A052542 -id:A052542 - cp's OEIS frontend

A098790 a(n) = 2*a(n-1) + a(n-2) + 1, a(0) = 1, a(1) = 2.

1, 2, 6, 15, 37, 90, 218, 527, 1273, 3074, 7422, 17919, 43261, 104442, 252146, 608735, 1469617, 3547970, 8565558, 20679087, 49923733, 120526554, 290976842, 702480239, 1695937321, 4094354882, 9884647086, 23863649055, 57611945197

Offset: 0

Creighton Dement, Oct 30 2004

nonn

Previous name was: a(n) = A048739(n) - A000129(n).

Partial sums of Pell numbers A000129 except omit next-to-last Pell number. E.g., 37 = 0+1+2+5+12+29 - 12.

M. Bicknell-Johnson and G. E. Bergum, The Generalized Fibonacci Numbers {C(n)}, C(n)=C(n-1)+C(n-2)+K, Applications of Fibonacci Numbers, 1986, pp. 193-205.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434.
Hermann Stamm-Wilbrandt, 4 interlaced bisections
Index entries for linear recurrences with constant coefficients, signature (3, -1, -1).

Cf. A006451, A000129, A048739, A124124, A024537, A074323, A052542.

a[0] = 1; a[1] = 2; a[n_] := a[n] = 2a[n - 1] + a[n - 2] + 1; Table[ a[n], {n, 0, 28}] (* Robert G. Wilson v, Nov 04 2004 *)
LinearRecurrence[{3,-1,-1},{1,2,6},31] (* Harvey P. Dale, Oct 15 2011 *)
CoefficientList[Series[(x^2 - x + 1)/((1 - x) (1 - 2 x - x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 14 2014 *)

a(n) = 2*a(n-1) + a(n-2) + 1, a(0) = 1, a(1) = 2.
G.f.: (x^2-x+1)/((1-x)(1-2x-x^2)).
a(n+1) = - A024537(n+1) + 2*A048739(n+1) - 2*A048739(n).
a(n) = - A024537(n) + A052542(n+1).
Partial sums of A074323. - Paul Barry, Mar 11 2007
a(n) = (sqrt(2)+1)^n*(3/4+sqrt(2)/4)+(sqrt(2)-1)^n*(3/4-sqrt(2)/4)*(-1)^n-1/2; - Paul Barry, Mar 11 2007
a(0)=1, a(1)=2, a(2)=6, a(n)=3*a(n-1)-a(n-2)-a(n-3). [Harvey P. Dale, Oct 15 2011]
a(2*n) = A124124(2*n+1). - Hermann Stamm-Wilbrandt, Aug 03 2014
a(2*n+1) = A006451(2*n+1). - Hermann Stamm-Wilbrandt, Aug 26 2014
a(n) = 7*a(n-2) - 7*a(n-4) + a(n-6), for n>5. - Hermann Stamm-Wilbrandt, Aug 26 2014
2*a(n) = A135532(n+1)-1. - R. J. Mathar, Jan 13 2023

More terms from Robert G. Wilson v, Nov 04 2004
Definition edited by N. J. A. Sloane, Aug 03 2014
New name from existing formula by Joerg Arndt, Aug 13 2014

A119016 Numerators of "Farey fraction" approximations to sqrt(2).

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 7, 10, 17, 24, 41, 58, 99, 140, 239, 338, 577, 816, 1393, 1970, 3363, 4756, 8119, 11482, 19601, 27720, 47321, 66922, 114243, 161564, 275807, 390050, 665857, 941664, 1607521, 2273378, 3880899, 5488420, 9369319, 13250218, 22619537, 31988856

Offset: 0

Joshua Zucker, May 08 2006

"Add" (meaning here to add the numerators and add the denominators, not to add the fractions) 1/0 to 1/1 to make the fraction bigger: 2/1. Now 2/1 is too big, so add 1/1 to make the fraction smaller: 3/2, 4/3. Now 4/3 is too small, so add 3/2 to make the fraction bigger: 7/5, 10/7, ... Because the continued fraction for sqrt(2) is all 2s, it will always take exactly two terms here to switch from a number that's bigger than sqrt(2) to one that's less. a(n+2) = A082766(n).
a(2n) are the interleaved values of m such that 2*m^2-2 and 2*m^2+2 are squares, respectively; a(2n+1) are the corresponding integer square roots. - Richard R. Forberg, Aug 19 2013
Apart from the first two terms, this is the sequence of numerators of the convergents of the continued fraction expansion sqrt(2) = 1/(1 - 1/(2 + 1/(1 - 1/(2 + 1/(1 - ....))))). - Peter Bala, Feb 02 2017

