A048739 -id:A048739 - cp's OEIS frontend

A210561 Triangle of coefficients of polynomials u(n,x) jointly generated with A210562; see the Formula section.

1, 1, 2, 1, 3, 4, 1, 3, 8, 8, 1, 3, 9, 20, 16, 1, 3, 9, 26, 48, 32, 1, 3, 9, 27, 72, 112, 64, 1, 3, 9, 27, 80, 192, 256, 128, 1, 3, 9, 27, 81, 232, 496, 576, 256, 1, 3, 9, 27, 81, 242, 656, 1248, 1280, 512, 1, 3, 9, 27, 81, 243, 716, 1808, 3072, 2816, 1024, 1, 3, 9

Offset: 1

Clark Kimberling, Mar 22 2012

Last term in row n:  2^(n-1)
Limiting row:  3^(k-1)
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
1...3...4
1...3...8...8
1...3...9...20...16
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 4x^2.

P. Bala, A note on the diagonals of a proper Riordan Array

Cf. A210562, A208510, A004070, A048739.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, 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[%]  (* A210559 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210560 *)

u(n,x)=x*u(n-1,x)+x*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.
From Peter Bala, Mar 06 2017: (Start)
T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1).
E.g.f. for n-th subdiagonal: exp(2*x)*(1 + x + x^2/2! + x^3/3! + ... + x^n/n!). Cf. A004070.
Riordan array (1/(1 - x), x*(2 + x)).
Row sums A048739.
(End)

A213688 a(n) = Sum_{i=0..n} A000129(i)^3.

Original entry on oeis.org

0, 1, 9, 134, 1862, 26251, 369251, 5196060, 73113372, 1028784997, 14476099149, 203694183170, 2866194639170, 40330419190351, 567492063162119, 7985219303802744, 112360562315573112, 1581033091723823881, 22246823846444284881, 313036566941955454910

Offset: 0

N. J. A. Sloane, Jun 18 2012

Bruno Berselli, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (13,18,-42,11,1).

Cf. A000129, A048739 (partial sums of A000129), A084158 (sum of squares of A000129).

Cf. A110272 (cubes of the Pell numbers).

A110272:=func; [&+[A110272(i): i in [0..n]]: n in [0..19]]; // Bruno Berselli, Jun 21 2012

LinearRecurrence[{13, 18, -42, 11, 1}, {0, 1, 9, 134, 1862}, 20] (* Bruno Berselli, Jun 21 2012 *)

G.f.: x*(1-4*x-x^2)/((1-x)*(1+2*x-x^2)*(1-14*x-x^2)). [Bruno Berselli, Jun 18 2012]

a(n) = ((3+sqrt(2))*(1+sqrt(2))^(3n+1)+(3-sqrt(2))*(1-sqrt(2))^(3n+1)-21*(-1)^n*((1+sqrt(2))^n+(1-sqrt(2))^n)+32)/224. [Bruno Berselli, Jun 18 2012]

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

A238375 Row sums of triangle in A152719.

Original entry on oeis.org

1, 2, 4, 6, 11, 16, 28, 40, 69, 98, 168, 238, 407, 576, 984, 1392, 2377, 3362, 5740, 8118, 13859, 19600, 33460, 47320, 80781, 114242, 195024, 275806, 470831, 665856, 1136688, 1607520, 2744209, 3880898, 6625108, 9369318, 15994427, 22619536, 38613964, 54608392

Offset: 0

Philippe Deléham, Feb 25 2014

			Triangle A152719 and row sums:
  1;  ............................. sum =  1
  1, 1;  .......................... sum =  2
  1, 2, 1;  ....................... sum =  4
  1, 2, 2,  1;  ................... sum =  6
  1, 2, 5,  2,  1;  ............... sum = 11
  1, 2, 5,  5,  2, 1;  ............ sum = 16
  1, 2, 5, 12,  5, 2, 1;  ......... sum = 28
  1, 2, 5, 12, 12, 5, 2, 1;  ...... sum = 40

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

Cf. A000129, A002203, A005409, A048739, A135153 (first differences), A152719.

Table[Sum[Fibonacci[1+Min[k, n-k], 2], {k,0,n}], {n,0,45}] (* G. C. Greubel, May 21 2021 *)

my(x='x+O('x^44)); Vec((1+x)/((1-2*x^2-x^4)*(1-x))) \\ Joerg Arndt, May 22 2021

def Pell(n): return n if (n<2) else 2*Pell(n-1) + Pell(n-2)
def a(n): return sum(Pell(1+min(k, n-k)) for k  in (0..n))
[a(n) for n in (0..45)] # G. C. Greubel, May 21 2021

a(n) = Sum_{k=0..n} A152719(n,k).
G.f.: (1+x)/((1-2*x^2-x^4)*(1-x)).
a(2*n) = A005409(n+2).
a(2*n+1) = 2*A048739(n).
a(n) = (-4 + 2*(1+(-1)^n)*Pell((n+4)/2) + (1-(-1)^n)*Q((n+3)/2))/4, where Pell(n) = A000129(n) and Q(n) = A002203(n). - G. C. Greubel, May 21 2021
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)+a(n-4)-a(n-5). - Wesley Ivan Hurt, May 22 2021

