, 35872268, 107616803, 322850408, 968551223, 2905653668, 8716961003, 26150883008, 78452649023, 235357947068, 706073841203, 2118221523608

Offset: 1

Colin Mallows and N. J. A. Sloane, Sep 16 2000

It appears that if s(n) is a first-order rational sequence of the form s(0)=4, s(n) = (2*s(n-1)+1)/(s(n-1)+2), n > 0, then s(n) = a(n)/(a(n)-1), n > 0.

			G.f. = 3*x + 8*x^2 + 23*x^3 + 68*x^4 + 203*x^5 + 608*x^6 + 1823*x^7 + 5468*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv:1611.08244 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Related to A046901.
Equals A060816 + 1.
Cf. A135423 (bisection), A191450 (2nd row).

[(5*3^(n-1) + 1)/2: n in [1..30]]; // Vincenzo Librandi, Oct 11 2012

Table[(5*3^(n-1) + 1)/2, {n, 30}] (* T. D. Noe, Oct 11 2012 *)

a(n)=(5*3^(n-1)+1)/2 \\ Charles R Greathouse IV, Oct 11 2012

a(n+1) = 3*a(n) - 1 for n > 1. - Reinhard Zumkeller, Jan 22 2011
G.f.: (5/2)*U(0) where U(k) = 1 + 2/(5*3^k + 5*3^k/(1 - 30*x*3^k/(15*x*3^k - 1/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 01 2012
E.g.f.: (5/2)*U(0) where U(k) = 1 + 2/(5*3^k + 5*3^k/(1 - 30*x*3^k/(15*x*3^k - (k+1)/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 01 2012
G.f.: x*(3-4*x) / ( (3*x-1)*(x-1) ). - R. J. Mathar, Jan 25 2015
E.g.f.: (5*exp(3*x) + 3*exp(x) - 8)/6. - Stefano Spezia, Aug 28 2023

Incorrect zeroth term removed by Jon Perry, Oct 11 2012

A084215 Expansion of g.f.: (1+x^2)/(1-2*x).

Original entry on oeis.org

1, 2, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 0

Paul Barry, May 19 2003

Associated with a math magic problem.

Elements are the sums of consecutive pairs of elements of A084214.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A020714, A060816, A084214.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x^2)/(1-2*x))); // G. C. Greubel, Oct 08 2018

Join[{1, 2, a = 5}, Table[a = 2*a, {n,0,40}]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
Table[Int[2^(n-2)*5],{n,0,40}] (* Taher Jamshidi, Sep 15 2012 *)
CoefficientList[Series[(1 + x^2)/(1 - 2 x), {x, 0, 30}], x] (* G. C. Greubel, Oct 08 2018 *)

x='x+O('x^30); Vec((1+x^2)/(1-2*x)) \\ G. C. Greubel, Oct 08 2018

a(n) = Sum_{k=0..n} 2^(n-k)*binomial(1, k/2)*(1+(-1)^k)/2. - Paul Barry, Oct 15 2004
a(n) = A020714(n-2), n > 1. - R. J. Mathar, Dec 19 2008
From Gary W. Adamson, Aug 26 2011: (Start)
a(n) is the sum of top row terms of M^n, M is an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  0, 0, 0, 0, 0, 0, ...
  0, 0, 0, 0, 0, 0, ...
  ...
E.g.: a(4) = 20 = (8 + 8 + 4) since the top row of M^4 = (8, 8, 4, 0, 0, 0, ...). (End)
a(n) = floor(2^(n-2)*5). - Taher Jamshidi, Sep 15 2012
a(n) = 2*a(n-1) for n >= 3, a(0) = 1, a(1) = 2, a(2) = 5. - Philippe Deléham, Mar 13 2013
E.g.f.: (5*exp(2*x) - 2*x - 1)/4. - Stefano Spezia, Feb 20 2023

A237930 a(n) = 3^(n+1) + (3^n-1)/2.

Original entry on oeis.org

3, 10, 31, 94, 283, 850, 2551, 7654, 22963, 68890, 206671, 620014, 1860043, 5580130, 16740391, 50221174, 150663523, 451990570, 1355971711, 4067915134, 12203745403, 36611236210, 109833708631, 329501125894, 988503377683, 2965510133050, 8896530399151

Offset: 0

Philippe Deléham, Feb 16 2014

a(n-1) agrees with the graph radius of the n-Sierpinski carpet graph for n = 2 to at least n = 5. See A100774 for the graph diameter of the n-Sierpinski carpet graph.
The inverse binomial transform gives 3, 7, 14, 28, 56, ... i.e., A005009 with a leading 3. - R. J. Mathar, Jan 08 2020
First differences of A108765. The digital root of a(n) for n > 1 is always 4. a(n) is never divisible by 7 or by 12. a(n) == 10 (mod 84) for odd n. a(n) == 31 (mod 84) for even n > 0. Conjecture: This sequence contains no prime factors p == {11, 13, 23, 61 71, 73} (mod 84). - Klaus Purath, Apr 13 2020
This is a subsequence of A017209 for n > 1. See formula. - Klaus Purath, Jul 03 2020

			Ternary....................Decimal
10...............................3
101.............................10
1011............................31
10111...........................94
101111.........................283
1011111........................850
10111111......................2551
101111111.....................7654, etc.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Graph Radius.
Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A003462, A005009, A005032 (first differences), A017209, A060816, A100774, A108765 (partial sums), A199109, A329774.

