A060816 -id:A060816 - cp's OEIS frontend

A208524 Triangle of coefficients of polynomials u(n,x) jointly generated with A208525; see the Formula section.

1, 1, 1, 1, 3, 3, 1, 6, 10, 5, 1, 10, 22, 23, 11, 1, 15, 40, 65, 60, 21, 1, 21, 65, 145, 195, 137, 43, 1, 28, 98, 280, 490, 518, 322, 85, 1, 36, 140, 490, 1050, 1484, 1372, 723, 171, 1, 45, 192, 798, 2016, 3570, 4368, 3447, 1624, 341, 1, 55, 255, 1230, 3570

Offset: 1

Clark Kimberling, Feb 29 2012

Alternating row sums: 1,0,1,0,1,0,1,0,...

			First five rows:
1
1...1
1...3....3
1...6....10...5
1...10...22...23...11
First five polynomials u(n,x):
1
1 + x
1 + 3x + 3x^2
1 + 6x + 10x^2 + 5x^3
1 + 10x + 22x^2 + 23x^3 + 11x^4

Cf. A208525.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A208524 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208525 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (*A060816*)
Table[v[n, x] /. x -> 1, {n, 1, z}] (*|A084244|*)
Table[u[n, x] /. x -> -1, {n, 1, z}](*alt. row sums*)
Table[v[n, x] /. x -> -1, {n, 1, z}](*alt. row sums*)

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A208525 Triangle of coefficients of polynomials v(n,x) jointly generated with A208524; see the Formula section.

Original entry on oeis.org

1, 2, 3, 3, 7, 5, 4, 12, 18, 11, 5, 18, 42, 49, 21, 6, 25, 80, 135, 116, 43, 7, 33, 135, 295, 381, 279, 85, 8, 42, 210, 560, 966, 1050, 638, 171, 9, 52, 308, 966, 2086, 2996, 2724, 1453, 341, 10, 63, 432, 1554, 4032, 7182, 8688, 6921, 3240, 683, 11, 75

Offset: 1

Clark Kimberling, Feb 29 2012

Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...

			First five rows:
1
2...3
3...7....5
4...12...18...11
5...18...42...49...21
First five polynomials v(n,x):
1
2 + 3x
3 + 7x + 5x^2
4 + 12x + 18x^2 + 11x^3
5 + 18x + 42x^2 + 49x^3 + 21x^4

Cf. A208524.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A208524 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208525 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (*A060816*)
Table[v[n, x] /. x -> 1, {n, 1, z}] (*|A084244|*)
Table[u[n, x] /. x -> -1, {n, 1, z}] (*alt. row sums*)
Table[v[n, x] /. x -> -1, {n, 1, z}] (*alt. row sums*)

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A329774 a(n) = n+1 for n <= 2; otherwise a(n) = 3*a(n-3)+1.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 13, 22, 31, 40, 67, 94, 121, 202, 283, 364, 607, 850, 1093, 1822, 2551, 3280, 5467, 7654, 9841, 16402, 22963, 29524, 49207, 68890, 88573, 147622, 206671, 265720, 442867, 620014, 797161, 1328602, 1860043, 2391484, 3985807

Offset: 0

N. J. A. Sloane, Nov 27 2019

Robert Fathauer observed that if the "warp and woof" construction used by Jim Conant in his recursive dissection of a square (see A328078) is applied to a triangle, one obtains the Sierpinski gasket.

The present sequence gives the number of regions after the n-th generation of this dissection of a triangle.

Robert Fathauer, Email to N. J. A. Sloane, Oct 14 2019.

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3).

A mixture of A003462, A060816, and A237930. Cf. A328078.

f:=proc(n) option remember;
if n<=2 then n+1 else 3*f(n-3)+1; fi; end;
[seq(f(n),n=0..50)];

Vec((1 + x + x^2 - 2*x^3) / ((1 - x)*(1 - 3*x^3)) + O(x^40)) \\ Colin Barker, Nov 27 2019

From Colin Barker, Nov 27 2019: (Start)
G.f.: (1 + x + x^2 - 2*x^3) / ((1 - x)*(1 - 3*x^3)).
a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) for n>3.
(End)

