A169634 -id:A169634 - cp's OEIS frontend

A222444 T(n,k) = number of n X k 0..3 arrays with entries increasing mod 4 by 0, 1 or 2 rightwards and downwards, starting with upper left zero.

1, 3, 3, 9, 21, 9, 27, 147, 147, 27, 81, 1029, 2403, 1029, 81, 243, 7203, 39285, 39285, 7203, 243, 729, 50421, 642249, 1500183, 642249, 50421, 729, 2187, 352947, 10499787, 57289767, 57289767, 10499787, 352947, 2187, 6561, 2470629, 171655443

Offset: 1

R. H. Hardin, Feb 20 2013

1/4 the number of 4-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

			Table starts
......1..........3...............9..................27.......................81
......3.........21.............147................1029.....................7203
......9........147............2403...............39285...................642249
.....27.......1029...........39285.............1500183.................57289767
.....81.......7203..........642249............57289767...............5110723191
....243......50421........10499787..........2187822609.............455924913093
....729.....352947.......171655443.........83550197745...........40672916404629
...2187....2470629......2806303725.......3190677470643.........3628419487925547
...6561...17294403.....45878770089.....121847980727187.......323690312271131451
..19683..121060821....750047661027....4653221950068669.....28876324830999722133
..59049..847425747..12262131106083..177700725073710285...2576049100980154511889
.177147.5931980229.200467073061765.6786168386579878383.229808641254065144560647
...
Some solutions for n=3, k=4:
..0..0..0..2....0..0..2..0....0..2..0..0....0..2..0..2....0..0..2..3
..1..2..2..3....0..2..3..1....2..2..2..0....0..0..0..2....0..2..3..1
..2..2..3..1....2..0..1..3....2..2..0..0....2..0..1..3....1..2..0..1

Andrew Howroyd, Table of n, a(n) for n = 1..496 (terms 1..180 from R. H. Hardin)
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Vertex Coloring
Wikipedia, Graph Coloring

Columns 1-7 are A000244(n-1), A169634(n-1), A222439, A222440, A222441, A222442, A222443.
Main diagonal is A068254.
Cf. A078099 (3 colorings), A198715 (unlabeled 4 colorings), A222144 (5 colorings), A222281 (6 colorings), A222340 (7 colorings), A222462 (8 colorings).

T(n,k) = 6*A198715(n,k) - 3 for n*k>1. - Andrew Howroyd, Jun 27 2017
Empirical for column k:
k=1: a(n) = 3*a(n-1).
k=2: a(n) = 7*a(n-1).
k=3: a(n) = 18*a(n-1) - 27*a(n-2).
k=4: a(n) = 45*a(n-1) - 267*a(n-2) + 263*a(n-3).
k=5: a(n) = 118*a(n-1) - 2811*a(n-2) + 22255*a(n-3) - 53860*a(n-4) - 54747*a(n-5) + 269406*a(n-6) - 175392*a(n-7).
k=6: [order 13]
k=7: [order 32]

A232955 T(n,k)=Number of nXk 0..3 arrays with no element x(i,j) adjacent to value 3-x(i,j) horizontally, diagonally or antidiagonally, and top left element zero.

Original entry on oeis.org

1, 3, 4, 9, 21, 16, 27, 129, 147, 64, 81, 771, 1881, 1029, 256, 243, 4629, 22971, 27441, 7203, 1024, 729, 27771, 283131, 685251, 400329, 50421, 4096, 2187, 166629, 3484893, 17429409, 20442651, 5840289, 352947, 16384, 6561, 999771, 42904365, 442227825

Offset: 1

R. H. Hardin, Dec 02 2013

Table starts
......1.........3............9..............27.................81
......4........21..........129.............771...............4629
.....16.......147.........1881...........22971.............283131
.....64......1029........27441..........685251...........17429409
....256......7203.......400329........20442651.........1074244299
...1024.....50421......5840289.......609853251........66226131273
...4096....352947.....85202361.....18193384251......4082986991091
..16384...2470629...1242993681....542752261251....251727862281441
..65536..17294403..18133691049..16191600916251..15519780149309307
.262144.121060821.264547403649.483034266181251.956841601733733945

			Some solutions for n=3 k=4
..0..2..3..2....0..1..3..3....0..1..0..2....0..1..0..0....0..2..0..2
..2..2..3..3....0..1..1..1....3..2..0..1....0..2..0..0....2..0..2..2
..0..2..3..3....0..1..0..0....2..2..3..2....0..2..0..2....1..0..2..0

R. H. Hardin, Table of n, a(n) for n = 1..199

Column 1 is A000302(n-1)
Column 2 is A169634(n-1)
Row 1 is A000244(n-1)