			The fractions are 1/0, 0/1, 1/1, 2/1, 3/2, 4/3, 7/5, 10/7, 17/12, ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Dave Rusin, Farey fractions on sci.math [Broken link]
Dave Rusin, Farey fractions on sci.math [Cached copy]
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Cf. A097545, A097546 gives the similar sequence for Pi. A119014, A119015 gives the similar sequence for e. A002965 gives the denominators for this sequence. Also very closely related to A001333, A052542 and A000129.

See A082766 for another version.

f:= gfun:-rectoproc({a(n+4)=2*a(n+2) +a(n),a(0)=1,a(1)=0,a(2)=1,a(3)=2}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Jun 10 2015

f[x_, n_] := (m = Floor[x]; f0 = {m, m+1/2, m+1}; r = ({a___, b_, c_, d___} /; b < x < c) :> {b, (Numerator[b] + Numerator[c]) / (Denominator[b] + Denominator[c]), c}; Join[{m, m+1}, NestList[# /. r &, f0, n-3][[All, 2]]]); Join[{1, 0 }, f[Sqrt[2], 38]] // Numerator (* Jean-François Alcover, May 18 2011 *)
LinearRecurrence[{0, 2, 0, 1}, {1, 0, 1, 2}, 40] (* and *) t = {1, 2}; Do[AppendTo[t, t[[-2]] + t[[-1]]]; AppendTo[t, t[[-3]] + t[[-1]]], {n, 30}]; t (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
a0 := LinearRecurrence[{2, 1}, {1, 1}, 20]; (*     A001333 *)
a1 := LinearRecurrence[{2, 1}, {0, 2}, 20]; (* 2 * A000129 *)
Flatten[MapIndexed[{a0[[#]],a1[[#]]} &, Range[20]]] (* Gerry Martens, Jun 09 2015 *)

x='x+O('x^50); Vec((1 - x^2 + 2*x^3)/(1 - 2*x^2 - x^4)) \\ G. C. Greubel, Oct 20 2017

From Joerg Arndt, Feb 14 2012: (Start)
a(0) = 1, a(1) = 0, a(2n) = a(2n-1) + a(2n-2), a(2n+1) = a(2n) + a(2n-2).
G.f.: (1 - x^2 + 2*x^3)/(1 - 2*x^2 - x^4). (End)
a(n) = 1/4*(1-(-1)^n)*(-2+sqrt(2))*(1+sqrt(2))*((1-sqrt(2))^(1/2*(n-1))-(1+sqrt(2))^(1/2*(n-1)))+1/4*(1+(-1)^n)*((1-sqrt(2))^(n/2)+(1+sqrt(2))^(n/2)). - Gerry Martens, Nov 04 2012
a(2n) = A001333(n). a(2n+1) = A052542(n) for n>0. - R. J. Mathar, Feb 05 2024

A193737 Mirror of the triangle A193736.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 4, 3, 1, 3, 8, 8, 4, 1, 5, 15, 19, 13, 5, 1, 8, 28, 42, 36, 19, 6, 1, 13, 51, 89, 91, 60, 26, 7, 1, 21, 92, 182, 216, 170, 92, 34, 8, 1, 34, 164, 363, 489, 446, 288, 133, 43, 9, 1, 55, 290, 709, 1068, 1105, 826, 455, 184, 53, 10, 1, 89, 509, 1362, 2266, 2619, 2219, 1414, 682, 246, 64, 11, 1

Offset: 0

Clark Kimberling, Aug 04 2011

This triangle is obtained by reversing the rows of the triangle A193736.

			First six rows:
  1;
  1,  1;
  1,  2,  1;
  2,  4,  3,  1;
  3,  8,  8,  4,  1;
  5, 15, 19, 13,  5,  1;

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

Cf. A000007, A011782 (diagonal sums), A019590, A052542 (row sums).

