A015551
Expansion of x/(1 - 6*x - 5*x^2).
Original entry on oeis.org
0, 1, 6, 41, 276, 1861, 12546, 84581, 570216, 3844201, 25916286, 174718721, 1177893756, 7940956141, 53535205626, 360916014461, 2433172114896, 16403612761681, 110587537144566, 745543286675801, 5026197405777636
Offset: 0
Cf.
A001076,
A006190,
A007482,
A015520,
A015521,
A015523,
A015524,
A015525,
A015528,
A015529,
A015530,
A015531,
A015532,
A015533,
A015534,
A015535,
A015536,
A015537,
A015440,
A015441,
A015443,
A015444,
A015445,
A015447,
A015548,
A030195,
A053404,
A057087,
A057088,
A057089,
A083858,
A085939,
A090017,
A091914,
A099012,
A135030,
A135032,
A180222,
A180226,
A180250.
-
I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011
-
Join[{a=0,b=1},Table[c=6*b+5*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
CoefficientList[Series[x/(1-6x-5x^2),{x,0,20}],x] (* or *) LinearRecurrence[ {6,5},{0,1},30] (* Harvey P. Dale, Oct 30 2017 *)
-
a(n)=([0,1; 5,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016
-
[lucas_number1(n,6,-5) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009
A189800
a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.
Original entry on oeis.org
0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176
Offset: 0
Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
-
I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011
-
LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)
-
a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016
A387480
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(k,n-k)^2.
Original entry on oeis.org
1, 3, 15, 99, 603, 3807, 24759, 162243, 1072683, 7147359, 47887767, 322330995, 2178055899, 14765637663, 100380161655, 684061007139, 4671543976587, 31962145170015, 219043736154711, 1503380943222867, 10332034575214779, 71092843087100319, 489712662842798007
Offset: 0
-
[&+[3^k * 2^(n-k) * Binomial(k, n-k)^2: k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 01 2025
-
Table[Sum[3^k*2^(n-k)*Binomial[k,n-k]^2,{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Sep 01 2025 *)
-
a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(k, n-k)^2);
A015541
Expansion of x/(1 - 5*x - 7*x^2).
Original entry on oeis.org
0, 1, 5, 32, 195, 1199, 7360, 45193, 277485, 1703776, 10461275, 64232807, 394392960, 2421594449, 14868722965, 91294775968, 560554940595, 3441838134751, 21133075257920, 129758243232857, 796722742969725, 4891921417478624, 30036666288181195
Offset: 0
Cf.
A001076,
A006190,
A007482,
A015520,
A015521,
A015523,
A015524,
A015525,
A015528,
A015529,
A015530,
A015531,
A015532,
A015533,
A015534,
A015535,
A015536,
A015537,
A015443,
A015447,
A030195,
A053404,
A057087,
A057088,
A083858,
A085939,
A090017,
A091914,
A099012,
A180222,
A180226.
-
[n le 2 select n-1 else 5*Self(n-1) + 7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012
-
Join[{a=0,b=1},Table[c=5*b+7*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{5, 7}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)
-
x='x+O('x^30); concat([0], Vec(x/(1-5*x-7*x^2))) \\ G. C. Greubel, Jan 24 2018
-
[lucas_number1(n,5,-7) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009
A015544
Lucas sequence U(5,-8): a(n+1) = 5*a(n) + 8*a(n-1), a(0)=0, a(1)=1.
Original entry on oeis.org
0, 1, 5, 33, 205, 1289, 8085, 50737, 318365, 1997721, 12535525, 78659393, 493581165, 3097180969, 19434554165, 121950218577, 765227526205, 4801739379641, 30130517107845, 189066500576353, 1186376639744525, 7444415203333449, 46713089134623445
Offset: 0
Cf.
A001076,
A006190,
A007482,
A015520,
A015521,
A015523,
A015524,
A015525,
A015528,
A015529,
A015530,
A015531,
A015532,
A015533,
A015534,
A015535,
A015536,
A015537,
A015441,
A015443,
A015447,
A030195,
A053404,
A057087,
A057088,
A083858,
A085939,
A090017,
A091914,
A099012,
A180222,
A180226,
A015555 (binomial transform).