Empirical for column k:
k=1: a(n) = 4*a(n-1)
k=2: a(n) = 7*a(n-1)
k=3: a(n) = 15*a(n-1) -6*a(n-2)
k=4: a(n) = 31*a(n-1) -35*a(n-2) +5*a(n-3)
k=5: [order 10]
k=6: [order 21]
Empirical for row n:
n=1: a(n) = 3*a(n-1)
n=2: a(n) = 5*a(n-1) +6*a(n-2)
n=3: a(n) = 12*a(n-1) +7*a(n-2) -40*a(n-3) +12*a(n-4)
n=4: [order 10]
n=5: [order 26] for n>27
n=6: [order 86] for n>87

A169604 a(n) = 3*6^n.

Original entry on oeis.org

3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368, 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408

Offset: 0

Klaus Brockhaus, Apr 04 2010

a(n) = A081341(n+1).
Essentially first differences of A125682.
Binomial transform of A005053 without initial term 1.
Second binomial transform of A164346.
Inverse binomial transform of A169634.
Second inverse binomial transform of A103333 without initial term 1.
Contribution from Reinhard Zumkeller, May 02 2010: (Start)
a(n) = 3*A000400(n) = A000400(n+1)/2;
subsequence of A003586; a(n)=A003586(A014105(n)) for n<6. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (6).

Cf. A081341, A125682 ((6^n-1)*3/5), A005053 (expand (1-2x)/(1-5x)), A164346 (3*4^n), A169634 (3*7^n), A103333 (expand (1-5x)/(1-8x)).

Magma
```
[ 3*6^n: n in [0..19] ];
```

3*6^Range[0, 25] (* Paolo Xausa, Jan 17 2025 *)

a(n)=3*6^n \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 6*a(n-1) for n > 0; a(0) = 3.

G.f.: 3/(1-6*x).

A217983 If n = floor(p/2) * p^e, for some (by necessity unique) prime p and exponent e > 0, then a(n) = p, otherwise a(n) = 1.

Original entry on oeis.org

1, 2, 3, 2, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Johannes W. Meijer, Oct 25 2012

nonn

a(A130290(n) * A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n) = 1 elsewhere. - The original name of the sequence.
The a(n) are related to the prime numbers A000040 and the number of nonzero quadratic residues modulo the n-th prime A130290, see the first formula and the Maple program.
This sequence resembles the exponential of the von Mangoldt function A014963; for the latter sequence a(A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n) = 1 elsewhere.
Positions of the first occurrence of each successive noncomposite number (and also the records) is given by the union of {2} and A008837. - Antti Karttunen, Jan 17 2025

Antti Karttunen, Table of n, a(n) for n = 1..100949

Cf. A000040, A014963, A128060, A130290, A160479.
Cf. A000079, A000244 (after their initial 1's, the positions of 2's and 3's respectively), A020699 (positions of 5's from its third term 10 onward), A169634 (positions of 7's from the second term onward), A379956 (positions of terms > 1).
Cf. A008837, A121057.

nmax := 78: A000040 := proc(n): ithprime(n) end: A130290 := proc(n): if n =1 then 1 else (A000040(n)-1)/2 fi: end: for n from 1 to nmax do A217983(n) := 1 od: for n from 1 to nmax do for n1 from 1 to floor(log[A000040(n)](nmax)) do A217983(A130290(n) * A000040(n)^n1) := A000040(n) od: od: seq(A217983(n), n=1..nmax);

A217983(n) = { my(f=factor(n)); for(i=1,#f~,if((n/(f[i,1]^f[i,2])) == (f[i,1]\2), return(f[i,1]))); (1); }; \\ Antti Karttunen, Jan 16 2025

a(A130290(n) * A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n)= 1 elsewhere.

a(n) = (A160479(n+1) * A128060(n+1))/(2*n+1) for n >= 2.

Definition simplified, original definition moved to comments; more terms added by Antti Karttunen, Jan 16 2025

A249457 The numerator of curvatures of touching circles inscribed in a special way in the larger segment of a unit circle divided by a chord of length sqrt(84)/5.