Cf. A034856, A193736, A321620, A324969, A330793.

function T(n,k) // T = A193737
  if k lt 0 or n lt 0 then return 0;
  elif n lt 3 then return Binomial(n,k);
  else return T(n - 1, k) + T(n - 1, k - 1) + T(n - 2, k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 24 2023

(* First program *)
z=20;
p[0, x_]:= 1;
p[n_, x_]:= Fibonacci[n+1, x] /; n > 0
q[n_, x_]:= (x + 1)^n;
t[n_, k_]:= Coefficient[p[n, x], x^(n-k)];
t[n_, n_]:= p[n, x] /. x -> 0;
w[n_, x_]:= Sum[t[n, k]*q[n-k+1, 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}]] (* A193736 *)
TableForm[Table[g[n], {n,-1,z}]]
Flatten[Table[g[n], {n,-1,z}]]          (* A193737 *)
(* Additional programs *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[1 + 1/(1 - x - x^2), 11]//Flatten  (* Peter Luschny, Feb 27 2021 *)
T[n_, k_]:= T[n, k]= If[n<3, Binomial[n, k], T[n-1,k] + T[n-1,k-1] + T[n-2,k]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 24 2023 *)

def T(n,k): # T = A193737
    if (n<3): return binomial(n,k)
    else: return T(n-1,k) +T(n-1,k-1) +T(n-2,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

Write w(n,k) for the triangle at A193736.  This is then given by w(n,n-k).
T(0,0) = T(1,0) = T(1,1) = T(2,0) = 1; T(n,k) = 0 if k<0 or k>n; T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k). - Philippe Deléham, Feb 13 2020
From G. C. Greubel, Oct 24 2023: (Start)
T(n, 0) = Fibonacci(n) + [n=0] = A324969(n+1).
T(n, n-1) = n, for n >= 1.
T(n, n-2) = A034856(n-1), for n >= 2.
T(2*n, n) = A330793(n).
Sum_{k=0..n} T(n,k) = A052542(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A011782(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k,k) = A019590(n). (End)

A094024 Alternating 1 with one less than the powers of 2.

Original entry on oeis.org

1, 1, 1, 3, 1, 7, 1, 15, 1, 31, 1, 63, 1, 127, 1, 255, 1, 511, 1, 1023, 1, 2047, 1, 4095, 1, 8191, 1, 16383, 1, 32767, 1, 65535, 1, 131071, 1, 262143, 1, 524287, 1, 1048575, 1, 2097151, 1, 4194303, 1, 8388607, 1, 16777215, 1, 33554431, 1, 67108863, 1

Offset: 0

Paul Barry, Apr 22 2004

Inverse binomial transform of A052542. Partial sums are A075427.

Let F(x) = product {n >= 0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number F(1/2) = 0.64227 25013 85234 96714 ... = 1/(1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + 1/(7 + 1/(1 + 1/(15 + ...)))))))). See A111317. - Peter Bala, Dec 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (-1,2,2).

Cf. A000129, A016116.

Cf. A111317.

[Ceiling((-1)^n+((Sqrt(2))^n-(-Sqrt(2))^n)/Sqrt(2)): n in [0..50]]; // Vincenzo Librandi, Aug 17 2011

LinearRecurrence[{-1, 2, 2}, {1, 1, 1}, 60] (* Jean-François Alcover, Jul 02 2018 *)

PARI
```
a(n)=(1-(-1)^n)*2^floor(n/2)+(-1)^n
```

G.f.: (1+2*x) / ((1+x) * (1-2*x^2)).
E.g.f.: exp(-x) + 2*sinh(sqrt(2)*x) / sqrt(2).
a(n) = (-1)^n + ((sqrt(2))^n - (-sqrt(2))^n) / sqrt(2).
a(n) = (1-(-1)^n) * 2^floor(n/2) + (-1)^n. - Ralf Stephan, Aug 19 2013
a(n) = -a(n-1) + 2*a(n-2) + 2*a(n-3). - Andrew Howroyd, Feb 21 2018

