A083099 -id:A083099 - cp's OEIS frontend

A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 1, 2, 5, 8, 5, 0, 1, 2, 6, 12, 16, 6, 0, 1, 2, 7, 16, 29, 32, 7, 0, 1, 2, 8, 20, 44, 70, 64, 8, 0, 1, 2, 9, 24, 61, 120, 169, 128, 9, 0, 1, 2, 10, 28, 80, 182, 328, 408, 256, 10, 0, 1, 2, 11, 32, 101, 256, 547, 896, 985, 512, 11

Offset: 0

Ralf Stephan, Oct 13 2004

			Square array, A(n, k), begins as:
  0, 1, 2,  3,  4,   5,    6,    7,     8, ... A001477;
  0, 1, 2,  4,  8,  16,   32,   64,   128, ... A000079;
  0, 1, 2,  5, 12,  29,   70,  169,   408, ... A000129;
  0, 1, 2,  6, 16,  44,  120,  328,   896, ... A002605;
  0, 1, 2,  7, 20,  61,  182,  547,  1640, ... A015518;
  0, 1, 2,  8, 24,  80,  256,  832,  2688, ... A063727;
  0, 1, 2,  9, 28, 101,  342, 1189,  4088, ... A002532;
  0, 1, 2, 10, 32, 124,  440, 1624,  5888, ... A083099;
  0, 1, 2, 11, 36, 149,  550, 2143,  8136, ... A015519;
  0, 1, 2, 12, 40, 176,  672, 2752, 10880, ... A003683;
  0, 1, 2, 13, 44, 205,  806, 3457, 14168, ... A002534;
  0, 1, 2, 14, 48, 236,  952, 4264, 18048, ... A083102;
  0, 1, 2, 15, 52, 269, 1110, 5179, 22568, ... A015520;
  0, 1, 2, 16, 56, 304, 1280, 6208, 27776, ... A091914;
Antidiagonal triangle, T(n, k), begins as:
  0;
  0,  1;
  0,  1,  2;
  0,  1,  2,  3;
  0,  1,  2,  4,  4;
  0,  1,  2,  5,  8,  5;
  0,  1,  2,  6, 12, 16,   6;
  0,  1,  2,  7, 16, 29,  32,   7;
  0,  1,  2,  8, 20, 44,  70,  64,   8;
  0,  1,  2,  9, 24, 61, 120, 169, 128,   9;
  0,  1,  2, 10, 28, 80, 182, 328, 408, 256,  10;

G. C. Greubel, Antidiagonals n = 0..50, flattened
Ralf Stephan, Prove or disprove. 100 Conjectures from the OEIS, #16, arXiv:math/0409509 [math.CO], 2004.

Rows m: A001477 (m=0), A000079 (m=1), A000129 (m=2), A002605 (m=3), A015518 (m=4), A063727 (m=5), A002532 (m=6), A083099 (m=7), A015519 (m=8), A003683 (m=9), A002534 (m=10), A083102 (m=11), A015520 (m=12), A091914 (m=13).
Columns q: A000004 (q=0), A000012 (q=1), A009056 (q=2), A008586 (q=3).
Main diagonal gives A357502.

A099173:= func< n,k | (&+[n^j*Binomial(k,2*j+1): j in [0..Floor(k/2)]]) >;
[A099173(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2023

A[k_, n_]:= Which[k==0, n, n==0, 0, True, ((1+Sqrt[k])^n - (1-Sqrt[k])^n)/(2 Sqrt[k])]; Table[A[k-n, n]//Simplify, {k, 0, 12}, {n, 0, k}]//Flatten (* Jean-François Alcover, Jan 21 2019 *)

A(k,n)=sum(i=0,n\2,k^i*binomial(n,2*i+1))

def A099173(n,k): return sum( n^j*binomial(k, 2*j+1) for j in range((k//2)+1) )
flatten([[A099173(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 17 2023