Cf. A024023, A029858, A057198, A116952, A171498.

[3^(n+1) + (3^n-1)/2: n in [0..40]]; // Vincenzo Librandi, Jan 09 2020

(* Start from Eric W. Weisstein, Mar 13 2018 *)
Table[(7 3^n - 1)/2, {n, 0, 20}]
(7 3^Range[0, 20] - 1)/2
LinearRecurrence[{4, -3}, {10, 31}, {0, 20}]
CoefficientList[Series[(3 - 2 x)/((x - 1) (3 x - 1)), {x, 0, 20}], x]
(* End *)

Vec((3 - 2*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 27 2019

G.f.: (3-2*x)/((1-x)*(1-3*x)).
a(n) = A000244(n+1) + A003462(n).
a(n) = 3*a(n-1) + 1 for n > 0, a(0)=3. (Note that if a(0) were 1 in this recurrence we would get A003462, if it were 2 we would get A060816. - N. J. A. Sloane, Dec 06 2019)
a(n) = 4*a(n-1) - 3*a(n-2) for n > 1, a(0)=3, a(1)=10.
a(n) = 2*a(n-1) + 3*a(n-2) + 2 for n > 1.
a(n) = A199109(n) - 1.
a(n) = (7*3^n - 1)/2. - Eric W. Weisstein, Mar 13 2018
From Klaus Purath, Apr 13 2020: (Start)
a(n) = A057198(n+1) + A024023(n).
a(n) = A029858(n+2) - A024023(n).
a(n) = A052919(n+1) + A029858(n+1).
a(n) = (A000244(n+1) + A171498(n))/2.
a(n) = 7*A003462(n) + 3.
a(n) = A116952(n) + 2. (End)
a(n) = A017209(7*(3^(n-2)-1)/2 + 3), n > 1. - Klaus Purath, Jul 03 2020
E.g.f.: exp(x)*(7*exp(2*x) - 1)/2. - Stefano Spezia, Aug 28 2023

A198643 a(n) = 5*3^n-1.

Original entry on oeis.org

4, 14, 44, 134, 404, 1214, 3644, 10934, 32804, 98414, 295244, 885734, 2657204, 7971614, 23914844, 71744534, 215233604, 645700814, 1937102444, 5811307334, 17433922004, 52301766014, 156905298044, 470715894134, 1412147682404, 4236443047214

Offset: 0

Vincenzo Librandi, Oct 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 11.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A027107, A048473, A171498.

Magma
```
[5*3^n-1: n in [0..30]];
```

5*3^Range[0, 30] - 1 (* or *)
NestList[3*# + 2 &, 4, 30] (* Paolo Xausa, Aug 28 2024 *)

a(n)=5*3^n-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*a(n-1)+2 = 2*A060816(n+1).

G.f.: ( 4-2*x ) / ( (3*x-1)*(x-1) ). - R. J. Mathar, Nov 17 2011

A008343 a(1)=1; thereafter a(n+1) = a(n)-n if a(n) >= n otherwise a(n+1) = a(n)+n.

Original entry on oeis.org

1, 0, 2, 5, 1, 6, 0, 7, 15, 6, 16, 5, 17, 4, 18, 3, 19, 2, 20, 1, 21, 0, 22, 45, 21, 46, 20, 47, 19, 48, 18, 49, 17, 50, 16, 51, 15, 52, 14, 53, 13, 54, 12, 55, 11, 56, 10, 57, 9, 58, 8, 59, 7, 60, 6, 61, 5, 62, 4, 63, 3

Offset: 1

N. J. A. Sloane and J. H. Conway

nonn

For n>0, a(A060816(n))=0. - Dmitry Kamenetsky, Feb 14 2017

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A008344, A060816.