A182950 Joint-rank array of the numbers (3i+2)3^j, where i>=0, j>=0, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 9, 7, 4, 27, 22, 12, 5, 81, 67, 36, 16, 6, 243, 202, 108, 49, 20, 8, 729, 607, 324, 148, 62, 25, 10, 2187, 1822, 972, 445, 188, 76, 30, 11, 6561, 5467, 2916, 1336, 566, 229, 90, 34, 13, 19683, 16402, 8748, 4009, 1700, 688, 270, 103, 39, 14

Offset: 1

Clark Kimberling, Dec 15 2010

Joint-rank arrays are defined in the first comment at A182801.  As for any joint-rank array, A182950 is a permutation of the positive integers, but, a fortiori, A182950 is an interspersion: after initial terms every row is interspersed with all other rows.  The numbers (3*i+2)*3^j as an array comprise A182830; and sorted, possibly A026179.
(row 1)=A000244.
(row 2)=A060816.
(row 3)=A003946.
(row 4)=A052909.
(row 5)=A027107?

			Northwest corner:
1....3....9....27...
2....7...22....67...
4...12...36...108...
5...16...49...148...

Clark Kimberling, Interspersions and Dispersions.

Cf. A182801, A182830, A026179, A182949.

 M[i_,j_]:=j+Floor[Log[3*i/2+1]/Log[3]];
 T[i_,j_]:=Sum[Floor[1/3+(3*i+2)*3^(j-k-1)],{k,0,M[i,j]}];
 TableForm[Table[T[i,j],{i,0,9},{j,0,9}]]

A117137 Same as triangle in A117136, but omit final 1 from each row.

Original entry on oeis.org

1, 1, 2, 4, 1, 3, 6, 2, 7, 1, 4, 8, 3, 9, 2, 10, 1, 5, 10, 4, 11, 3, 12, 2, 13, 1, 6, 12, 5, 13, 4, 14, 3, 15, 2, 16, 1, 7, 14, 6, 15, 5, 16, 4, 17, 3, 18, 2, 19, 1

Offset: 0

N. J. A. Sloane, Apr 21 2006

Row n has length 2n+1.

			Triangle begins:
Row 0: 1
Row 1: 1 2 4
Row 2: 1 3 6 2 7
Row 3: 1 4 8 3 9 2 10
Row 4: 1 5 10 4 11 3 12 2 13
...

Cf. A117136, A046901, A008344.

The segments of A046901 appear as rows 2, 7, 22, 67, ... (A060816) of this array.

A238055 a(n) = (13*3^n-1)/2.

Original entry on oeis.org

6, 19, 58, 175, 526, 1579, 4738, 14215, 42646, 127939, 383818, 1151455, 3454366, 10363099, 31089298, 93267895, 279803686, 839411059, 2518233178, 7554699535, 22664098606, 67992295819, 203976887458, 611930662375, 1835791987126, 5507375961379, 16522127884138

Offset: 0

Philippe Deléham, Feb 17 2014

			Ternary....................Decimal
20...............................6
201.............................19
2011............................58
20111..........................175
201111.........................526
2011111.......................1579
20111111......................4738
201111111....................14215, etc.

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

Cf. A000244, A003462, A052909, A060816, A237930.

a(n) = 3*a(n-1) + 1, a(0)=6.
a(n) = 4*a(n-1) - 3*a(n-2), a(0)=6, a(1)=19.
a(n) = 2*A237930(n) - A003462(n).
a(n) = A052909(n+1) + A000244(n).
a(n) = A237930(n) + A000244(n+1).
a(n) = 13*A003462(n) + 6.
G.f.: (6-5*x)/((1-x)*(1-3*x)).
E.g.f.: exp(x)*(13*exp(2*x) - 1)/2. - Stefano Spezia, Aug 28 2023

A244762 a(n) = (53^n-2n-1)/4.

Original entry on oeis.org

1, 3, 10, 32, 99, 301, 908, 2730, 8197, 24599, 73806, 221428, 664295, 1992897, 5978704, 17936126, 53808393, 161425195, 484275602, 1452826824, 4358480491, 13075441493, 39226324500, 117678973522, 353036920589, 1059110761791, 3177332285398, 9531996856220, 28595990568687, 85787971706089, 257363915118296

Offset: 0

Franklin T. Adams-Watters, Jul 06 2014

Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A060816 (first differences).

Cf. A108765, A164039, A047926, A000340.