A(n, k) = Sum_{i=0..floor(k/2)} n^i * C(k, 2*i+1) (array).
Recurrence: A(n, k) = 2*A(n, k-1) + (n-1)*A(n, k-2), with A(n, 0) = 0, A(n, 1) = 1.
T(n, k) = A(n-k, k) (antidiagonal triangle).
T(2*n, n) = A357502(n).
A(n, k) = ((1+sqrt(n))^k - (1-sqrt(n))^k)/(2*sqrt(n)). - Jean-François Alcover, Jan 21 2019

A274520 a(n) = ((1 + sqrt(7))^n - (1 - sqrt(7))^n)/sqrt(7).

Original entry on oeis.org

0, 2, 4, 20, 64, 248, 880, 3248, 11776, 43040, 156736, 571712, 2083840, 7597952, 27698944, 100985600, 368164864, 1342243328, 4893475840, 17840411648, 65041678336, 237125826560, 864501723136, 3151758405632, 11490527150080, 41891604733952, 152726372368384

Offset: 0

Ilya Gutkovskiy, Jun 26 2016

Number of zeros in substitution system {0 -> 111, 1 -> 1001} at step n from initial string "1" (see example).

			Evolution from initial string "1": 1 -> 1001 -> 10011111111001 -> 1001111111100110011001100110011001100110011111111001 -> ...
Therefore, number of zeros at step n:
a(0) = 0;
a(1) = 2;
a(2) = 4;
a(3) = 20, etc.

Ilya Gutkovskiy, Illustration (substitution system {0 -> 111, 1 -> 1001})
Eric Weisstein's World of Mathematics, Substitution System
Index entries for linear recurrences with constant coefficients, signature (2,6)

Cf. A010465, A083099.

Mathematica
```
LinearRecurrence[{2, 6}, {0, 2}, 27]
```

a(n)=([0,1; 6,2]^n*[0;2])[1,1] \\ Charles R Greathouse IV, Jul 26 2016

O.g.f.: 2*x/(1 - 2*x - 6*x^2).
E.g.f.: 2*exp(x)*sinh(sqrt(7)*x)/sqrt(7).
Dirichlet g.f.: (PolyLog(s,1+sqrt(7)) - PolyLog(s,1-sqrt(7)))/sqrt(7), where PolyLog(s,x) is the polylogarithm function.
a(n) = 2*a(n-1) + 6*a(n-2).
a(n) = 2*A083099(n).
Lim_{n->infinity} a(n+1)/a(n) = 1 + sqrt(7) = 1 + A010465.

A291008 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 7*S^2.

Original entry on oeis.org

0, 7, 14, 70, 224, 868, 3080, 11368, 41216, 150640, 548576, 2000992, 7293440, 26592832, 96946304, 353449600, 1288577024, 4697851648, 17127165440, 62441440768, 227645874176, 829940392960, 3025756030976, 11031154419712, 40216845025280, 146620616568832

Offset: 0

Clark Kimberling, Aug 22 2017

Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A291000 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,6).

Cf. A000012, A083099, A289780, A291000.

[n le 2 select 7*(n-1) else 2*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 01 2023

z = 60; s = x/(1 - x); p = 1 - s^7;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291008 *)
LinearRecurrence[{2,6}, {0,7}, 40] (* G. C. Greubel, Jun 01 2023 *)

A291008=BinaryRecurrenceSequence(2,6,0,7)
[A291008(n) for n in range(41)] # G. C. Greubel, Jun 01 2023

G.f.: 7*x/(1 - 2*x - 6*x^2).
a(n) = 2*a(n-1) + 6*a(n-2) for n >= 3.
a(n) = 7*A083099(n).
a(n) = (sqrt(7)*((1+sqrt(7))^n - (1-sqrt(7))^n)) / 2. - Colin Barker, Aug 23 2017
a(n) = 7*i^(n-1)*6^((n-1)/2)*ChebyshevU(n-1, -i/sqrt(6)). - G. C. Greubel, Jun 01 2023

A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.