Original entry on oeis.org

10, 100, 2890, 96100, 3237610, 109202500, 3683712490, 124263300100, 4191798484810, 141402777864100, 4769968258260490, 160906295771812900, 5427884341892493610, 183099910962324064900, 6176546013641762558890, 208354665265158340802500, 7028469704892605715408010

Offset: 0

Kival Ngaokrajang, Oct 29 2014

The denominators are conjectured to be A005032.
Refer to comments and links of A240926. Consider a unit circle with a chord of length sqrt(84)/5. This has been chosen such that the larger sagitta has length 7/5. The input, besides the unit circle C, is the circle C_0 with radius R_0 = 7/10, touching the chord and circle C. The following sequence of circles C_n with radii R_n, n >= 1, is obtained from the conditions that C_n touches (i) the circle C, (ii) the chord and (iii) the circle C_(n-1). The curvature of the n-th circle is C_n = 1/R_n, n >= 0, and its numerator is conjectured to be a(n).
If one considers the curvature of touching circles inscribed in the smaller segment (sagitta length 3/5), the rational sequence would be A249458/A169634. See an illustration given in the link.
For the proof and the formula for the rational curvatures of the circles in the larger segment see a comment under A249862. C_n = (5/7)*(S(n, 34/3) - (17/3)*S(n-1, 34/3) + 1), n >= 0, with Chebyshev's S polynomials (A049310). - Wolfdieter Lang, Nov 07 2014

Kival Ngaokrajang, Illustration of initial terms.
Eric Weisstein's World of Mathematics, Sagitta.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (37,-111,27).

Cf. A005032, A049310, A078986, A097315, A169364, A240926, A247335, A247512, A248834, A249458, A249862.

I:=[10,100,2890]; [n le 3 select I[n] else 37*Self(n-1) - 111*Self(n-2) + 27*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 20 2017

LinearRecurrence[{37, -111, 27},{10, 100, 2890},16] (* Ray Chandler, Aug 11 2015 *)
CoefficientList[Series[10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x)), {x, 0, 50}], x] (* G. C. Greubel, Dec 20 2017 *)

{
r=0.7;dn=7;print1(round(dn/r),", ");r1=r;
for (n=1,40,
     if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
     ac=sqrt(ab^2-r^2);
     if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
     b=acos(r/ab)-z;
     r=r*(1-cos(b))/(1+cos(b)); dn=dn*3;
     print1(round(dn/r),", ");
)
}

x='x+O('x^30); Vec(10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x))) \\ G. C. Greubel, Dec 20 2017

Empirical g.f.: -10*(30*x^2-27*x+1) /((3*x - 1)*(9*x^2-34*x+1)). - Colin Barker, Oct 29 2014
From Wolfdieter Lang, Nov 07 2014: (Start)
a(n) = 5*(A249862(n) + 3^n) = 5*3^n*(S(n, 34/3) - (17/3)*S(n-1, 34/3) + 1), n >= 0, with Chebyshev's S polynomials (A049310). See the comments on A249862 for the proof.
O.g.f.: 5*((1 - 17*x)/(1 - 34*x + 9*x^2) + 1/(1-3*x)) = 10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x)) proving the conjecture of Colin Barker above. (End)
E.g.f.: 5*exp(3*x)*(1 + exp(14*x)*cosh(2*sqrt(70)*x)). - Stefano Spezia, Mar 24 2023

Edited. Name and comment small changes, keyword easy added. - Wolfdieter Lang, Nov 07 2014

a(16) from Stefano Spezia, Mar 24 2023

A193577 a(n) = 5*7^n.

Original entry on oeis.org

5, 35, 245, 1715, 12005, 84035, 588245, 4117715, 28824005, 201768035, 1412376245, 9886633715, 69206436005, 484445052035, 3391115364245, 23737807549715, 166164652848005, 1163152569936035, 8142067989552245, 56994475926865715

Offset: 0

Vincenzo Librandi, Sep 15 2011

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

Cf. A005055.

Cf. A169634, A000420.

Magma
```
[5*7^n: n in [0..20]];
```

5 7^Range[0,20] (* or *) NestList[7#&,5,20] (* Harvey P. Dale, Dec 10 2017 *)

G.f.: 5/(1-7*x).

a(n) = 7*a(n-1), a(0)=5.

A249458 The numerators of curvatures of touching circles inscribed in a special way in the smaller segment of unit circle divided by a chord of length sqrt(84)/5.

