cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

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

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Examples

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

Links

Crossrefs

Rows give A059502, A059505, A059506, A059507, A059508; main diagonal = A059509.
Cf. A000667, A059216, A059219.

Programs

  • Mathematica
    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 *)

Formula

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

A059505 Transform of A059502 applied to sequence 2,3,4,...

Original entry on oeis.org

2, 5, 14, 40, 114, 323, 910, 2551, 7120, 19796, 54852, 151525, 417434, 1147145, 3145394, 8606848, 23507190, 64093031, 174474790, 474261691, 1287398452, 3490267820, 9451319304, 25565098825, 69080289074
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The second row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502.

Programs

  • Mathematica
    LinearRecurrence[{6,-11,6,-1},{2,5,14,40}, 50] (* or *) Rest[CoefficientList[Series[x*(2 - 7*x + 6*x^2 - x^3)/(1 - 3*x + x^2)^2, {x,0,50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    x='x+O('x^50); Vec(x*(2-7*x+6*x^2-x^3)/(1-3*x+x^2)^2) \\ G. C. Greubel, Sep 10 2017

Formula

G.f.: x*(2 - 7*x + 6*x^2 - x^3)/(1 - 3*x + x^2)^2.
From G. C. Greubel, Sep 10 2017: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
a(n) = ((3 - n)*Fibonacci(2*n) + (5 + 3*n)*Fibonacci(2*n - 1))/5. (End)

A059506 Transform of A059502 applied to sequence 3,4,5,...

Original entry on oeis.org

3, 7, 19, 53, 148, 412, 1143, 3161, 8717, 23977, 65798, 180182, 492459, 1343563, 3659623, 9953117, 27031768, 73320496, 198632607, 537507677, 1452978593, 3923762257, 10586222474, 28536313898, 76859031123
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The third row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502.

Programs

  • Mathematica
    LinearRecurrence[{6,-11,6,-1},{3,7,19,53},30] (* Harvey P. Dale, Jul 30 2015 *)
Rest[CoefficientList[Series[x*(1 - x)*(2*x^2 - 8*x + 3)/(x^2 - 3*x + 1)^2, {x,0,50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    Vec(x*(1-x)*(2*x^2-8*x+3)/(x^2-3*x+1)^2 + O(x^30)) \\ Michel Marcus, Sep 09 2017

Formula

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(2*x^2-8*x+3)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (10 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017

A059507 Transform of A059502 applied to sequence 4,5,6,...

Original entry on oeis.org

4, 9, 24, 66, 182, 501, 1376, 3771, 10314, 28158, 76744, 208839, 567484, 1539981, 4173852, 11299386, 30556346, 82547961, 222790424, 600753663, 1618558734, 4357256694, 11721125644, 31507528971, 84637773172
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The fourth row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502.

Programs

  • Mathematica
    Rest[CoefficientList[Series[x*(1 - x)*(3*x^2 - 11*x + 4)/(x^2 - 3*x + 1)^2, {x, 0, 50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    Vec(x*(1-x)*(3*x^2-11*x+4)/(x^2-3*x+1)^2 + O(x^40)) \\ Michel Marcus, Sep 09 2017

Formula

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(3*x^2-11*x+4)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (15 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017

A059508 Transform of A059502 applied to sequence 5,6,7,...

Original entry on oeis.org

5, 11, 29, 79, 216, 590, 1609, 4381, 11911, 32339, 87690, 237496, 642509, 1736399, 4688081, 12645655, 34080924, 91775426, 246948241, 663999649, 1784138875, 4790751131, 12856028814, 34478744044, 92416515221
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The fifth row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502, A059503.

Programs

  • Mathematica
    Rest[CoefficientList[Series[x*(1-x)*(4*x^2 - 14*x + 5)/(x^2 - 3*x + 1)^2, {x, 0, 50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    Vec(-x*(x-1)*(4*x^2-14*x+5)/(x^2-3*x+1)^2 + O(x^40)) \\ Michel Marcus, Sep 09 2017

Formula

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(4*x^2-14*x+5)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (20 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017

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

Views

Author

Clark Kimberling

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Formula

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

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

Views

Author

Floor van Lamoen, Jan 21 2001

Keywords

Comments

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

Crossrefs

Cf. A000667, A059216, A059219, A059502, A027994.
a(1-2n)=A005207(2n), a(-2n)=A056014(2n+1).