CoefficientList[Series[(1-2*x+2*x^2)/((1-3*x)*(1-x)^2), {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 06 2014 *)

a(n+1) = 3*a(n) + n.
G.f.: (1-2*x+2*x^2) / ((1-3*x)*(1-x)^2).
E.g.f.: exp(x)*(5*exp(2*x) - 2*x - 1)/4. - Stefano Spezia, Aug 28 2023

A220946 Expansion of (1+2x+2x^2-x^3)/((1-x)(1+x)(1-3x^2)).

Original entry on oeis.org

1, 2, 6, 7, 21, 22, 66, 67, 201, 202, 606, 607, 1821, 1822, 5466, 5467, 16401, 16402, 49206, 49207, 147621, 147622, 442866, 442867, 1328601, 1328602, 3985806, 3985807, 11957421, 11957422, 35872266, 35872267, 107616801, 107616802, 322850406, 322850407

Offset: 0

Philippe Deléham, Apr 14 2013

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

Cf. A060816, A134931.

LinearRecurrence[{0, 4, 0, -3}, {1, 2, 6, 7}, 40] (* T. D. Noe, Apr 17 2013 *)

a(n) = a(n-1)+1 if n odd, a(n) = a(n-1)*3 if n even.
a(2n) = A134931(n), a(2n+1) = A060816(n+1).
a(n) = 4*a(n-2) - 3*a(n-4) with a(0)=1, a(1)=2, a(2)=6, a(3)=7.

A238206 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) is A007494(k) and T(n,k) = 3*T(n-1,k) + 1 for n>0.

Original entry on oeis.org

0, 2, 1, 3, 7, 4, 5, 10, 22, 13, 6, 16, 31, 67, 40, 8, 19, 49, 94, 202, 121, 9, 25, 58, 148, 283, 607, 364, 11, 28, 76, 175, 445, 850, 1822, 1093, 12, 34, 85, 229, 526, 1336, 2551, 5467, 3280, 14, 37, 103, 256, 688, 1579, 4009, 7654, 16402, 9841, 15, 43, 112, 310

Offset: 0

Philippe Deléham, Feb 20 2014

Permutation of nonnegative integers.

			Square array begins:
0, 2, 3, 5, 6, 8, 9, ...
1, 7, 10, 16, 19, 25, 28, ...
4, 22, 31, 49, 58, 76, 85, ...
13, 67, 94, 148, 175, 229, 256, ...
40, 202, 283, 445, 523, 688, 769, ...
121, 607, 850, 1336, 1579, 2065, 2308, ...
364, 1822, 2551, 4009, 4738, 6196, 6925, ...
1093, 5467, 7654, 12028, 14215, 18589, 20776, ...
3280, 16402, 22963, 36085, 42646, 55768, 62329, ...
9841, 49207, 68890, 108256, 127939, 167305, 186988, ...
...

Cf. A003462, A007494, A052909, A060816, A237930, A238055

T(n,k) = T(0,k)*3^n + T(n,0) where T(0,k) = (6*k + 1 -(-1)^k)/4 = A007494(k) and T(n,0) = (3^n - 1)/2 = A003462(n).

cp's OEIS Frontend

A208524 Triangle of coefficients of polynomials u(n,x) jointly generated with A208525; see the Formula section.

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A208525 Triangle of coefficients of polynomials v(n,x) jointly generated with A208524; see the Formula section.

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A329774 a(n) = n+1 for n <= 2; otherwise a(n) = 3*a(n-3)+1.

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A182950 Joint-rank array of the numbers (3*i+2)*3^j, where i>=0, j>=0, by antidiagonals.

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A117137 Same as triangle in A117136, but omit final 1 from each row.

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A238055 a(n) = (13*3^n-1)/2.

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A244762 a(n) = (5*3^n-2*n-1)/4.

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A220946 Expansion of (1+2*x+2*x^2-x^3)/((1-x)*(1+x)*(1-3x^2)).

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A238206 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) is A007494(k) and T(n,k) = 3*T(n-1,k) + 1 for n>0.

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A182950 Joint-rank array of the numbers (3i+2)3^j, where i>=0, j>=0, by antidiagonals.

A244762 a(n) = (53^n-2n-1)/4.

A220946 Expansion of (1+2x+2x^2-x^3)/((1-x)(1+x)(1-3x^2)).