A059216 -id:A059216 - cp's OEIS frontend

A059217 The array in A059216 read by antidiagonals in 'up' direction.

1, 1, 2, 5, 3, 1, 1, 6, 10, 14, 45, 37, 26, 15, 1, 1, 46, 84, 121, 150, 169, 740, 686, 592, 471, 321, 170, 1, 1, 741, 1428, 2111, 2704, 3183, 3532, 3721, 21142, 20347, 18826, 16685, 13953, 10777, 7255, 3722, 1, 1, 21143, 41491, 61798, 80598

Offset: 1

Floor van Lamoen, Jan 18 2001

			The array begins
   1  2  1 14  1 ...
   1  3 10 15 ...
   5  6 26 ...
   1 37 ...
  45 ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000667, A059216, A059219, A059220, A059234.

Maple
```
See A059216 for Maple code.
```

max = 9; t[0, 0] = 1; t[0, ?EvenQ] = 1; t[?OddQ, 0] = 1; t[n_, k_] /; OddQ[n + k](*up*):= t[n, k] = t[n+1, k-1] + Sum[t[n, j], {j, 0, k-1}]; t[n_, k_] /; EvenQ[n + k](*down*):= t[n, k] = t[n-1, k+1] + Sum[t[j, k], {j, 0, n-1}]; Table[t[n-k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 16 2013 *)

A059718 Triangle T(n,k), 0<=k<=n, giving coefficients when output sequence O_0, O_1, O_2, ... from transformation described in A059216 is expressed in terms of input sequence I_0, I_1, I_2, ...

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 9, 16, 15, 1, 5, 14, 35, 59, 55, 1, 6, 20, 64, 152, 258, 239, 1, 7, 27, 105, 319, 767, 1296, 1199, 1, 8, 35, 160, 590, 1820, 4356, 7362, 6810, 1, 9, 44, 231, 1000, 3751, 11514, 27583, 46609, 43108, 1, 10, 54, 320, 1589

Offset: 0

N. J. A. Sloane, Feb 08 2001

			The triangle begins as:
  1;
  1, 1;
  1, 2, 2;
  1, 3, 5,  5;
  1, 4, 9, 16, 15;
  ...
For example, O_4 = I_4 + 4*I_3 + 9*I_2 + 16*I_1 + 15*I_0.

Cf. A059720, A059216.

A059234 The array in A059216 read by antidiagonals in the direction in which it was constructed.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 6, 10, 14, 1, 15, 26, 37, 45, 1, 46, 84, 121, 150, 169, 1, 170, 321, 471, 592, 686, 740, 1, 741, 1428, 2111, 2704, 3183, 3532, 3721, 1, 3722, 7255, 10777, 13953, 16685, 18826, 20347, 21142, 1, 21143, 41491, 61798, 80598, 97345, 111419

Offset: 1

Floor van Lamoen, Jan 18 2001

			The array begins
   1  2  1 14  1 ...
   1  3 10 15 ...
   5  6 26 ...
   1 37 ...
  45 ...

Cf. A000667, A059216, A059219, A059220, A059217.

More terms from Larry Reeves (larryr(AT)acm.org), Jan 23 2001

A059219 Variation of Boustrophedon transform applied to sequence 1,0,0,0,...: fill an array by diagonals in alternating directions - 'up' and 'down'. The first element of each diagonal after the first is 0. When 'going up', add to the previous element the elements of the row the new element is in. When 'going down', add to the previous element the elements of the column the new element is in. The final element of the n-th diagonal is a(n).

Original entry on oeis.org

1, 1, 2, 5, 15, 55, 239, 1199, 6810, 43108, 300731, 2291162, 18923688, 168402163, 1606199354, 16345042652, 176758631046, 2024225038882, 24471719797265, 311446235344127, 4162172487402027, 58275220793611957, 853045299274146032

Offset: 0

N. J. A. Sloane, Jan 18 2001

			The array begins
1 1 0 5 0 55 0 ...
0 1 3 5 48 55 ...
2 2 8 39 103 ...
0 12 27 152 ...
15 15 190 ...
0 221 ...

