A077925 -id:A077925 - cp's OEIS frontend

A175722 a(n) = -a(n-1) + a(n-2) - F(-n) + 1, a(0) = 1, a(1) = -1, where F() = Fibonacci numbers A000045.

1, -1, 4, -6, 14, -24, 47, -83, 152, -268, 476, -832, 1453, -2517, 4348, -7474, 12810, -21880, 37275, -63335, 107376, -181656, 306744, -517056, 870169, -1462249, 2453812, -4112478, 6884102, -11510808, 19226951, -32084027, 53489288, -89097892, 148290068

Offset: 0

Roger L. Bagula, Dec 04 2010

			G.f. = 1 - x + 4*x^2 - 6*x^3 + 14*x^4 - 24*x^5 + 47*x^6 - 83*x^7 + 152*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,3,1,-3,1).

Cf. m=1: A077899, m large: A077925.

Cf. A000032, A000045, A006478.

List([0..40], n-> 1 + (-1)^n*(n*Lucas(1,-1,n+1)[2] + 7*Fibonacci(n))/5 ); # G. C. Greubel, Dec 04 2019

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1+x-x^2)^2))); // G. C. Greubel, Aug 14 2018

with(combinat); seq( 1 + (-1)^n*(n*fibonacci(n+2) + (n+7)*fibonacci(n))/5, n=0..40); # G. C. Greubel, Dec 04 2019

f[x_, m_] = ExpandAll[(x -x^(m+1))*(1-x-x^2) -(1 -2*x +x^(m+1))];
g[x_, n_] = ExpandAll[x^(m + 3)*f[1/x, m]];
a = Table[Table[SeriesCoefficient[Series[1/g[x, m], {x, 0, 20}], n], {n, 0, 20}], {m, 1, 20}]
CoefficientList[Series[1/((1-x)(1+x-x^2)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 13 2014 *)
RecurrenceTable[{a[0]==1,a[1]==-1,a[n]==-a[n-1]+a[n-2]-Fibonacci[-n]+1},a,{n,40}] (* Harvey P. Dale, May 12 2018 *)
Table[1 + (-1)^n*(n*LucasL[n+1] + 7*Fibonacci[n])/5, {n,0,40}] (* G. C. Greubel, Dec 04 2019 *)

{a(n) = if( n<0, polcoeff( x^5 / ((1 - x) * (1 - x - x^2)^2) + x * O(x^-n), -n), polcoeff( 1 / ((1 - x) * (1 + x - x^2)^2) + x * O(x^n), n))}; /* Michael Somos, Mar 11 2014 */

vector(41, n, my(f=fibonacci); 1 -(-1)^n*((n-1)*f(n+1) +(n+6)*f(n-1))/5 ) \\ G. C. Greubel, Dec 04 2019

[1 + (-1)^n*(n*lucas_number2(n+1, 1,-1) + 7*fibonacci(n))/5 for n in (0..40)] # G. C. Greubel, Dec 04 2019

G.f.: 1/(- x^m + 1 - x^(1 + m) + x + 3*x^(2 + m) - 2*x^2 - x^(3 + m)) for m=2.
G.f.: 1 / ((1 - x) * (1 + x - x^2)^2). - Michael Somos, Mar 11 2014
a(n) = A006478(-2-n) for all n in Z. - Michael Somos, Mar 11 2014
a(n) = 1 + (-1)^n*(n*Lucas(n+1) + 7*Fibonacci(n))/5. - G. C. Greubel, Dec 04 2019
E.g.f.: exp(-x/2)*(25*exp(3*x/2) - 15*x*cosh(sqrt(5)*x/2) + sqrt(5)*(5*x - 14)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Jul 24 2022

A283642 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 678", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, 178956971, 357913941, 715827883, 1431655765, 2863311531, 5726623061, 11453246123