Original entry on oeis.org

10, 100, 1690, 36100, 835210, 19802500, 472931290, 11318832100, 271066588810, 6492762648100, 155527144782490, 3725543446072900, 89243180863948810, 2137770243127864900, 51209104645650371290, 1226685938180259902500

Offset: 0

Kival Ngaokrajang, Oct 29 2014

The denominators are conjectured to be A169634.
Refer to comments and links of A240926. Consider a unit circle with a chord of length sqrt(84)/5. This has been chosen such that the smaller sagitta has length 3/5. The input, besides the circle C, is the circle C_0 with radius R_0 = 3/10, touching the chord and circle C. The following sequence of circles C_n with radii R_n, n >= 1, is obtained from the conditions that C_n touches (i) the circle C, (ii) the chord and (iii) the circle C_(n-1). The curvature of the n-th circle is C_n = 1/R_n, n >= 0, and its numerator is conjectured to be a(n). If one considers the curvature of touching circles inscribed in the larger segment (sagitta length 7/5), the sequence would be A249457/A005032. See an illustration given in the link.
For the proof and the formula for the rational curvatures of the circles in the smaller segment see a comment under A249864. C_n = (5/(3*7))*(7*S(n, 26/7) - 13*S(n-1, 26/7) + 7), n >= 0, with Chebyshev's S polynomials (A049310). - Wolfdieter Lang, Nov 08 2014

Kival Ngaokrajang, Illustration of initial terms.
Eric Weisstein's World of Mathematics, Sagitta.
Index entries for linear recurrences with constant coefficients, signature (33,-231,343).
Index entries for sequences related to Chebyshev polynomials.

Cf. A240926, A078986, A097315, A247512, A247335, A247512, A248834, A169634, A249457, A049310, A249863, A249864.

I:=[10, 100, 1690]; [n le 3 select I[n] else 33*Self(n-1) - 231*Self(n-2) + 343*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 20 2017

LinearRecurrence[{33, -231, 343},{10, 100, 1690},16] (* Ray Chandler, Aug 11 2015 *)
CoefficientList[Series[10*(1 - 23*x + 70*x^2)/((1 - 26*x + (7*x)^2)*(1 - 7*x)), {x, 0, 50}], x] (* G. C. Greubel, Dec 20 2017 *)

{
r=0.3;dn=3;print1(round(dn/r),", ");r1=r;
for (n=1,40,
     if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
     ac=sqrt(ab^2-r^2);
     if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
     b=acos(r/ab)-z;
     r=r*(1-cos(b))/(1+cos(b)); dn=dn*7;
     print1(round(dn/r),", ");
)
}

x='x+O('x^30); Vec(10*(1 - 23*x + 70*x^2)/((1 - 26*x + (7*x)^2)*(1 - 7*x))) \\ G. C. Greubel, Dec 20 2017

Empirical g.f.: -10*(70*x^2-23*x+1) / ((7*x-1)*(49*x^2-26*x+1)). - Colin Barker, Oct 29 2014
From Wolfdieter Lang, Nov 09 2014 (Start)
a(n) = 5*(A249864(n) + 7^n) = (5*7^n)*(S(n, 26/7) - (13/7)*S(n-1, 26/7) + 1), n >= 0, with Chebyshev's S polynomials (A049310). See the comments on A249864 for the proof.
O.g.f.: 5*((1 - 13*x)/(1 - 26*x + (7*x)^2) + 1/(1-7*x)) = 10*(1 - 23*x + 70*x^2)/((1 - 26*x + (7*x)^2)*(1 - 7*x)) proving the conjecture of Colin Barker above. (End)

Edited. In name and comment small changes, keyword easy and crossrefs added. - Wolfdieter Lang, Nov 08 2014

A193726 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.

Original entry on oeis.org

1, 1, 2, 2, 9, 10, 4, 28, 65, 50, 8, 76, 270, 425, 250, 16, 192, 920, 2200, 2625, 1250, 32, 464, 2800, 9000, 16250, 15625, 6250, 64, 1088, 7920, 32000, 77500, 112500, 90625, 31250, 128, 2496, 21280, 103600, 315000, 612500, 743750, 515625, 156250

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 (1,1,0,0,0,0,0,0,0,...) DELTA (2,3,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
   1;
   1,   2;
   2,   9,  10;
   4,  28,  65,   50;
   8,  76, 270,  425,  250;
  16, 192, 920, 2200, 2625, 1250;

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

Cf. A002532, A011782, A020699, A045873, A081040, A084938.

Cf. A133494, A141413, A169634, A193722, A193727.

function T(n, k) // T = A193726
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return 2*T(n-1, k) + 5*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 02 2023

(* First program *)
z = 8; a = 1; b = 2; c = 1; d = 2;
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}]]  (* A193726 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193727 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 2*T[n-1, k] + 5*T[n -1, k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2023 *)

def T(n, k): # T = A193726
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return 2*T(n-1, k) + 5*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 02 2023

T(n,k) = 5*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-x-3*x*y)/(1-2*x-5*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 02 2023: (Start)
T(n, 0) = A011782(n).
T(n, n) = A020699(n).
T(n, n-1) = A081040(n-1).
Sum_{k=0..n} T(n, k) = A169634(n-1) + (4/7)*[n=0].
Sum_{k=0..n} (-1)^k * T(n, k) = (-1)^n*A133494(n) = -A141413(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A002532(n) + 2*A002532(n-1) + (3/5)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A045873(n) - 2*A045873(n-1) + (3/5)*[n=0]. (End)

A193727 Mirror of the triangle A193726.