Vincenzo Librandi, Table of n, a(n) for n = 0..120
Index entries for sequences related to boustrophedon transform

Cf. A000667, A059216, A059217, A059220, A059237, A059720, A059718.

aaa := proc(m,n) option remember; local j,s,t1; if m=0 and n=0 then RETURN(1); fi; if n = 0 and m mod 2 = 1 then RETURN(0); fi; if m = 0 and n mod 2 = 0 then RETURN(0); fi; s := m+n; if s mod 2 = 1 then t1 := aaa(m+1,n-1); for j from 0 to n-1 do t1 := t1+aaa(m,j); od: else t1 := aaa(m-1,n+1); for j from 0 to m-1 do t1 := t1+aaa(j,n); od: fi; RETURN(t1); end; # the n-th antidiagonal in the up direction is aaa(n,0), aaa(n-1,1), aaa(n-2,2), ..., aaa(0,n)

max = 22; t[0, 0] = 1; t[0, ?EvenQ] = 0; t[?OddQ, 0] = 0; t[n_, k_] /; OddQ[n+k](* up *):= t[n, k] = t[n+1, k-1] + Sum[t[n, j], {j, 0, k-1}]; t[n_, k_] /; EvenQ[n+k](* down *):= t[n, k] = t[n-1, k+1] + Sum[t[j, k], {j, 0, n-1}]; tnk = Table[t[n, k], {n, 0, max}, {k, 0, max-n}]; Join[{1},  Rest[Union[tnk[[1]], tnk[[All, 1]]]]](* Jean-François Alcover, May 16 2012 *)

More terms from Floor van Lamoen, Jan 19 2001; and from N. J. A. Sloane Jan 20 2001.

A059502 a(n) = (3nF(2n-1) + (3-n)*F(2n))/5 where F() = Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 3, 9, 27, 80, 234, 677, 1941, 5523, 15615, 43906, 122868, 342409, 950727, 2631165, 7260579, 19982612, 54865566, 150316973, 411015705, 1121818311, 3056773383, 8316416134, 22593883752, 61301547025, 166118284299, 449639574897, 1215751720491, 3283883157848

Offset: 0

Floor van Lamoen, Jan 19 2001

Substituting x(1-x)/(1-2x) into x/(1-x)^2 yields g.f. of sequence.

Variation of A059216 (and of Boustrophedon transform) applied to 1,2,3,4,...: fill an array by diagonals, each time in the same direction, say the 'up' direction. The first column is 1,2,3,4,... For the next element of a diagonal, add to the previous element the elements of the row the new element is in. The first row gives a(n).

			The array (see A059503) begins
  1 3  9 27 80 ...
  2 5 14 40 ...
  3 7 19 ...
  4 9  5 ...

Harry J. Smith, Table of n, a(n) for n = 0..200
Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 17, 19.
Index entries for two-way infinite sequences
Index entries for sequences related to boustrophedon transform
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A000045, A000667, A059216, A059219, A027994, A059512, A059503.

[(3*n*Fibonacci(2*n-1)+(3-n)*Fibonacci(2*n))/5: n in [0..100]]; // Vincenzo Librandi, Apr 23 2011

Table[(3n Fibonacci[2n-1]+(3-n)Fibonacci[2n])/5,{n,0,30}] (* or *) CoefficientList[Series[x(1-x)(1-2x)/(1-3x+x^2)^2,{x,0,30}],x] (* Harvey P. Dale, Apr 23 2011 *)

a(n)=(3*n*fibonacci(2*n-1)+(3-n)*fibonacci(2*n))/5

a(n) = 2*a(n-1) + Sum{m<=n-2} a(m) + A001519(n-2).
G.f.: x*(1 - x)*(1 - 2*x)/(1 - 3*x + x^2)^2. - Emeric Deutsch, Oct 07 2002
a(n) = A147703(n,1). - Philippe Deléham, Nov 29 2008
a(n) = A001871(n-1) - 3*A001871(n-2) + 2*A001871(n-3). - R. J. Mathar, Apr 09 2019
E.g.f.: 2*exp(3*x/2)*(5*x*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Mar 04 2025

A059220 The array in A059219 read by antidiagonals in 'up' direction.

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 0, 2, 3, 5, 15, 12, 8, 5, 0, 0, 15, 27, 39, 48, 55, 239, 221, 190, 152, 103, 55, 0, 0, 239, 460, 680, 871, 1025, 1137, 1199, 6810, 6553, 6062, 5374, 4493, 3471, 2336, 1199, 0, 0, 6810, 13363, 19903, 25958, 31351, 35884, 39399, 41847

Offset: 1

N. J. A. Sloane, Jan 18 2001

			The array begins
   1   1   0   5   0  55   0 ...
   0   1   3   5  48  55   ...
   2   2   8  39 103  ...
   0  12  27 152 ...
  15  15 190 ...
   0 221 ...