Programs

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

Formula

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]

A059578 Variation of Boustrophedon transform applied to 1,1,1,1,... Fill an array by diagonals, all in the 'up' direction. The first column is 1,1,1,1,.... For the next element of a diagonal, add to the previous element the elements of the row and the column the new element is in. The first row gives a(n).

Original entry on oeis.org

1, 2, 7, 30, 147, 792, 4559, 27500, 171645, 1099388, 7185101, 47724494, 321225165, 2186177302, 15018795171, 104011496474, 725373340023, 5089785834004, 35907469451787, 254541483884544, 1812185157383017, 12951828431246472, 92893383046741073, 668383820775639066
Offset: 1

Views

Author

Floor van Lamoen, Jan 23 2001

Keywords

Examples

			The array begins
1 2 7 30 ...
1 4 20 ...
1 8 ...
1 ...

Crossrefs

Cf. A000667, A035002, A059216, A059219, A059502, A059513.

Extensions

More terms from Sean A. Irvine, Sep 29 2022

A116845 Number of permutations of length n which avoid the patterns 231, 12534.

Original entry on oeis.org

1, 2, 5, 14, 41, 121, 355, 1032, 2973, 8496, 24111, 68017, 190885, 533294, 1484021, 4115186, 11375765, 31358377, 86223943, 236540916, 647556621, 1769374932, 4826148315, 13142564449, 35736448201, 97037995226, 263156279525, 712795854422, 1928547574913
Offset: 1

Views

Author

Lara Pudwell, Feb 26 2006

Keywords

Links

Crossrefs

Cf. A059502 (first differences).

Programs

  • Mathematica
    Rest[CoefficientList[Series[x*(1 - 5*x + 8*x^2 - 4*x^3 + x^4) / ((1 - x)*(1 - 3*x + x^2)^2), {x, 0, 30}], x]] (* Vaclav Kotesovec, Aug 04 2018 *)
RecurrenceTable[{a[1] == 1, a[2] == 2, a[3] == 5, a[4] == 14, a[5] == 41, a[n] == 7*a[n-1] - 17*a[n-2] + 17*a[n-3] - 7*a[n-4] + a[n-5]}, a, {n, 1, 30}] (* Vaclav Kotesovec, Aug 04 2018 *)
Table[1 + Fibonacci[2*n]/5 + LucasL[2*n - 3]*n/5, {n, 1, 30}] (* Vaclav Kotesovec, Aug 04 2018 *)
  • PARI
    Vec(x*(1 - 5*x + 8*x^2 - 4*x^3 + x^4) / ((1 - x)*(1 - 3*x + x^2)^2) + O(x^40)) \\ Colin Barker, Oct 19 2017

Formula

G.f.: x*(1 - 5*x + 8*x^2 - 4*x^3 + x^4) / ((1 - x)*(1 - 3*x + x^2)^2). [restored by Michael D. Weiner, Jul 05 2018]
a(n) = 7*a(n-1) - 17*a(n-2) + 17*a(n-3) - 7*a(n-4) + a(n-5) for n>5. - Colin Barker, Oct 19 2017
a(n) = 1 + Fibonacci(2*n)/5 + Lucas(2*n - 3)*n/5. - Vaclav Kotesovec, Aug 04 2018

A199479 Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,0,0,...) DELTA (1,1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 7, 20, 27, 13, 1, 9, 35, 73, 80, 34, 1, 11, 54, 151, 252, 234, 89, 1, 13, 77, 269, 597, 837, 677, 233, 1, 15, 104, 435, 1199, 2225, 2702, 1941, 610, 1, 17, 135, 657, 2158, 4956, 7943, 8533, 5523, 1597
Offset: 0

Views

Author

Philippe Deléham, Nov 06 2011

Keywords

Comments

Mirror image of triangle in A147703.

Examples

			Triangle begins:
  1;
  1,  1;
  1,  3,  2;
  1,  5,  9,  5;
  1,  7, 20, 27, 13;
  1,  9, 35, 73, 80, 34;

Crossrefs

Cf. A001519, A059502, A000012, A005408, A014107.

Formula

Sum_{k=0..n} T(n,k)*x^k = A152620(n), A152594(n), A000007(n), A000012(n), A006012(n), A152596(n), A152599(n) for x=-3,-2,-1,0,1,2,3 respectively.
T(n,n) = A001519(n).
G.f.: (1-2y*x)/(1-(1+3y)*x+y*(1+y)*x^2).
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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