A247311 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 2, 1, 4, 5, 3, 9, 12, 8, 21, 29, 20, 50, 70, 49, 120, 169, 119, 289, 408, 288, 697, 985, 696, 1682, 2378, 1681, 4060, 5741, 4059, 9801, 13860, 9800, 23661, 33461, 23660, 57122, 80782, 57121, 137904, 195025, 137903, 332929, 470832, 332928

Offset: 0

Clark Kimberling, Sep 12 2014

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, and for 0 < i <= n, s(i) is in {0,1,2}, and s(i) - s(i-1) is in {-1,0,1}.
(row 0, the bottom row): A024537;
(row 1, the middle row): A000129;
(row 2, the top row): A048739;
(n-th column sum): A000129.

			First 10 columns:
  0 .. 0 .. 1 .. 3 .. 8 ... 20 .. 49 .. 119 .. 288 .. 696
  0 .. 1 .. 2 .. 5 .. 12 .. 29 .. 70 .. 169 .. 408 .. 985
  1 .. 1 .. 2 .. 4 .. 9 ... 21 .. 50 .. 120 .. 289 .. 697
T(3,2) counts these 3 paths, given as vector sums applied to (0,0):
  (1,1) + (1,1) + (1,0); (1,1) + (1,0) + (1,1); (1,0) + (1,1) + (1,1).

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A247049, A247309, A247310, A000129, A024537, A048739.

t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[1, 2] = 0;
t[n_, 0] := t[n, 0] = t[n - 1, 0] + t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 1, 2]
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)
Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (* A247311 *)

A048771 Partial sums of A048695.

Original entry on oeis.org

1, 9, 26, 68, 169, 413, 1002, 2424, 5857, 14145, 34154, 82460, 199081, 480629, 1160346, 2801328, 6763009, 16327353, 39417722, 95162804, 229743337, 554649485, 1339042314, 3232734120, 7804510561, 18841755249, 45488021066, 109817797388, 265123615849

Offset: 0

Barry E. Williams

			a(n)=[ {(8+(9/2)*sqrt(2))(1+sqrt(2))^n -(8-(9/2)*sqrt(2))(1-sqrt(2))^n}/ 2*sqrt(2) ]-7/2.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1)

Cf. A001333, A000129, A048694, A048695.

Table[6*Fibonacci[n, 2] + Fibonacci[n+1, 2], {n, 0, 22}] // Accumulate (* Jean-François Alcover, Mar 25 2013 *)
Accumulate[LinearRecurrence[{2,1},{1,8},40]] (* or *) LinearRecurrence[ {3,-1,-1},{1,9,26},40] (* Harvey P. Dale, May 01 2013 *)

a(n)=2*a(n-1)+a(n-1)+7; a(0)=1, a(1)=9.
G.f. ( 1+6*x ) / ( (x-1)*(x^2+2*x-1) ). a(n)=A048739(n)+6*A048739(n-1). - R. J. Mathar, Nov 08 2012
a(0)=1, a(1)=9, a(2)=26, a(n)=3*a(n-1)-a(n-2)-a(n-3). - Harvey P. Dale, May 01 2013

More terms from Harvey P. Dale, May 01 2013

A088015 Expansion of e.g.f. cosh(sqrt(2)x) + exp(x)(cosh(sqrt(2)*x) - 1).

Original entry on oeis.org

1, 0, 4, 6, 20, 40, 106, 238, 592, 1392, 3394, 8118, 19664, 47320, 114370, 275806, 666112, 1607520, 3881410, 9369318, 22620560, 54608392, 131838370, 318281038, 768402496, 1855077840, 4478562274, 10812186006, 26102942480, 63018038200

Offset: 0

Paul Barry, Sep 18 2003

This sequence is A000079 (with interpolated zeros) + 2*(A048739 (with two leading zeros)).

G. C. Greubel, Table of n, a(n) for n = 0..2600
Index entries for linear recurrences with constant coefficients, signature (3,1,-7,2,2).

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 -3*x +3*x^2 +x^3 -4*x^4)/((1-x)*(1-2*x-3*x^2+4*x^3+2*x^4)))); // G. C. Greubel, Sep 27 2018

LinearRecurrence[{3,1,-7,2,2},{1,0,4,6,20},30] (* Harvey P. Dale, May 05 2018 *)

x='x+O('x^30); Vec((1 -3*x +3*x^2 +x^3 -4*x^4)/((1-x)*(1-2*x-3*x^2+4*x^3+2*x^4))) \\ G. C. Greubel, Sep 27 2018

a(n) = A088014(n)-1.
G.f.: (1 -3*x +3*x^2 +x^3 -4*x^4)/((1-x)*(1-2*x-3*x^2+4*x^3+2*x^4)).
E.g.f. : cosh(sqrt(2)x)+exp(x)(cosh(sqrt(2)x)-1);
a(n) = ((sqrt(2))^n +(-sqrt(2))^n +(1+sqrt(2))^n +(1-sqrt(2))^n)/2 -1.
G.f.: ( -1-3*x^2-x^3+4*x^4+3*x ) / ( (x-1)*(2*x^2-1)*(x^2+2*x-1) ). - R. J. Mathar, Dec 10 2014

A099846 An Alexander sequence for the knot 8_5.