Better name from Ralf Stephan, Aug 19 2013

Even terms for n >= 60 corrected in b-file by Andrew Howroyd, Feb 21 2018

A082766 Series ratios converge alternately to sqrt(2) and 1+sqrt(1/2).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 17, 24, 41, 58, 99, 140, 239, 338, 577, 816, 1393, 1970, 3363, 4756, 8119, 11482, 19601, 27720, 47321, 66922, 114243, 161564, 275807, 390050, 665857, 941664, 1607521, 2273378, 3880899, 5488420, 9369319, 13250218, 22619537

Offset: 1

Gary W. Adamson, May 24 2003

nonn

a(2n+2)/a(2n+1) converges to sqrt(2).
a(2n+1)/a(2n) converges to 1+sqrt(1/2).
a(n+2)/a(n) converges to 1+sqrt(2).
a(2n) is A001333, a(2n+1) is A052542.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Haocong Song and Wen Wu, Hankel determinants of a Sturmian sequence, arXiv:2007.09940 [math.CO], 2020. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Cf. A001333, A052542. See A119016 for another version.

import Data.List (transpose)
a082766 n = a082766_list !! (n-1)
a082766_list = concat $ transpose [a052542_list, tail a001333_list]
-- Reinhard Zumkeller, Feb 24 2015

Rest[CoefficientList[Series[x (1 - x^2 + x) (x^2 + 1)/(1 - 2 x^2 - x^4), {x, 0, 50}], x]] (* G. C. Greubel, Nov 28 2017 *)
LinearRecurrence[{0,2,0,1},{1,1,2,3,4},50] (* Harvey P. Dale, Dec 15 2022 *)

x='x+O('x^30); Vec(x*(1+x-x^2)*(x^2+1)/(1-2*x^2-x^4)) \\ G. C. Greubel, Nov 28 2017

a(2n) = a(2n-1) + a(2n-2); a(2n+1) = a(2n) + a(2n-2)

O.g.f.: x*(1+x-x^2)*(x^2+1)/(1-2*x^2-x^4). - R. J. Mathar, Aug 08 2008

Edited by Don Reble, Nov 04 2005

A117918 Difference row triangle of the Pell sequence.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 2, 4, 7, 12, 4, 6, 10, 17, 29, 4, 8, 14, 24, 41, 70, 8, 12, 20, 34, 58, 99, 169, 8, 16, 28, 48, 82, 140, 239, 408, 16, 24, 40, 68, 116, 198, 338, 577, 985, 16, 32, 56, 96, 164, 280, 478, 816, 1393, 2378, 32, 48, 80, 136, 232, 396, 676, 1154, 1970, 3363, 5741

Offset: 1

Gary W. Adamson, Apr 02 2006

			First few rows of the triangle are:
  1;
  1,  2;
  2,  3,  5;
  2,  4,  7, 12;
  4,  6, 10, 17, 29;
  4,  8, 14, 24, 41, 70;
  8, 12, 20, 34, 58, 99, 169;
  ...

Raymond Lebois, "Le théorème de Pythagore et ses implications", p. 123, Editions PIM, (1979).

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Column 1 is A016116(n-1).

Diagonals include A000129, A001333, A052542, A002203, A371596.

Pell:= func< n | Round(((1+Sqrt(2))^n -(1-Sqrt(2))^n)/(2*Sqrt(2))) >;
T:= func< n,k | (&+[ (-1)^j*Binomial(n-k,j)*Pell(n-j): j in [0..n-k]]) >;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 23 2021

T[n_, k_]:= T[n, k]= If[k==1, 2^Floor[(n-1)/2], T[n, k-1] + T[n-1, k-1]];
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 22 2021 *)

def A117918(n,k): return sum( (-1)^j*binomial(n-k, j)*lucas_number1(n-j, 2,-1) for j in (0..n) )
flatten([[A117918(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 23 2021

Difference rows of the Pell sequence A000129 starting (1, 2, 5, 12, ...) become the diagonals of the triangle.
T(n, n) = A000129(n).
T(n, n-1) = A000129(n) - A000129(n-1).
From G. C. Greubel, Oct 23 2021: (Start)
T(n, k) = T(n, k-1) + T(n-1, k-1) with T(n, 1) = 2^floor((n-1)/2).
T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n-k, j)*Pell(n-j), where Pell(n) = A000129(n).
Sum_{k=1..n} T(n, k) = Pell(n+1) -2^floor(n/2)*((1 + (-1)^n)/2) - 2^floor((n - 1)/2)*((1 - (-1)^n)/2). (End)