Cf. A000667, A059216, A059219, A059217, A059235.

Maple
```
See A059219 for Maple code.
```

max = 9; t[0, 0] = 1; t[0, ?EvenQ] = 0; t[?OddQ, 0] = 0; t[n_, k_] /; OddQ[n + k] (* up *) := t[n, k] = t[n+1, k-1] + Sum[t[n, j], {j, 0, k-1}]; t[n_, k_] /; EvenQ[n + k] (* down *) := t[n, k] = t[n-1, k+1] + Sum[t[j, k], {j, 0, n-1}]; Table[t[n - k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 19 2013 *)

More terms from Floor van Lamoen, Jan 19 2001

A027994 a(n) = (F(2n+3) - F(n))/2 where F() = Fibonacci numbers A000045.

Original entry on oeis.org

1, 2, 6, 16, 43, 114, 301, 792, 2080, 5456, 14301, 37468, 98137, 256998, 672946, 1761984, 4613239, 12078110, 31621701, 82787980, 216743836, 567446112, 1485598681, 3889356696, 10182482353, 26658108074, 69791870526, 182717549872, 478360854115, 1252365133866, 3278734743901, 8583839415648, 22472784017272

Offset: 0

Clark Kimberling

Substituting x*(1-x)/(1-2x) into x^2/(1-x^2) yields x^2*(g.f. of sequence).
The number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+1, s(0) = 2, s(n+1) = 3. - Herbert Kociemba, Jun 02 2004
Diagonal sums of triangle in A125171. - Philippe Deléham, Jan 14 2014

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Cf. A000667, A059216, A059219, A059502, A027926.

[(Fibonacci(2*n+3)-Fibonacci(n))/2 : n in [0..40]]; // Vincenzo Librandi, Jan 01 2025

Table[(Fibonacci[2n+3]-Fibonacci[n])/2,{n,0,30}] (* or *) LinearRecurrence[{4,-3,-2,1},{1,2,6,16},30] (* Harvey P. Dale, Apr 28 2022 *)

PARI
```
a(n)=(fibonacci(2*n+3)-fibonacci(n))/2
```

G.f.: (1-x)^2/((1-x-x^2)*(1-3*x+x^2)). - Floor van Lamoen and N. J. A. Sloane, Jan 21 2001
a(n) = Sum_{k=0..n} T(n, k)*T(n, n+k), T given by A027926.
a(n) = 2*a(n-1) + Sum_{m < n-1} a(m) + F(n-1) = A059512(n+2) - F(n) where F(n) is the n-th Fibonacci number (A000045). - Floor van Lamoen, Jan 21 2001
a(n) = (2/5)*Sum_{k=1..4} sin(2*Pi*k/5)*sin(3*Pi*k/5)*(1+2*cos(Pi*k/5))^(n+1). - Herbert Kociemba, Jun 02 2004
a(-1-2n) = A056014(2n), a(-2n) = A005207(2n-1).
E.g.f.: exp(3*x/2)*cosh(sqrt(5)*x/2) + exp(x/2)*(2*exp(x) - 1)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Jan 01 2025

A059503 The array in A059502 read by antidiagonals in 'up' direction.

Original entry on oeis.org

1, 2, 3, 3, 5, 9, 4, 7, 14, 27, 5, 9, 19, 40, 80, 6, 11, 24, 53, 114, 234, 7, 13, 29, 66, 148, 323, 677, 8, 15, 34, 79, 182, 412, 910, 1941, 9, 17, 39, 92, 216, 501, 1143, 2551, 5523, 10, 19, 44, 105, 250, 590, 1376, 3161, 7120, 15615, 11, 21, 49, 118

Offset: 0

Floor van Lamoen, Jan 19 2001

			The array begins
1 3 9 27 80 ...
2 5 14 40 ...
3 7 19 ...
4 9 5 ...

Rows give A059502, A059505, A059506, A059507, A059508; main diagonal = A059509.

Cf. A000667, A059216, A059219.