Original entry on oeis.org

1, 2, 1, 3, 4, 4, 1, 10, 5, 6, 20, 6, 1, 21, 35, 7, 8, 56, 56, 8, 1, 36, 126, 84, 9, 10, 120, 252, 120, 10, 1, 55, 330, 462, 165, 11, 12, 220, 792, 792, 220, 12, 1, 78, 715, 1716, 1287, 286, 13, 14, 364, 2002, 3432, 2002, 364, 14, 1, 105, 1365, 5005, 6435, 3003, 455, 15

Offset: 1

Philippe Deléham, Aug 27 2005

The same as A119900 without 0's. A reflected version of A034867 or A202064. - Alois P. Heinz, Feb 07 2014
From Vladimir Shevelev, Feb 07 2014: (Start)
Also table of coefficients of polynomials  P_1(x)=1, P_2(x)=2, for n>=2, P_(n+1)(x) = 2*P_n(x)+(x-1)* P_(n-1)(x).  The polynomials P_n(x)/2^(n-1) are connected with sequences A000045 (x=5), A001045 (x=9), A006130 (x=13), A006131 (x=17), A015440 (x=21), A015441 (x=25), A015442 (x=29), A015443 (x=33), A015445 (x=37), A015446 (x=41), A015447 (x=45), A053404 (x=49); also the polynomials P_n(x) are connected with sequences A000129, A002605, A015518, A063727, A085449, A002532, A083099, A015519, A003683, A002534, A083102, A015520. (End)

			Starred terms in Pascal's triangle (A007318), read by rows:
1;
1*, 1;
1, 2*, 1;
1*, 3, 3*, 1;
1, 4*, 6, 4*, 1;
1*, 5, 10*, 10, 5*, 1;
1, 6*, 15, 20*, 15, 6*, 1;
1*, 7, 21*, 35, 35*, 21, 7*, 1;
1, 8*, 28, 56*, 70, 56*, 28, 8*, 1;
1*, 9, 36*, 84, 126*, 126, 84*, 36, 9*, 1;
Triangle T(n,k) begins:
1;
2;
1,    3;
4,    4;
1,   10,  5;
6,   20,  6;
1,   21,  35,   7;
8,   56,  56,   8;
1,   36, 126,  84,  9;
10, 120, 252, 120, 10;

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A109446.

T:= (n, k)-> binomial(n, 2*k+1-irem(n, 2)):
seq(seq(T(n, k), k=0..ceil((n-2)/2)), n=1..20);  # Alois P. Heinz, Feb 07 2014

Flatten[ Table[ If[ OddQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Aug 30 2005

Corrected offset by Alois P. Heinz, Feb 07 2014

A127226 a(n) = 2a(n-1) + 6a(n-2) with a(0)=2, a(1)=2.

Original entry on oeis.org

2, 2, 16, 44, 184, 632, 2368, 8528, 31264, 113696, 414976, 1512128, 5514112, 20100992, 73286656, 267179264, 974078464, 3551232512, 12946935808, 47201266688, 172084148224, 627375896576, 2287256682496, 8338768744448, 30401077583872, 110834767634432, 404076000772096

Offset: 0

Miklos Kristof, Mar 26 2007

Andrew Howroyd, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (2,6).

Cf. A083098, A083099.

a[0]=2;a[1]=2;a[n_]:=2a[n-1]+6a[n-2];Table[a[n],{n,0,22}] (* James C. McMahon, Dec 30 2024 *)

