A124400 -id:A124400 - cp's OEIS frontend

A131322 Row sums of triangle A131321.

1, 1, 3, 5, 12, 23, 51, 103, 221, 456, 965, 2009, 4227, 8833, 18540, 38803, 81363, 170399, 357145, 748176, 1567849, 3284833, 6883059, 14421533, 30218028, 63314735, 132664227, 277968871, 582428789, 1220356440, 2557009709

Offset: 0

Gary W. Adamson, Jun 28 2007

nonn

Equals INVERT transform of (1, 2, 0, 1, 0, 1, 0, 1, ...). - Gary W. Adamson, Apr 28 2009
The sequence is also the INVERT transform of the aerated odd-indexed Fibonacci numbers (i.e., of (1, 0, 2, 0, 5, 0, ...)). Sequence A124400 is the INVERT transform of the aerated even-indexed Fibonacci numbers. - Gary W. Adamson, Feb 07 2014
a(n) is the number of tilings of a 4 X 2n rectangle into L tetrominoes (no reflections, only rotations). - Nicolas Bělohoubek, Jan 21 2025

			a(4) = 12 = 5 + 0 + 6 + 0 + 1.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 3.
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).

Cf. A166482, A131321, A000045.

Cf. A124400.

LinearRecurrence[{1, 3, -1, -1}, {1, 1, 3, 5}, 50] (* Paolo Xausa, Jan 28 2025 *)

G.f.: (1-x^2)/(1 - x - 3x^2 + x^3 + x^4). - Philippe Deléham, Jan 21 2012
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - a(n-4), a(0)=1, a(1)=1, a(2)=3, a(3)=5. - Philippe Deléham, Jan 21 2012
a(n) = Sum_{m=0..ceiling(n/2)} binomial(n-m,n-2*m)*Fibonacci(n-2*m+1). - Vladimir Kruchinin, Jan 26 2013
From Nicolas Bělohoubek, Jan 21 2025: (Start)
a(n) = Sum_{m=1..4} (alpha_m * x_m^n). For x_m and alpha_m values see "L-tetromino tilings" article in links.
a(2*n) = A166482(n). (End)

a(10)-a(30) from Philippe Deléham, Jan 21 2012

A102756 Triangle T(n,k), 0<=k<=n, read by rows defined by: T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 10, 10, 3, 5, 20, 31, 20, 5, 6, 35, 76, 78, 40, 8, 7, 56, 161, 232, 184, 76, 13, 8, 84, 308, 582, 636, 406, 142, 21, 9, 120, 546, 1296, 1831, 1604, 861, 260, 34, 10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55

Offset: 0

Philippe Deléham, Dec 18 2006

Rising and falling diagonals are A008999, A124400.
Subtriangle of triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 17 2012
Jointly generated with A209130 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Mar 05 2012

			Triangle begins:
  1;
  2, 1;
  3, 4, 2;
  4, 10, 10, 3;
  5, 20, 31, 20, 5;
  6, 35, 76, 78, 40, 8;
  7, 56, 161, 232, 184, 76, 13;
  8, 84, 308, 582, 636, 406, 142, 21;
  9, 120, 546, 1296, 1831, 1604, 861, 260, 34;
  10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55;
Triangle (1, 1, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
  1
  1, 0
  2, 1, 0
  3, 4, 2, 0
  4, 10, 10, 3, 0
  5, 20, 31, 20, 5, 0
  6, 35, 76, 78, 40, 8, 0
  7, 56, 161, 232, 184, 76, 13, 0

Cf. A209130.
Cf. A008999, A124400, A084938, A209130.
Cf. A254006, A000012, A000027, A000244, A015530, A015544.
Cf. A001629, A000045, A000292, A051747.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*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[%]    (* A102756 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209130 *)
(* Clark Kimberling, Mar 05 2012 *)

Sum_{k=0..n} x^k*T(n,k) = A254006(n), A000012(n), A000027(n+1), A000244(n), A015530(n+1), A015544(n+1) for x = -2, -1, 0, 1, 2, 3 respectively.
T(n,n-1) = 2*A001629(n+1) for n>=1.
T(n,n) = Fibonacci(n+1) = A000045(n+1).
T(n,0) = n+1.
T(n,1) = A000292(n) for n>=1.
T(n+1,2) = binomial(n+4,n-1)+binomial(n+2,n-1)= A051747(n) for n>=1.
G.f.: 1/(1-(2+y)*x+(1+y)*(1-y)*x^2). - Philippe Deléham, Feb 17 2012

A127963 Number of 1's in A127962(n).