Original entry on oeis.org

1, 3, 5, 8, 15, 29, 55, 104, 196, 368, 692, 1304, 2457, 4627, 8713, 16408, 30899, 58189, 109583, 206368, 388632, 731872, 1378264, 2595552, 4887953, 9205011, 17334909, 32645160, 61477479, 115774605, 218027143, 410589480, 773223548, 1456137296

Offset: 0

Paul Barry, Oct 27 2004

The g.f. is a transformation of the g.f. 1/((1-x)(1-2x-x^2)) of A048739 under the mapping G(x)->(1/(1+x^2)^3)G(x/(1+x^2)). The denominator of the g.f. is a parameterization of the Alexander polynomial of the knot 8_5. Relates 8_5 to the Pell numbers.

Index entries for linear recurrences with constant coefficients, signature (3,-4,5,-4,3,-1).

Cf. A099854.

CoefficientList[Series[1/(1-3x+4x^2-5x^3+4x^4-3x^5+x^6),{x,0,40}],x] (* or *) LinearRecurrence[{3,-4,5,-4,3,-1},{1,3,5,8,15,29},41] (* Harvey P. Dale, Sep 25 2011 *)

G.f.: 1/(1-3x+4x^2-5x^3+4x^4-3x^5+x^6).

a(0)=1, a(1)=3, a(2)=5, a(3)=8, a(4)=15, a(5)=29, a(n)=3*a(n-1)- 4*a(n-2)+ 5*a(n-3)-4*a(n-4)+3*a(n-5)-a(n-6). - Harvey P. Dale, Sep 25 2011

A181658 Row sums of A181657.

Original entry on oeis.org

1, 2, 3, 7, 8, 19, 20, 48, 49, 118, 119, 287, 288, 695, 696, 1680, 1681, 4058, 4059, 9799, 9800, 23659, 23660, 57120, 57121, 137902, 137903, 332927, 332928, 803759, 803760, 1940448, 1940449, 4684658, 4684659, 11309767, 11309768, 27304195, 27304196

Offset: 0

Paul Barry, Nov 03 2010

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1,0,-1)

Cf. A048739 (bisection).

CoefficientList[Series[(1+2x+x^3)/(1-3x^2+x^4+x^6),{x,0,40}],x] (* or *) LinearRecurrence[{0,3,0,-1,0,-1},{1,2,3,7,8,19},40] (* Harvey P. Dale, Mar 29 2022 *)

G.f.: (1+2x+x^3)/(1-3x^2+x^4+x^6)=(1+2x+x^3)/((1-x^2)(1-2x^2-x^4)).

A270345 Composite integers n such that the sum of the Pell numbers A000129(0) + ... + A000129(n-1) is divisible by n.

Original entry on oeis.org

4, 8, 16, 24, 32, 48, 64, 72, 96, 120, 128, 144, 168, 169, 192, 216, 240, 256, 264, 272, 288, 336, 360, 384, 385, 432, 480, 504, 512, 528, 544, 576, 600, 648, 672, 720, 768, 792, 816, 840, 864, 960, 961, 1008, 1024, 1056, 1080, 1088, 1105, 1121, 1152, 1176, 1200, 1296, 1320, 1344

Offset: 1

Altug Alkan, Mar 15 2016

nonn

Nonprime terms of A270342.

Terms that are not divisible by 4 are 169, 385, 961, 1105, 1121, 3827, 4901, 6265, 6441, 6601, 7107, 7801, 8119, ...

			4 is a term because 0 + 1 + 2 + 5 = 8 is divisible by 4.
8 is a term because 0 + 1 + 2 + 5 + 12 + 29 + 70 + 169 = 288 is divisible by 8.

Cf. A000129, A048739, A270342.

a048739(n) = local(w=quadgen(8)); -1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n;
for(n=1, 1e3, if(a048739(n-1) % (n+1) == 0 && !isprime(n+1), print1(n+1, ", ")));

cp's OEIS Frontend

A210561 Triangle of coefficients of polynomials u(n,x) jointly generated with A210562; see the Formula section.

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A213688 a(n) = Sum_{i=0..n} A000129(i)^3.

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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.

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A238375 Row sums of triangle in A152719.

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A247311 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1).

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A048771 Partial sums of A048695.

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A088015 Expansion of e.g.f. cosh(sqrt(2)*x) + exp(x)*(cosh(sqrt(2)*x) - 1).

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A099846 An Alexander sequence for the knot 8_5.

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A088015 Expansion of e.g.f. cosh(sqrt(2)x) + exp(x)(cosh(sqrt(2)*x) - 1).