Vec(2*(1 - x)/(1 - 2*x - 6*x^2) + O(x^31)) \\ Andrew Howroyd, Dec 30 2024

[lucas_number2(n,2,-6) for n in range(0, 23)] # Zerinvary Lajos, Apr 30 2009

G.f.: 2*(1 - x)/(1 - 2*x - 6*x^2).
E.g.f.: (exp((1+sqrt(7))*x) + exp((1-sqrt(7))*x));
a(n) = A083099(n) + 6*A083099(n-2).
G.f.: G(0), where G(k)= 1 + 1/(1 - x*(7*k-1)/(x*(7*k+6) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
a(n) = 2*A083098(n). - Andrew Howroyd, Dec 30 2024

a(23) onwards from Andrew Howroyd, Dec 30 2024

A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 6, 10, 16, 28, 44, 76, 120, 208, 328, 568, 896, 1552, 2448, 4240, 6688, 11584, 18272, 31648, 49920, 86464, 136384, 236224, 372608, 645376, 1017984, 1763200, 2781184, 4817152, 7598336, 13160704, 20759040, 35955712, 56714752

Offset: 1

Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 14 2009

This sequence is the case k=3 of a family of sequences with recurrences a(2*n+1) = a(2*n) + a(2*n-1), a(2*n+2) = k*a(2*n-1) + a(2*n), a(1)=0, a(2)=1. Values of k, for k >= 0, are given by A057979 (k=0), A158780 (k=1), A002965 (k=2), this sequence (k=3). See "Family of sequences for k" link for other connected sequences.
It seems that the ratio of two successive numbers with even, or two successive numbers with odd, indices approaches sqrt(k) for these sequences as n-> infinity.
This algorithm can be found in a historical figure named "Villardsche Figur" of the 13th century. There you can see a geometrical interpretation.

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Beinert, Villardscher Teilungskanon, Lexikon der Typographie
W. Limbrunner, Das Quadrat, ein Wunder der Geometrie. (in German)
Willibald Limbrunner, Family of sequences for k
M-T. Zenner, Villard de Honnecourt and Euclidean Geoometry, Nexus Network Journal 4 (2002) 65-78.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,2).

Cf. A002532, A002533, A002534, A002535, A002605, A002965, A003665.
Cf. A003683, A015518, A015519, A026150, A046717, A057979, A063727.
Cf. A083098, A083099, A083100, A084057, A158780.

I:=[0,1,1,1]; [n le 4 select I[n] else 2*(Self(n-2) +Self(n-4)): n in [1..40]]; // G. C. Greubel, Feb 18 2023

LinearRecurrence[{0,2,0,2}, {0,1,1,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

@CachedFunction
def a(n): # a = A160444
    if (n<5): return ((n+1)//3)
    else: return 2*(a(n-2) + a(n-4))
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 18 2023

a(n) = 2*a(n-2) + 2*a(n-4).
a(2*n+1) = A002605(n).
a(2*n) = A026150(n-1).

Edited by R. J. Mathar, May 14 2009

A120711 Expansion of 2x(7+16x-2x^2-14x^3)/(1-11x^2-12x^3+10x^4+12*x^5).

Original entry on oeis.org

0, 14, 32, 150, 492, 1894, 6724, 24854, 89972, 329238, 1197972, 4372054, 15930580, 58096214, 211770452, 772129110, 2814859092, 10262536534, 37414140244, 136403674454, 497291840852, 1813006427478, 6609762501972, 24097566365014

Offset: 0

Roger L. Bagula, Aug 12 2006

Former title: 7 X 7 matrix Matrov of seven vertex Fano Plane: Characteristic polynomial: 12 + 10*x - 24*x^2 - 21*x^3 + 12*x^4 + 12*x^5 - x^7.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Fano Plane
Index entries for linear recurrences with constant coefficients, signature (0,11,12,-10,-12).

Cf. A083099, A111384.