T[n_, k_] := ((3 - k)*Fibonacci[2*k] + (5*n + 3*k)*Fibonacci[2*k - 1])/5;
TableForm[Table[T[n, k], {n, 0, 5}, {k, 1, 5}]]
Table[T[n - k, k + 1], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 10 2017 *)

T(n,k) = ((3 - k)*Fibonacci(2*k) + (5*n + 3*k)*Fibonacci(2*k - 1))/5. - G. C. Greubel, Sep 10 2017

A062704 Di-Boustrophedon transform of all 1's sequence: Fill in an array by diagonals alternating in the 'up' and 'down' directions. Each diagonal starts with a 1. When going in the 'up' direction the next element is the sum of the previous element of the diagonal and the previous two elements of the row the new element is in. When going in the 'down' direction the next element is the sum of the previous element of the diagonal and the previous two elements of the column the new element is in. The final element of the n-th diagonal is a(n).

Original entry on oeis.org

1, 2, 5, 13, 40, 145, 616, 3017, 16752, 103973, 713040, 5352729, 43645848, 384059537, 3626960272, 36585357429, 392545057280, 4463791225145, 53622168102640, 678508544425721, 9020035443775264, 125684948107190045, 1831698736650660952, 27866044704218390113

Offset: 1

Floor van Lamoen, Jul 11 2001

			The array begins:
   1   2   1  13   1
   1   3  10  14
   5   6  25
   1  34
  40

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A000667, A059216, A063179.

T:= proc(n, k) option remember;
      if n<1 or k<1 then 0
    elif n=1 and irem(k, 2)=1 or k=1 and irem(n, 2)=0 then 1
    elif irem(n+k, 2)=0 then T(n-1, k+1)+T(n-1, k)+T(n-2, k)
                        else T(n+1, k-1)+T(n, k-1)+T(n, k-2)
      fi
    end:
a:= n-> `if`(irem (n, 2)=0, T(1, n), T(n, 1)):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 08 2011

T[n_, k_] := T[n, k] = Which[n < 1 || k < 1, 0
     , n == 1 && Mod[k, 2] == 1 || k == 1 && Mod[n, 2] == 0, 1
     , Mod[n + k, 2] == 0, T[n - 1, k + 1] + T[n - 1, k] + T[n - 2, k]
     , True,               T[n + 1, k - 1] + T[n, k - 1] + T[n, k - 2]];
a[n_] := If[Mod [n, 2] == 0, T[1, n], T[n, 1]];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

More terms from Alois P. Heinz, Feb 08 2011

A059512 For n>=2, the number of (s(0), s(1), ..., s(n-1)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n-1, s(0) = 2, s(n-1) = 2.

Original entry on oeis.org

0, 1, 1, 3, 7, 18, 46, 119, 309, 805, 2101, 5490, 14356, 37557, 98281, 257231, 673323, 1762594, 4614226, 12079707, 31624285, 82792161, 216750601, 567457058, 1485616392, 3889385353, 10182528721, 26658183099, 69791991919

Offset: 0

Floor van Lamoen, Jan 21 2001

Substituting x(1-x)/(1-2x) into x/(1-x^2) yields g.f. of sequence.

Cf. A000667, A059216, A059219, A059502, A027994.

a(1-2n)=A005207(2n), a(-2n)=A056014(2n+1).

CoefficientList[Series[x(1-x)(1-2x)/((1-x-x^2)(1-3x+x^2)), {x,0,30}], x]  (* Harvey P. Dale, Apr 23 2011 *)

a(n)=(fibonacci(2*n-1)+fibonacci(n-2))/2

a(n) = 2a(n-1) + Sum{mA000045).
G.f.: x(1-x)(1-2x)/((1-x-x^2)(1-3x+x^2)).
a(n+1)=sum{k=0..floor(n/2), C(n,2k)*F(2k+1)}. [From Paul Barry, Oct 14 2009]

cp's OEIS Frontend

A059217 The array in A059216 read by antidiagonals in 'up' direction.

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A059718 Triangle T(n,k), 0<=k<=n, giving coefficients when output sequence O_0, O_1, O_2, ... from transformation described in A059216 is expressed in terms of input sequence I_0, I_1, I_2, ...

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A059234 The array in A059216 read by antidiagonals in the direction in which it was constructed.

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A059502 a(n) = (3*n*F(2n-1) + (3-n)*F(2n))/5 where F() = Fibonacci numbers A000045.

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A059220 The array in A059219 read by antidiagonals in 'up' direction.

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A027994 a(n) = (F(2n+3) - F(n))/2 where F() = Fibonacci numbers A000045.

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A059503 The array in A059502 read by antidiagonals in 'up' direction.

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A059502 a(n) = (3nF(2n-1) + (3-n)*F(2n))/5 where F() = Fibonacci numbers A000045.