Offset: 0

Robert Price, Mar 12 2017

Initialized with a single black (ON) cell at stage zero.
Similar to A001045.
It is not difficult to prove that one has indeed a(n) = round(4*2^n/3) = A001045(n+2) for all n. The proof as well as the growth of the pattern is nearly identical to that of the toothpick sequence A139250. - M. F. Hasler, Feb 13 2020
The decimal representations of the n-th interval of elementary cellular automata rules 28 and 156 (see A266502 and A266508) generate this sequence. - Karl V. Keller, Jr., Sep 03 2021

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A283641, A266502, A266508, A086893, A001045.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 678; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]

print([(4*2**n + 1)//3 for n in range(50)]) # Karl V. Keller, Jr., Sep 03 2021

From Colin Barker, Mar 14 2017: (Start)
G.f.: (1 + 2*x) / ((1 + x)*(1 - 2*x)).
a(n) = (2^(n+2) - 1) / 3 for n even.
a(n) = (2^(n+2) + 1) / 3 for n odd.
a(n) = a(n-1) + 2*a(n-2) for n>1.
(End)
I.e., a(n) = A001045(n+2) = A154917(n+2) = A167167(n+2) = |A077925(n+1)| = A328284(n+5) = round(4*2^n/3), cf. comments. - M. F. Hasler, Feb 13 2020
E.g.f.: (4*exp(2*x) - exp(-x))/3. - Stefano Spezia, Feb 13 2020
a(n) = floor((4*2^n + 1)/3). - Karl V. Keller, Jr., Sep 03 2021

A305098 Triangle read by rows: T(0,0) = 1; T(n,k) = -T(n-1,k) + 2 T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, -1, 1, 2, -1, -4, 1, 6, 4, -1, -8, -12, 1, 10, 24, 8, -1, -12, -40, -32, 1, 14, 60, 80, 16, -1, -16, -84, -160, -80, 1, 18, 112, 280, 240, 32, -1, -20, -144, -448, -560, -192, 1, 22, 180, 672, 1120, 672, 64, -1, -24, -220, -960, -2016, -1792, -448

Offset: 0

Shara Lalo, May 25 2018

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2*x)^n).
The coefficients in the expansion of 1/(1+x-2x^2) are given by the sequence generated by the row sums.
When n is even the numbers in the row are positive, and when n is odd the numbers in the row are negative.

			Triangle begins:
   1;
  -1;
   1,   2;
  -1,  -4;
   1,   6,    4;
  -1,  -8,  -12;
   1,  10,   24,     8;
  -1, -12,  -40,   -32;
   1,  14,   60,    80,     16;
  -1, -16,  -84,  -160,    -80;
   1,  18,  112,   280,    240,     32;
  -1, -20, -144,  -448,   -560,   -192;
   1,  22,  180,   672,   1120,    672,     64;
  -1, -24, -220,  -960,  -2016,  -1792,   -448;
   1,  26,  264,  1320,   3360,   4032,   1792,    128;
  -1, -28, -312, -1760,  -5280,  -8064,  -5376,  -1024;
   1,  30,  364,  2288,   7920,  14784,  13440,   4608,   256;
  -1, -32, -420, -2912, -11440, -25344, -29568, -15360, -2304;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 389-391.

Signed version of A128099.
Row sums give A077925.
Cf. A303872, A033999 (column 0).

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, -t[n - 1, k] + 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, -T(n-1, k) + 2*T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 26 2018

G.f.: 1 / (1 + t*x - 2t^2).

A345034 a(n) = Sum_{k=1..n} (-2)^(floor(n/k) - 1).

Original entry on oeis.org

1, -1, 6, -8, 17, -27, 70, -136, 255, -491, 1046, -2082, 4063, -8131, 16476, -32882, 65423, -130845, 262372, -524818, 1048149, -2096045, 4195412, -8390820, 16775029, -33550477, 67113210, -134225588, 268427597, -536854983, 1073757754, -2147517076

Offset: 1

Seiichi Manyama, Jun 06 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..3000

Column k=2 of A345033.