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( 2*x*(7+16*x-2*x^2-14*x^3)/(1-11*x^2-12*x^3+10*x^4+12*x^5) )); // G. C. Greubel, Jul 22 2023

M = {{0,1,0,0,0,1,1}, {1,0,1,0,0,0,1}, {0,1,0,1,0,0,1}, {0,0,1,0,1,0, 1}, {0,0,0,1,0,1,1}, {1,0,0,0,1,0,1}, {1,1,1,1,1,1,0}};
v[1] = {0,1,1,2,3,5,8}; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n,50}]
LinearRecurrence[{0,11,12,-10,-12}, {0,14,32,150,492}, 40] (* G. C. Greubel, Jul 22 2023 *)

A083099=BinaryRecurrenceSequence(2,6,0,1)
def A120711(n): return (1/3)*(-1 -3*(-1)^n +(-2)^(n+1) +6*(A083099(n+1) +4*A083099(n)))
[A120711(n) for n in range(41)] # G. C. Greubel, Jul 22 2023

a(n) = 11*a(n-2) + 12*a(n-3) - 10*a(n-4) - 12*a(n-5).
G.f.: 2*x*(7+16*x-2*x^2-14*x^3)/((1-x)*(1+x)*(1+2*x)*(1-2*x-6*x^2)). - Colin Barker, Mar 26 2012
a(n) = (1/3)*(-1 - 3*(-1)^n + (-2)^(n+1) + 6*(A083099(n+1) + 4*A083099(n))). - G. C. Greubel, Jul 22 2023

Edited by G. C. Greubel, Jul 22 2023

A247584 a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + 3*a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 13, 43, 113, 253, 509, 969, 1849, 3719, 8009, 18027, 40897, 91257, 198697, 423777, 894081, 1886011, 4007301, 8594411, 18560081, 40181493, 86872293, 187197193, 402060793, 861827743, 1846685729, 3960390059, 8504658049, 18283290609, 39325827729

Offset: 0

Alexander Samokrutov, Sep 20 2014

a(n)/a(n-1) tends to 2.1486... = 1 + 2^(1/5), the real root of the polynomial x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 3.
If x^5 = 2 and n >= 0, then there are unique integers a, b, c, d, g  such that (1 + x)^n = a + b*x + c*x^2 + d*x^3 + g*x^4. The coefficient a is a(n) (from A052102). - Alexander Samokrutov, Jul 11 2015
If x=a(n), y=a(n+1), z=a(n+2), s=a(n+3), t=a(n+4) then x, y, z, s, t satisfies Diophantine equation (see link). - Alexander Samokrutov, Jul 11 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..48 from Alexander Samokrutov)
Alexander Samokrutov, Diophantine equation
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,3).

The following sequences belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A052101, A052102, A052103, A063727, A083098, A083099, A083100, A084057, A093406, A247344.

Cf. A005531.

[n le 5 select 1 else 5*Self(n-1) -10*Self(n-2) +10*Self(n-3) -5*Self(n-4) +3*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Jul 11 2015

m:=50; S:=series( (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 15 2021

LinearRecurrence[{5,-10,10,-5,3}, {1,1,1,1,1}, 50] (* Vincenzo Librandi, Jul 11 2015 *)

makelist(sum(2^k*binomial(n,5*k), k, 0, floor(n/5)), n, 0, 50); /* Alexander Samokrutov, Jul 11 2015 */

Vec((1-x)^4/(1-5*x+10*x^2-10*x^3+5*x^4-3*x^5) + O(x^100)) \\ Colin Barker, Sep 22 2014

[sum(2^j*binomial(n, 5*j) for j in (0..n//5)) for n in (0..50)] # G. C. Greubel, Apr 15 2021

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 3*a(n-5).
a(n) = Sum_{k=0...floor(n/5)} (2^k*binomial(n,5*k)). - Alexander Samokrutov, Jul 11 2015
G.f.: (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5). - Colin Barker, Sep 22 2014

A307469 a(n) = 2a(n-1) + 6a(n-2) for n >= 2, a(0) = 1, a(1) = 5.

Original entry on oeis.org

1, 5, 16, 62, 220, 812, 2944, 10760, 39184, 142928, 520960, 1899488, 6924736, 25246400, 92041216, 335560832, 1223368960, 4460102912, 16260419584, 59281456640, 216125430784, 787939601408, 2872631787520, 10472901183488, 38181593092096, 139200593285120

Offset: 0

Armend Shabani, Apr 09 2019

a(n) is the number of words of length n over alphabet {1,2,3,4,5} such that no odd letter is followed by an odd letter.