A193736 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (n+1)-st Fibonacci polynomial and q(n,x) = (x+1)^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 4, 8, 8, 3, 1, 5, 13, 19, 15, 5, 1, 6, 19, 36, 42, 28, 8, 1, 7, 26, 60, 91, 89, 51, 13, 1, 8, 34, 92, 170, 216, 182, 92, 21, 1, 9, 43, 133, 288, 446, 489, 363, 164, 34, 1, 10, 53, 184, 455, 826, 1105, 1068, 709, 290, 55, 1, 11, 64, 246, 682, 1414, 2219, 2619, 2266, 1362, 509, 89

Offset: 0

Clark Kimberling, Aug 04 2011

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

			First six rows:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  4,  2;
  1,  4,  8,  8,  3;
  1,  5, 13, 19, 15,  5;

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

Cf. A000007, A005314 (diagonal sums), A052542 (row sums), A077962.

Cf. A029907, A193722, A193737, A324969.

function T(n,k) // T = A193736
  if k lt 0 or n lt 0 then return 0;
  elif n lt 3 then return Binomial(n,k);
  else return T(n-1, k) + T(n-1, k-1) + T(n-2, k-2);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 24 2023

(* First program *)
p[0, x_] := 1
p[n_, x_] := Fibonacci[n + 1, x] /; n > 0
q[n_, x_] := (x + 1)^n
t[n_, k_] := Coefficient[p[n, x], x^(n - k)];
t[n_, n_] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n - k + 1, 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}]] (* A193736 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]          (* A193737 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[n<3, Binomial[n, k], T[n-1,k] +T[n-1,k-1] +T[n -2,k-2]];
Table[T[n, k], {n,0,12}, {k,0,n}]//TableForm (* G. C. Greubel, Oct 24 2023 *)

def T(n,k): # T = A193736
    if (n<3): return binomial(n,k)
    else: return T(n-1,k) +T(n-1,k-1) +T(n-2,k-2)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

T(0,0) = T(1,0) = T(1,1) = T(2,0) = T(2,2) = 1; T(n,k) = 0 if k<0 or k>n; T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2). - Philippe Deléham, Feb 13 2020
From G. C. Greubel, Oct 24 2023: (Start)
T(n, k) = A193737(n, n-k).
T(n, n) = Fibonacci(n) + [n=0] = A324969(n+1).
T(n, n-1) = A029907(n).
Sum_{k=0..n} T(n, k) = A052542(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A005314(n) + [n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = [n=0] + A077962(n-1). (End)

A201509 Irregular triangle read by rows: T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = 0, T(n,0) = T(1,1) = 1 and T(n,k) = 0 if k < 0 or if n < k.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 1, 8, 12, 4, 16, 28, 13, 1, 32, 64, 38, 6, 64, 144, 104, 25, 1, 128, 320, 272, 88, 8, 256, 704, 688, 280, 41, 1, 512, 1536, 1696, 832, 170, 10, 1024, 3328, 4096, 2352, 620, 61, 1, 2048, 7168

Offset: 0

Paul Curtz, Dec 02 2011

This is the pseudo-triangle whose successive lines are of the type T(n,0), T(n,1)+T(n-1,0), T(n,2)+T(n-1,1), ... T(n,k)+T(n-1,k-1), without 0's, with T=A201701. [e-mail, Philippe Deléham, Dec 04 2011]

			Triangle starts:
    1   1
    2   2
    4   5   1
    8  12   4
   16  28  13  1
   32  64  38  6
   64 144 104 25 1
  128 320 272 88 8
  ...
Triangle begins (full version):
    0
    1,   1
    2,   2,   0
    4,   5,   1,  0
    8,  12,   4,  0, 0
   16,  28,  13,  1, 0, 0
   32,  64,  38,  6, 0, 0, 0
   64, 144, 104, 25, 1, 0, 0, 0
  128, 320, 272, 88, 8, 0, 0, 0, 0

Cf. A052542 (row sums).
Cf. A039991, A034867.
Cf. A201780, A263633.