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 10, 12, 16, 22, 31, 40, 51, 64, 84, 96, 100, 157, 174, 351, 855, 1309, 1770, 2904, 5251, 5346, 5640, 6196, 7240, 21369, 41670, 47685, 58620, 63516, 69469, 70540, 133509, 134994, 187161, 493096, 2015700

Offset: 1

Artur Jasinski, Feb 09 2007

Cf. A000120, A000978, A000979, A007953, A124400, A126614, A127936, A127955, A127956, A127957, A127958, A127961, A127962.

b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, DigitCount[a[[x]], 10, 1]], {x, 1, Length[a]}]; d (* Artur Jasinski, Feb 09 2007 *)
DigitCount[#, 2, 1]& /@ Select[Table[(2^p + 1)/3, {p, Prime[Range[300]]}], PrimeQ] (* Amiram Eldar, Jul 23 2023 *)

a(n) = A000120(A000979(n)). - Michel Marcus, Nov 07 2013

a(n) = A007953(A127962(n)). - Amiram Eldar, Jul 23 2023

a(22)-a(29) from Vincenzo Librandi, Mar 31 2012

a(30)-a(41) from Amiram Eldar, Jul 23 2023

A127964 Number of 0's in the binary expansion of A127962(n).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 14, 20, 29, 38, 49, 62, 82, 94, 98, 155, 172, 349, 853, 1307, 1768, 2902, 5249, 5344, 5638, 6194, 7238, 21367, 41668, 47683, 58618, 63514, 69467, 70538, 133507, 134992, 187159, 493094, 2015698

Offset: 1

Artur Jasinski, Feb 09 2007

Apparently numbers k such that (2^(2*k+3)+1)/3 is prime. - James R. Buddenhagen, Apr 14 2011 [This is true. See the second formula. - Amiram Eldar, Oct 13 2024]

Cf. A000978, A000979, A023416, A127962, A127961, A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936.

b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, DigitCount[a[[x]], 10, 0]], {x, 1, Length[a]}]; d
(Select[Prime[Range[200]], PrimeQ[(2^# + 1)/3] &] - 3)/2 (* Amiram Eldar, Oct 13 2024 *)

a(n) = A023416(A000979(n)). - Michel Marcus, Nov 07 2013

a(n) = (A000978(n)-3)/2. - Amiram Eldar, Oct 13 2024

a(22)-a(29) from Vincenzo Librandi, Mar 31 2012

a(30)-a(41) from Amiram Eldar, Oct 13 2024

A127965 Number of bits in A127962(n).

Original entry on oeis.org

2, 4, 6, 10, 12, 16, 18, 22, 30, 42, 60, 78, 100, 126, 166, 190, 198, 312, 346, 700, 1708, 2616, 3538, 5806, 10500, 10690, 11278, 12390, 14478, 42736, 83338, 95368, 117238, 127030, 138936, 141078, 267016, 269986, 374320, 986190, 4031398

Offset: 1

Artur Jasinski, Feb 09 2007

Cf. A127962, A127963, A127964, A127961, A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936.

b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, DigitCount[a[[x]], 10, 0]+DigitCount[a[[x]], 10, 1]], {x, 1, Length[a]}]; d

a(n) = A127964(n) + A127963(n).

a(n) = 1 + floor(log_2(A000979(n))) = 1 + floor(log_2(2^A000978(n)+1) - A020857) = A000978(n) - 1. - R. J. Mathar, Feb 01 2008

a(22)-a(29) from Vincenzo Librandi, Mar 30 2012

a(30)-a(41) from Amiram Eldar, Oct 19 2024

A077920 Expansion of (1-x)^(-1)/(1+2*x-x^2-x^3).

Original entry on oeis.org

1, -1, 4, -7, 18, -38, 88, -195, 441, -988, 2223, -4992, 11220, -25208, 56645, -127277, 285992, -642615, 1443946, -3244514, 7290360, -16381287, 36808421, -82707768, 185842671, -417584688, 938304280, -2108350576, 4737420745, -10644887785, 23918845740, -53745158519, 120764274994

Offset: 0

N. J. A. Sloane, Nov 17 2002

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

CoefficientList[Series[(1-x)^(-1)/(1+2x-x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{-1,3,0,-1},{1,-1,4,-7},40] (* Harvey P. Dale, Mar 13 2013 *)

Vec((1-x)^(-1)/(1+2*x-x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n) = (-1)^n*A124400(n). - Philippe Deléham, Dec 18 2006

a(n) = a(n-1) + 3*a(n-2) - a(n-4); a(0)=1, a(1)=-1, a(2)=4, a(3)=-7. - Harvey P. Dale, Mar 13 2013

A127959 Nonprime numbers of the form 1 + Sum_{k=1..m} 2^(2*k - 1).

Original entry on oeis.org

171, 10923, 699051, 11184811, 44739243, 178956971, 2863311531, 11453246123, 45812984491, 183251937963, 733007751851, 11728124029611, 46912496118443, 187649984473771, 750599937895083, 3002399751580331, 12009599006321323

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

Prime numbers of the form 1 + Sum_{k=1..m} 2^(2*n - 1) is A000979. Numbers x such that 1 + Sum_{k=1..m} 2^(2*n - 1) is prime for n=1,2,...,x is A127936. A127955 is probably a subset of the present sequence.

Cf. A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936.

a = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c] == False, AppendTo[a, c]], {x, 1, 50}]; a
Select[Table[Sum[2^(2k-1),{k,n}]+1,{n,50}],!PrimeQ[#]&] (* Harvey P. Dale, Dec 23 2017 *)

cp's OEIS Frontend

A131322 Row sums of triangle A131321.

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A102756 Triangle T(n,k), 0<=k<=n, read by rows defined by: T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

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A127963 Number of 1's in A127962(n).

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A127964 Number of 0's in the binary expansion of A127962(n).

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A127965 Number of bits in A127962(n).

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Extensions

A077920 Expansion of (1-x)^(-1)/(1+2*x-x^2-x^3).

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A127959 Nonprime numbers of the form 1 + Sum_{k=1..m} 2^(2*k - 1).

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