			For n=2 the a(2)=16 solutions are: 12, 14, 21, 22, 23, 24, 25, 32, 34, 41, 42, 43, 44, 45, 52, 54.

Index entries for linear recurrences with constant coefficients, signature (2,6).

The same over alphabet {1,2,3} gives A001045(n+2).

aseq:=proc(n) option remember;
if n<0 then return "seq not defined for negative indices";
elif n=0 then return 1;
elif n=1 then return 5;
else 2*aseq(n-1)+6*aseq(n-2);
end if;
end proc:
seq(aseq(n),n=0..26);

a[0] = 1; a[1] = 5;
a[n_] := a[n] = 2*a[n - 1] + 6*a[n - 2];
Table[a[n], {n, 0, 26}]
LinearRecurrence[{2,6},{1,5},30] (* Harvey P. Dale, Feb 20 2023 *)

a(n) = (-(2/7)*sqrt(7)+1/2)*(1-sqrt(7))^n+((2/7)*sqrt(7)+1/2)*(1+sqrt(7))^n.
G.f.: (1+3*x)/(1-2*x-6*x^2).
a(n) = 3*A083099(n) + A083099(n+1). - R. J. Mathar, Jan 27 2020

A136425 a(n) = floor((x^n-(1-x)^n)/sqrt(7)+1/2) where x = (sqrt(7)+1)/2.

Original entry on oeis.org

1, 1, 3, 4, 8, 14, 25, 46, 84, 153, 279, 509, 927, 1691, 3082, 5618, 10241, 18667, 34028, 62029, 113070, 206113, 375719, 684889, 1248467, 2275800, 4148501, 7562201, 13784953, 25128255, 45805684, 83498067, 152206593, 277453693, 505763582

Offset: 1

Cino Hilliard, Apr 01 2008

nonn

This is analogous to the formula for the n-th Fibonacci number. Even before truncation, these numbers are rational and the decimal part always ends in 5. For x = (sqrt(7)+1)/2, a(n)/a(n-1) -> x. The general form of x is (sqrt(r)+1)/2, r=1,2,3..

g(n,r) = for(m=1,n,print1(fib(m,r)",")) fib(n,r) = x=(sqrt(r)+1)/2;floor((x^n-(1-x)^n)/sqrt(r)+.5)

Asymptotically a(n) ~ A083099(n)/2^(n-1). - R. J. Mathar, Apr 20 2008

a(n) = floor(b(n)/2^n) where b(n) = 2*A083099(n)+2^(n-1). - R. J. Mathar, Sep 10 2016

Definition corrected by R. J. Mathar, Apr 20 2008

cp's OEIS Frontend

A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2*x-(k-1)*x^2).

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A274520 a(n) = ((1 + sqrt(7))^n - (1 - sqrt(7))^n)/sqrt(7).

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A291008 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 7*S^2.

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A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.

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A127226 a(n) = 2*a(n-1) + 6*a(n-2) with a(0)=2, a(1)=2.

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A160444 Expansion of g.f.: x^2*(1 + x - x^2)/(1 - 2*x^2 - 2*x^4).

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A120711 Expansion of 2*x*(7+16*x-2*x^2-14*x^3)/(1-11*x^2-12*x^3+10*x^4+12*x^5).

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A099173 Array, A(k,n), read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).

A127226 a(n) = 2a(n-1) + 6a(n-2) with a(0)=2, a(1)=2.

A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

A120711 Expansion of 2x(7+16x-2x^2-14x^3)/(1-11x^2-12x^3+10x^4+12*x^5).

A247584 a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + 3*a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

A307469 a(n) = 2a(n-1) + 6a(n-2) for n >= 2, a(0) = 1, a(1) = 5.