Cf. A077925, A345028.

a[n_] := Sum[(-2)^(Floor[n/k] - 1), {k, 1, n}]; Array[a, 30] (* Amiram Eldar, Jun 06 2021 *)

PARI
```
a(n) = sum(k=1, n, (-2)^(n\k-1));
```

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k*(1-x^k)/(1+2*x^k))/(1-x))

G.f.: (1/(1 - x)) * Sum_{k>=1} x^k * (1 - x^k)/(1 + 2*x^k).

A209998 Triangle of coefficients of polynomials v(n,x) jointly generated with A209996; see the Formula section.

Original entry on oeis.org

1, 2, 3, 2, 8, 9, 2, 10, 30, 27, 2, 10, 46, 108, 81, 2, 10, 50, 198, 378, 243, 2, 10, 50, 242, 810, 1296, 729, 2, 10, 50, 250, 1122, 3186, 4374, 2187, 2, 10, 50, 250, 1234, 4986, 12150, 14580, 6561, 2, 10, 50, 250, 1250, 5946, 21330, 45198, 48114, 19683

Offset: 1

Clark Kimberling, Mar 23 2012

Row n starts 2, 2*5, 2*5^2,... ; ends with 3^(n-1).
Conjecture: penultimate term in row n is A199923(n).
Alternating row sums: A077925
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2...3
2...8....9
2...10...30...27
2...10...46...108...81
First three polynomials v(n,x): 1, 2 + 3x , 2 + 8x + 9x^2.

Cf. A209996, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*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[%]    (* A209996 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209998 *)

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

a(55) corrected by Georg Fischer, Sep 03 2021

A210747 Triangle of coefficients of polynomials u(n,x) jointly generated with A210748; see the Formula section.

Original entry on oeis.org

1, 2, 3, 4, 9, 8, 7, 24, 33, 21, 12, 54, 109, 111, 55, 20, 114, 297, 435, 355, 144, 33, 228, 736, 1383, 1606, 1098, 377, 54, 441, 1697, 3912, 5813, 5625, 3316, 987, 88, 831, 3723, 10158, 18419, 22779, 18962, 9837, 2584, 143, 1536, 7859, 24798

Offset: 1

Clark Kimberling, Mar 25 2012

Row n starts with -1+F(n+2) and ends with F(2n), where F=A000045 (Fibonacci numbers).
Row sums: A002450
Alternating row sums:  A077925
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2....3
4....9....8
7....24...33....21
12...54...109...111...55
First three polynomials u(n,x): 1, 2+ 3x, 4 + 9x + 8x^2.

Cf. A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*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[%]    (* A210747 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210748 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A002450 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A002450 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)

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

A210748 Triangle of coefficients of polynomials v(n,x) jointly generated with A210747; see the Formula section.

Original entry on oeis.org

1, 3, 2, 6, 10, 5, 11, 29, 32, 13, 19, 71, 118, 99, 34, 32, 156, 352, 437, 299, 89, 53, 322, 919, 1521, 1526, 887, 233, 87, 636, 2205, 4559, 6036, 5117, 2595, 610, 142, 1218, 4979, 12373, 20320, 22591, 16653, 7508, 1597, 231, 2279, 10751, 31233

Offset: 1

Clark Kimberling, Mar 25 2012

Row n starts with -2+F(n+3) and ends with F(2n-1), where F=A000045 (Fibonacci numbers).
Row sums: A002450
Alternating row sums:  1,1,1,1,1,1,1,1,1,...(A000012)
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
3....2
6....10...5
11...29...32....13
19...71...118...99...34
First three polynomials v(n,x): 1, 3 + 2x, 6 + 10x +5x^2

Cf. A210747, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*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[%]   (* A210747 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210748 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A002450 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A002450 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)

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

A210793 Triangle of coefficients of polynomials u(n,x) jointly generated with A210794; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 6, 10, 8, 3, 9, 24, 27, 16, 5, 18, 51, 74, 62, 30, 8, 27, 108, 189, 200, 136, 56, 13, 54, 216, 450, 574, 488, 282, 102, 21, 81, 432, 1026, 1536, 1571, 1128, 569, 184, 34, 162, 837, 2268, 3864, 4598, 3967, 2486, 1118, 328, 55, 243, 1620