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = 0, T(n,0) = T(1,1) = 1 and T(n,k) = 0 if k < 0 or if n < k. - Philippe Deléham, Dec 05 2011
The n-th row polynomial appears to equal Sum_{k = 1..floor((n+1)/2)} binomial(n,2*k-1)*(1+t)^k. Cf. A034867. - Peter Bala, Sep 10 2012
Aside from the first two rows below, the signed coefficients appear in the expansion (b*x - 1)^2 / (a*b*x^2 - 2a*x + 1) = 1 + (2 a - 2 b)x + (4 a^2 - 5 a b + b^2)x^2 + (8 a^3 - 12 a^2b + 4 ab^2)x^3 + ..., the reciprocal of the derivative of x*(1-a*x) / (1-b*x). This is related to A263633 via the expansion (a*b*x^2 - 2a*x + 1) / (b*x - 1)^2  = 1 + (b - a) (2x + 3b x^2 + 4b^2 x^3 + ...). See also A201780. - Tom Copeland, Oct 30 2023

Edited and new name using Philippe Deléham's formula, Joerg Arndt, Dec 13 2023

A227972 Two column recursive array A(n,k), relating expressions based on half-squares (A007590) to each other and several other sequences, read by rows.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 4, 5, 7, 7, 10, 17, 24, 29, 41, 41, 58, 99, 140, 169, 239, 239, 338, 577, 816, 985, 1393, 1393, 1970, 3363, 4756, 5741, 8119, 8119, 11482, 19601, 27720, 33461, 47321, 47321, 66922, 114243, 161564, 195025, 275807, 275807, 390050, 665857, 941664, 1136689, 1607521

Offset: 1

Richard R. Forberg, Aug 01 2013

nonn

The first column (k=1) holds the interleaved integer square roots of these two "Half-Square" expressions in ascending order:  floor(m^2/2 + 1) for m=>0 and floor(m^2/2 - 1) for m=>1.  The second column (k=2) holds the value of m that yields the corresponding integer square root.
The value of m for row n (at n mod 3 = 2) is the value of the square root for the next row (at n mod 3 = 0) which uses the other expression.
There are twice as many results for the expression floor(m^2/2 + 1) as for floor(m^2/2 - 1), interleaved consistently as two of every three results (as shown in the example below).
The first column, for n mod 3 = 1, produces A001541.
The first column, for n mod 3 = 2, produces A001653.
  NOTE: Interleaving of the two sequences above is A079496.
The first column, for n mod 3 = 0, produces A002315 (NSW Numbers).
  NOTE: Interleaving of A001541 and A002315 is A001333.
The second column, for n mod 3 = 1, produces A005319.
The second column, for n mod 3 = 2, produces A002315 (again).
  NOTE: Interleaving of the two sequences above is A143608.
The second column, for n mod 3 = 0, produces A075870.
  NOTE: Interleaving of A005319 and A075870 is A052542 = 2*A000129 (Pell)
The row sums at n mod 3 = 1 and n mod 3 = 0 are used in the recursion to produce values in subsequent rows of the array for both columns.
For rows at n mod 3 = 2,  the ascending interleaved combination of A(n,1) and the row sum (of the same row) produces A000129 (Pell Numbers).
Row sums also hold all the integer square roots (as given in A001542) of the Half-Squares, (A007590), at n mod 3 = 2, and the corresponding values of m in the next row at n mod 3 = 0, corresponding to A001541.
The value of the floor of half the row sum, for n mod 3 =0 and n mod 3 = 1, produces A048739, giving the partial sums of A000129 (Pell Numbers), for the Pell Numbers produced through the prior row at  n mod 3 = 2.
The value of half the row sum, for n mod 3 = 2, produces A001109 (without its initial 0).  This subsequence is also produced from finding the integer square roots of A083374. The value of the indices of that sequence where these roots occur is given by A002315 (NSW Numbers).
The differences of two entries in row n equals the row sum for row n-3, consistently for all rows n > 3.
The ratio of the two entries in the same row converges to sqrt(2).
The ratio of two entries in the same column (either k=1 or k=2) converge as follows:
      A(k,n)/A(k,n-1)--> sqrt(2)       for n mod 3 = 0,
                     --> sqrt(2) + 1   for n mod 3 = 1,
                     --> sqrt(2)/2 + 1 for n mod 3 = 2.
      A(k,n)/A(k,n-3)--> sqrt(8) + 3   for n mod 3 = 0, 1, or 2,
  That last line means: A001541, A001653, A002315, A005319 and A075870 all have the convergence ratio of sqrt(8) + 3 for adjacent terms. In addition alternating Pell Numbers also converge to that ratio.