A008343 := proc(n) option remember; if n = 1 then 1 elif A008343(n-1) >= (n-1) then A008343(n-1)-(n-1) else A008343(n-1)+(n-1); fi; end;

nxt[{n_,a_}]:={n+1,If[a>=n,a-n,a+n]}; Transpose[NestList[nxt,{1,1},60]][[2]] (* Harvey P. Dale, May 04 2014 *)

a(n) = (n-1+a(n-1)) mod (2*(n-1)). - Jon Maiga, Jul 09 2021

Name edited by Dmitry Kamenetsky, Feb 14 2017

A191451 Dispersion of (3*n-2), for n>=2, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 13, 7, 3, 40, 22, 10, 5, 121, 67, 31, 16, 6, 364, 202, 94, 49, 19, 8, 1093, 607, 283, 148, 58, 25, 9, 3280, 1822, 850, 445, 175, 76, 28, 11, 9841, 5467, 2551, 1336, 526, 229, 85, 34, 12, 29524, 16402, 7654, 4009, 1579, 688, 256, 103, 37, 14, 88573

Offset: 1

Clark Kimberling, Jun 05 2011

Row 1:  A003462
Row 2:  A060816
Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...4....13...40...121
  2...7....22...67...202
  3...10...31...94...283
  5...16...49...148..445
  6...19...58...175..526

Cf. A114537, A035513, A035506, A191449.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=3n+1 (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191451 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191451 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A181655 Expansion of (1+2x-x^3+x^4)/(1-4x^2+3x^4).

Original entry on oeis.org

1, 2, 4, 7, 14, 22, 44, 67, 134, 202, 404, 607, 1214, 1822, 3644, 5467, 10934, 16402, 32804, 49207, 98414, 147622, 295244, 442867, 885734, 1328602, 2657204, 3985807, 7971614, 11957422, 23914844, 35872267, 71744534, 107616802, 215233604

Offset: 0

Paul Barry, Nov 03 2010

Row sums of A181654.

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-3).

Cf. A060816, A198643 (bisections).

CoefficientList[Series[(1+2x-x^3+x^4)/(1-4x^2+3x^4),{x,0,40}],x] (* or *) Join[{1},LinearRecurrence[{0,4,0,-3},{2,4,7,14},40]] (* Harvey P. Dale, Jan 11 2012 *)

A181655(n)=if(bitand(n,1), 3^(n\2)*5\2, n, 3^(n\2-1)*5-1, 1) \\ M. F. Hasler, Apr 06 2019

G.f.: (1+2*x-x^3+x^4)/((1-x^2)*(1-3*x^2)).
a(n) = 5*A038754(n+1)/6 - A040001(n)/2. - R. J. Mathar, May 14 2016
a(2n-1) = A060816(n-1), a(2n) = A198643(n-1); n >= 1. a(n+1) = 2*a(n) if n is odd. - M. F. Hasler, Apr 06 2019

A193728 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (x+2)^n and q(n,x) = (2*x+1)^n.

Original entry on oeis.org

1, 2, 1, 8, 10, 3, 32, 64, 42, 9, 128, 352, 360, 162, 27, 512, 1792, 2496, 1728, 594, 81, 2048, 8704, 15360, 14400, 7560, 2106, 243, 8192, 40960, 87552, 103680, 73440, 31104, 7290, 729, 32768, 188416, 473088, 677376, 604800, 344736, 122472, 24786, 2187

Offset: 0

Clark Kimberling, Aug 04 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Triangle T(n,k), read by rows, given by (2,2,0,0,0,0,0,0,0,...) DELTA (1,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
    1;
    2,    1;
    8,   10,    3;
   32,   64,   42,    9;
  128,  352,  360,  162,  27;
  512, 1792, 2496, 1728, 594, 81;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000012, A000420, A060816, A081038, A081294.

Cf. A084938, A133494, A193722, A193729.

function T(n, k) // T = A193728
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 4*T(n-1, k) + 3*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 28 2023

(* First program *)
z = 8; a = 1; b = 2; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193728 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193729 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 4*T[n-1,k] + 3*T[n-1,k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 28 2023 *)

def T(n, k): # T = A193728
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 4*T(n-1, k) + 3*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 28 2023

T(n,k) = 3*T(n-1,k-1) + 4*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x-2*x*y)/(1-4*x-3*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 28 2023: (Start)
T(n, n-k) = A193729(n, k).
T(n, 0) = A081294(n).
T(n, n-1) = 2*A081038(n-1).
T(n, n) = A133494(n).
Sum_{k=0..n} T(n, k) = (1/7)*(4*[n=0] + 3*A000420(n)).
Sum_{k=0..n} (-1)^k * T(n, k) = A000012(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (5*b(n) + 4*b(n-1))/14 + (2/3)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A060816(n),
where b(n) = (2 + sqrt(7))^n + (2 - sqrt(7))^n. (End)

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A084214 Inverse binomial transform of a math magic problem.

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A134931 a(n) = (5*3^n-3)/2.

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A057198 a(n) = (5*3^(n-1)+1)/2.

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A084215 Expansion of g.f.: (1+x^2)/(1-2*x).

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A237930 a(n) = 3^(n+1) + (3^n-1)/2.

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A198643 a(n) = 5*3^n-1.

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A008343 a(1)=1; thereafter a(n+1) = a(n)-n if a(n) >= n otherwise a(n+1) = a(n)+n.