Offset: 1

Clark Kimberling, Mar 26 2012

Row n starts with A038754(n) and ends with F(n), where F=A000045 (Fibonacci numbers).
Row sums: A000244 (powers of 3).
Alternating row sums: A000012 (1,1,1,1,1,1,1,1,1,1,1,...).
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 29 2012

			First five rows:
  1;
  2,  1;
  3,  4,  2;
  6, 10,  8,  3;
  9, 24, 27, 16,  5;
First three polynomials u(n,x):
  1
  2 + x
  3 + 4x + 2x^2.
From _Philippe Deléham_, Mar 29 2012: (Start)
(1, 1, -1, -1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,  0;
   3,  4,  2,  0;
   6, 10,  8,  3,  0;
   9, 24, 27, 16,  5,  0;
  18, 51, 74, 62, 30,  8,  0; (End)

Cf. A210794, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 1; c = 0; h = 2; p = -1; f = 0;
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[%]   (* A210793 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210794 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000244 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A000012 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A077925 *)

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = (x+2)*u(n-1,x) + (x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 29 2012: (Start)
As DELTA(triangle T(n,k) with 0 <= k <= n:
G.f.: (1 + x - y*x^2 - 2*y*x^2 - y^2*x^2)/(1 - y*x - 3*x^2 - 2*y*x^2 - y^2*x^2).
T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k <= n. (End)

A210794 Triangle of coefficients of polynomials v(n,x) jointly generated with A210793; see the Formula section.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 3, 11, 8, 5, 9, 18, 29, 17, 8, 9, 48, 67, 71, 35, 13, 27, 81, 180, 194, 158, 68, 21, 27, 189, 387, 575, 508, 338, 129, 34, 81, 324, 918, 1410, 1617, 1222, 695, 239, 55, 81, 702, 1890, 3606, 4471, 4222, 2793, 1393, 436, 89, 243, 1215

Offset: 1

Clark Kimberling, Mar 26 2012

Row n starts with a power of 3 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Row sums: A000244 (powers of 3)
Alternating row sums: A077925
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
3...3....3
3...11...8....5
9...18...29...17...8
First three polynomials v(n,x): 1, 1 + 2x, 3 + 3x + 3x^2

Cf. A210793, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 1; c = 0; h = 2; p = -1; f = 0;
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[%]   (* A210793 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210794 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000244 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)

u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 29 2012

A210870 Triangle of coefficients of polynomials u(n,x) jointly generated with A210871; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 4, 6, 5, 5, 1, 5, 8, 12, 8, 8, 1, 6, 12, 17, 23, 13, 13, 1, 7, 15, 29, 33, 43, 21, 21, 1, 8, 20, 38, 64, 63, 79, 34, 34, 1, 9, 24, 56, 86, 136, 117, 143, 55, 55, 1, 10, 30, 70, 140, 187, 279, 214, 256, 89, 89, 1, 11, 35, 95, 180, 332

Offset: 1

Clark Kimberling, Mar 29 2012

In row n, for n>1, the first two terms are 1 and n-1, and the last two are F(n) and F(n), where F = A000045 (Fibonacci numbers).
Row sums: A000975
Alternating row sums:  A113954
For a discussion and guide to related arrays, see A208510.

			First six rows:
1
1...1
1...2...2
1...3...3...3
1...4...6...5....5
1...5...8...12...8...8
First three polynomials u(n,x): 1, 1 + x, 1 + 2x + 2x^2.

Cf. A210871, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%]    (* A210870 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210871 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000975 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A001045 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]   (* A113954 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A077925 *)

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

cp's OEIS Frontend

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A210747 Triangle of coefficients of polynomials u(n,x) jointly generated with A210748; see the Formula section.

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