-
[n le 2 select n-1 else 5*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012
-
a[n_]:=(MatrixPower[{{1,2},{1,-6}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{5, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)
-
A015544(n)=imag((2+quadgen(57))^n) \\ M. F. Hasler, Mar 06 2009
-
x='x+O('x^30); concat([0], Vec(x/(1 - 5*x - 8*x^2))) \\ G. C. Greubel, Jan 01 2018
-
[lucas_number1(n,5,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009
A208328
Triangle of coefficients of polynomials u(n,x) jointly generated with A208329; see the Formula section.
Original entry on oeis.org
1, 1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 9, 11, 1, 1, 9, 13, 25, 21, 1, 1, 11, 17, 43, 53, 43, 1, 1, 13, 21, 65, 97, 125, 85, 1, 1, 15, 25, 91, 153, 255, 273, 171, 1, 1, 17, 29, 121, 221, 441, 597, 609, 341, 1, 1, 19, 33, 155, 301, 691, 1089, 1443, 1325, 683, 1, 1, 21
Offset: 1
First five rows:
1;
1, 1;
1, 1, 3;
1, 1, 5, 5;
1, 1, 7, 9, 11;
First five polynomials u(n,x):
1
1 + x
1 + x + 3x^2
1 + x + 5x^2 + 5x^3
1 + x + 7x^2 + 9x^3 + 11x^4.
From _Philippe Deléham_, Mar 07 2012: (Start)
(1, 0, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, ...) begins:
1;
1, 0;
1, 1, 0;
1, 1, 3, 0;
1, 1, 5, 5, 0;
1, 1, 7, 9, 11, 0;
1, 1, 9, 13, 25, 21, 0;
1, 1, 11, 17, 43, 53, 43, 0; (End)
-
u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x];
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[%] (* A208328 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208329 *)
A299146
Modified Pascal's triangle read by rows: T(n,k) = C(n+1,k) - n, 1 <= k <= n.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 15, 10, 1, 1, 15, 29, 29, 15, 1, 1, 21, 49, 63, 49, 21, 1, 1, 28, 76, 118, 118, 76, 28, 1, 1, 36, 111, 201, 243, 201, 111, 36, 1, 1, 45, 155, 320, 452, 452, 320, 155, 45, 1, 1, 55, 209, 484, 781, 913, 781, 484, 209, 55, 1
Offset: 1
The triangle T(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9 10
1 1;
2 1, 1;
3 1, 3, 1;
4 1, 6, 6, 1;
5 1, 10, 15, 10, 1;
6 1, 15, 29, 29, 15, 1;
7 1, 21, 49, 63, 49, 21, 1;
8 1, 28, 76, 118, 118, 76, 28, 1;
9 1, 36, 111, 201, 243, 201, 111, 36, 1;
10 1, 45, 155, 320, 452, 452, 320, 155, 45, 1; etc.
-
Flat(List([1..100],n->List([1..n],k->Binomial(n+1,k)-n))); # Muniru A Asiru, Feb 05 2018
-
[[Binomial(n+1, k)- 1*n: k in [1..n]]: n in [1..10]];
-
seq(seq(binomial(n+1,k)-n, k=1..n), n=1..10); # Muniru A Asiru, Feb 05 2018
-
Table[Binomial[n + 1, k] - n, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Feb 05 2018 *)
-
T(n, k) = binomial(n+1,k) - n;
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 01 2018
A249861
a(n) are values of k that yield "record-breaking" integer sequence lengths for the recursion: b(i) = 3*(b(i-1) + b(i-2))/2, with b(0) = 1 and b(1) = k.
Original entry on oeis.org
1, 5, 37, 101, 229, 485, 2533, 6629, 23013, 88549, 219621, 481765, 1006053, 3103205, 7297509, 24074725, 158292453, 1232034277, 3379517925, 7674485221, 282552392165, 1382064019941, 5780110531045, 14576203553253, 84944947730917, 647894901152229
Offset: 1
Subtracting 1 from Nmax gives the exponents of 2 needed to generate a(n) using the formula above, as:
a(1) = 1 (by definition)
a(2) = 1 + 2^(3-1) = 5
a(3) = 5 + 2^(6-1) = 37
a(4) = 37 + 2^(7-1) = 101
... etc.
Comments