Original entry on oeis.org

1, 2, 1, 10, 9, 2, 50, 65, 28, 4, 250, 425, 270, 76, 8, 1250, 2625, 2200, 920, 192, 16, 6250, 15625, 16250, 9000, 2800, 464, 32, 31250, 90625, 112500, 77500, 32000, 7920, 1088, 64, 156250, 515625, 743750, 612500, 315000, 103600, 21280, 2496, 128

Offset: 0

Clark Kimberling, Aug 04 2011

This triangle is obtained by reversing the rows of the triangle A193726.

Triangle T(n,k), read by rows, given by (2,3,0,0,0,0,0,0,0,...) DELTA (1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

			First six rows:
     1;
     2,    1;
    10,    9,    2;
    50,   65,   28,   4;
   250,  425,  270,  76,   8;
  1250, 2625, 2200, 920, 192; 16;

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

Cf. A011782, A015535, A020699, A081040, A084938.

Cf. A107839, A133494, A169634, A193722, A193726.

function T(n, k) // T = A193727
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 5*T(n-1, k) + 2*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 02 2023

(* First program *)
z = 8; a = 1; b = 2; c = 1; d = 2;
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}]]  (* A193726 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193727 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 5*T[n-1, k] + 2*T[n-1, k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2023 *)

def T(n, k): # T = A193727
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 5*T(n-1, k) + 2*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 02 2023

T(n,k) = A193726(n,n-k).
T(n,k) = 2*T(n-1,k-1) + 5*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-3*x-x*y)/(1-5*x-2*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 02 2023: (Start)
T(n, 0) = A020699(n).
T(n, 1) = A081040(n-1).
T(n, n) = A011782(n).
Sum_{k=0..n} T(n, k) = A169634(n-1) + (4/7)*[n=0].
Sum_{k=0..n} (-1)^k * T(n, k) = A133494(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = 2*A015535(n) + A015535(n-1) + (1/2)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = 2*A107839(n-1) - A107839(n-2) + (1/2)*[n=0]. (End)

A270471 Expansion of g.f. (1-3x)/(1-7x).

Original entry on oeis.org

1, 4, 28, 196, 1372, 9604, 67228, 470596, 3294172, 23059204, 161414428, 1129900996, 7909306972, 55365148804, 387556041628, 2712892291396, 18990246039772, 132931722278404, 930522055948828, 6513654391641796, 45595580741492572, 319169065190448004, 2234183456333136028

Offset: 0

Colin Barker, Mar 17 2016

After 1, is this A208704?

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7).

Cf. A208704.
Cf. A000420 (powers of 7), A083076 (partial sums).
Cf. A193577: (1-2*x)/(1-7*x); A169634: (1-4*x)/(1-7*x).

CoefficientList[Series[(1 - 3 x)/(1 - 7 x), {x, 0, 21}], x] (* Michael De Vlieger, Mar 18 2016 *)
Join[{1},NestList[7#&,4,20]] (* Harvey P. Dale, Dec 21 2019 *)

PARI
```
Vec((1-3*x)/(1-7*x) + O(x^30))
```

G.f.: (1-3*x)/(1-7*x).
a(n) = 7*a(n-1) for n>1.
a(n) = 4*7^(n-1) for n>0.
E.g.f.: (4*exp(7*x) + 3)/7. - Elmo R. Oliveira, Mar 25 2025

cp's OEIS Frontend

A222444 T(n,k) = number of n X k 0..3 arrays with entries increasing mod 4 by 0, 1 or 2 rightwards and downwards, starting with upper left zero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A232955 T(n,k)=Number of nXk 0..3 arrays with no element x(i,j) adjacent to value 3-x(i,j) horizontally, diagonally or antidiagonally, and top left element zero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A169604 a(n) = 3*6^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A217983 If n = floor(p/2) * p^e, for some (by necessity unique) prime p and exponent e > 0, then a(n) = p, otherwise a(n) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A249457 The numerator of curvatures of touching circles inscribed in a special way in the larger segment of a unit circle divided by a chord of length sqrt(84)/5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

A193577 a(n) = 5*7^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A249458 The numerators of curvatures of touching circles inscribed in a special way in the smaller segment of unit circle divided by a chord of length sqrt(84)/5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A270471 Expansion of g.f. (1-3x)/(1-7x).