			The two column array with row number n and the row sum. An extra column on the right shows which expression is applicable to get that row's values: either floor(m^2/2 + 1) indicated as "+1",  or floor(m^2/2 - 1) indicated as "-1". (NOTE: The value of n is immaterial, except as a row number).
The array begins:
Row         k=1         k=2                   Applicable "Half-Square"
n          (sqrt)       (m)         Row Sum        Expression
1            1           0               1             +1
2            1           1               2             +1
3            1           2               3             -1
4            3           4               7             +1
5            5           7              12             +1
6            7          10              17             -1
7           17          24              41             +1
8           29          41              70             +1
9           41          58              99             -1
10          99         140             239             +1
11         169         239             408             +1
12         239         338             577             -1
13         577         816            1393             +1
14         985        1393            2378             +1
15        1393        1970            3363             -1
16        3363        4756            8119             +1
17        5741        8119           13860             +1
18        8119       11482           19601             -1
19       19601       27720           47321             +1
20       33461       47321           80782             +1

Cf. A007590, A001541, A001653, A079496, A002315, A005319, A143608, A075870, A000129, A001109, A083374.

Initialize row 1 as A(1,1) = 1 and A(1,2) = 0, then:
  For rows at n mod 3 = 0:  A(n,1) = A(n-1, 2)
                            A(n,2) = A(n, 1) + A(n-2, 1)
  For rows at n mod 3 = 1:  A(n,1) = A(n-1, 1) + A(n-1, 2)
                            A(n,2) = A(n, 1) + A(n-1, 1)
  For rows at n mod 3 = 2:  A(n,1) = A(n-1,1) + A(n-3, 1)
                            A(n,2) = A(n-1,1) + A(n-1, 2)
Empirical g.f.: -x*(2*x^11-x^10-x^9+x^8-4*x^7+3*x^6-2*x^5-x^4-x^3-x^2-1) / ((x^6-2*x^3-1)*(x^6+2*x^3-1)). - Colin Barker, Aug 08 2013

Some additional comments by Richard R. Forberg, Aug 12 2013

A287144 Number of partitions of n such that the absolute difference between any part i and the sum of all other parts not larger than i is not larger than two.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 6, 10, 8, 15, 11, 17, 13, 23, 16, 24, 18, 33, 23, 34, 26, 43, 28, 41, 30, 53, 36, 52, 39, 63, 40, 58, 42, 75, 51, 74, 56, 90, 57, 83, 60, 103, 69, 97, 71, 112, 69, 99, 71, 124, 83, 119, 89, 141, 88, 127, 91, 154, 102, 142, 103, 161, 98

Offset: 0

Alois P. Heinz, May 20 2017

			a(7) = 6: 31111, 3211, 322, 331, 4111, 421.
a(8) = 10: 32111, 3221, 3311, 332, 41111, 4211, 422, 431, 5111, 521.
a(9) = 8: 42111, 4221, 4311, 432, 51111, 5211, 522, 531.

Alois P. Heinz, Table of n, a(n) for n = 0..16384

Cf. A002487, A052542, A182780, A265278, A287146.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n or abs(i-(n-i))>2, 0, b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..100);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n || Abs[i - (n - i)] > 2, 0, b[n - i, i]]]];
a[n_] := b[n, n];
a /@ Range[0, 100] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

a(2^n-1) = A265278(n) for n>0.
a(2^n) = A052542(n).
a(2^n+1) = A182780(n-1) for n>0.

cp's OEIS Frontend

A098790 a(n) = 2*a(n-1) + a(n-2) + 1, a(0) = 1, a(1) = 2.

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A119016 Numerators of "Farey fraction" approximations to sqrt(2).

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Maple

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A193737 Mirror of the triangle A193736.

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Magma

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A094024 Alternating 1 with one less than the powers of 2.

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A082766 Series ratios converge alternately to sqrt(2) and 1+sqrt(1/2).

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Haskell

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A117918 Difference row triangle of the Pell sequence.

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A193736 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (n+1)-st Fibonacci polynomial and q(n,x) = (x+1)^n.

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