A131300 -id:A131300 - cp's OEIS frontend

A131299 Triangle T(n,k) = 3*binomial(n-floor((k+1)/2), floor(k/2))-2 with k=0..n, read by rows.

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 7, 4, 1, 1, 1, 10, 7, 7, 1, 1, 1, 13, 10, 16, 7, 1, 1, 1, 16, 13, 28, 16, 10, 1, 1, 1, 19, 16, 43, 28, 28, 10, 1, 1, 1, 22, 19, 61, 43, 58, 28, 13, 1, 1, 1, 25, 22, 82, 61, 103, 58, 43, 13, 1, 1, 1, 28, 25, 106, 82, 166

Offset: 0

Gary W. Adamson, Jun 27 2007

Row sums are in A131300. Reversed row triangle = A131301.
From R. J. Mathar, Apr 08 2013: (Start)
The matrix inverse starts
   1;
  -1,     1;
   0,    -1,       1;
   0,     3,      -4,     1;
   0,    -6,       9,    -4,      1;
   0,    30,     -45,    21,     -7,    1;
   0,  -132,     198,   -93,     33,   -7,    1;
   0,   984,   -1476,   693,   -246,   54,  -10,   1;
   0, -6756,   10134, -4758,   1689, -372,   72, -10,   1;
   0, 66972, -100458, 47166, -16743, 3687, -714, 102, -13, 1;
(End)

			Triangle begins:
  1;
  1,  1;
  1,  1,  1;
  1,  1,  4,  1;
  1,  1,  7,  4,  1;
  1,  1, 10,  7,  7,  1;
  1,  1, 13, 10, 16,  7,  1;
  ...

Nathaniel Johnston, Rows n = 0..100, flattened

Cf. A065941, A000012, A131300, A131301.

A131299 := proc(n,k) 3*binomial(n-floor((k+1)/2),floor(k/2))-2 ; end proc; # Nathaniel Johnston, Jun 30 2011

3*A065941 - 2*A000012 as infinite lower triangular matrices.

Better definition from Bruno Berselli, May 03 2012

A131301 Regular triangle read by rows: T(n,k) = 3*binomial(floor((n+k)/2),k)-2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 7, 1, 1, 1, 7, 7, 10, 1, 1, 1, 7, 16, 10, 13, 1, 1, 1, 10, 16, 28, 13, 16, 1, 1, 1, 10, 28, 28, 43, 16, 19, 1, 1, 1, 13, 28, 58, 43, 61, 19, 22, 1, 1, 1, 13, 43, 58, 103, 61, 82, 22, 25, 1, 1, 1, 16, 43, 103, 103, 166, 82

Offset: 0

Gary W. Adamson, Jun 27 2007

Row sums = A131300: (1, 2, 3, 7, 14, 27, 49, 86, ...). Reversed triangle = A131299.

			First few rows of the triangle:
  1;
  1,  1;
  1,  1,  1;
  1,  4,  1,  1;
  1,  4,  7,  1,  1;
  1,  7,  7, 10,  1,  1;
  1,  7, 16, 10, 13,  1,  1;
  ...

Nathaniel Johnston, Rows 0..100, flattened

Cf. A046854, A000012, A131299, A131300.

for n from 0 to 6 do seq(3*binomial(floor((n+k)/2),k)-2,k=0..n); od; # Nathaniel Johnston, Jun 29 2011

t[n_, k_] := 3 Binomial[Floor[(n + k)/2], k] - 2; Table[t[n, k], {n, 11}, {k, 0, n}] // Flatten
(* to view triangle: Table[t[n, k], {n, 5}, {k, 0, n}] // TableForm *) (* Robert G. Wilson v, Feb 28 2015 *)

3*A046854 - 2*A000012 as infinite lower triangular matrices (former name).

T(n,k) = 3*binomial(floor((n+k)/2),k)-2. - Nathaniel Johnston, Jun 29 2011

A192953 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 2, 6, 13, 26, 48, 85, 146, 246, 409, 674, 1104, 1801, 2930, 4758, 7717, 12506, 20256, 32797, 53090, 85926, 139057, 225026, 364128, 589201, 953378, 1542630, 2496061, 4038746, 6534864, 10573669, 17108594, 27682326, 44790985, 72473378

Offset: 0

Clark Kimberling, Jul 13 2011

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 2n - 1, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A192232, A192744, A192951.

F:=Fibonacci;; List([0..40], n-> 3*F(n+2)-(2*n+3)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [3*F(n+2)-(2*n+3): n in [0..40]]; // G. C. Greubel, Jul 12 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + 2n - 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A111314 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192953 *)
(* Second program *)
With[{F=Fibonacci}, Table[3*F[n+2]-(2*n+3), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, n--; f=fibonacci; 3*f(n+2)-(2*n+3)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [3*f(n+2)-(2*n+3) for n in (0..40)] # G. C. Greubel, Jul 12 2019

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: x*(1 -x +2*x^2)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, Aug 01 2011
a(n) = -2*n - 3 + 3*A000045(n+2). - R. J. Mathar, Aug 01 2011
a(n) = A131300(n) - 1. - R. J. Mathar, Mar 24 2018
a(n) = 3*Fibonacci(n+2) - (2*n+3). - G. C. Greubel, Jul 12 2019

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A131299 Triangle T(n,k) = 3*binomial(n-floor((k+1)/2), floor(k/2))-2 with k=0..n, read by rows.

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A131301 Regular triangle read by rows: T(n,k) = 3*binomial(floor((n+k)/2),k)-2.

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A192953 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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