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 64 results. Next

A042950 Row sums of the Lucas triangle A029635.

Original entry on oeis.org

2, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944
Offset: 0

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Author

N. J. A. Sloane and J. H. Conway

Keywords

Comments

Map a binary sequence b=[ b_1,...] to a binary sequence c=[ c_1,...] so that C = 1/Product((1-x^i)^c_i == 1 + Sum b_i*x^i mod 2.
This produces 2 new sequences: d={i:c_i=1} and e=[ 1,e_1,... ] where C = 1 + Sum e_i*x^i.
This sequence is d when b=[ 0,1,1,1,1,...].
Number of rises after n+1 iterations of morphism A007413.
a(n) written in base 2: a(0) = 10, a(n) for n >= 1: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-1) times 0 (see A003953(n)). - Jaroslav Krizek, Aug 17 2009
Row sums of the Lucas triangle A029635. - Sergio Falcon, Mar 17 2014

Links

Crossrefs

Cf. A003945, A007283.

Programs

  • Magma
    [2] cat [2^(n+1) - 2^(n-1): n in [1..40]]; // Vincenzo Librandi, Aug 08 2015
  • Mathematica
    Table[ Ceiling[3*2^(n - 1)], {n, 0, 32}] (* Robert G. Wilson v, Jul 08 2006 *)
a[0] = 2; a[1] = 3; a[n_] := 2a[n - 1]; Table[a[n], {n, 0, 32}] (* Robert G. Wilson v, Jul 08 2006 *)
f[s_] := Append[s, 1 + Plus @@ s]; Nest[f, {2}, 32] (* Robert G. Wilson v, Jul 08 2006 *)
CoefficientList[Series[(2 - x)/(1 - 2x), {x, 0, 32}], x] (* Robert G. Wilson v, Jul 08 2006 *)
  • PARI
    a(n)=ceil(3*2^(n-1))
  • Python
    def A042950(n): return (3*2**n + int(n==0))//2 # G. C. Greubel, Jun 06 2025

Formula

G.f.: (2-x)/(1-2*x).
a(n) = 2*a(n-1), n > 1; a(0)=2, a(1)=3.
a(n) = A003945(n), for n > 0.
From Paul Barry, Dec 06 2004: (Start)
Binomial transform of 2, 1, 2, 1, 2, 1, ... = (3+(-1)^n)/2.
a(n) = (3*2^n + 0^n)/2. (End)
a(0) = 2, a(n) = 3*2^(n-1) = 2^n + 2^(n-1) for n >= 1. - Jaroslav Krizek, Aug 17 2009
a(n) = 2^(n+1) - 2^(n-1), for n > 0. - Ilya Gutkovskiy, Aug 08 2015
E.g.f.: (3*exp(2*x) + 1)/2. - G. C. Greubel, Jun 06 2025

A029651 Central elements of the (1,2)-Pascal triangle A029635.

Original entry on oeis.org

1, 3, 9, 30, 105, 378, 1386, 5148, 19305, 72930, 277134, 1058148, 4056234, 15600900, 60174900, 232676280, 901620585, 3500409330, 13612702950, 53017895700, 206769793230, 807386811660, 3156148445580, 12350146091400, 48371405524650
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Comments

If Y is a fixed 2-subset of a (2n+1)-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007

References

  • V. N. Smith and L. Shapiro, Catalan numbers, Pascal's triangle and mutators, Congressus Numerant., 205 (2010), 187-197.

Links

Crossrefs

Cf. A001700, A143398, A145460.
Essentially a duplicate of A003409.

Programs

  • Maple
    a := n -> (3/2)*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(1+n))-0^n/2;
seq(simplify(a(n)), n=0..24); # Peter Luschny, Dec 16 2015
  • Mathematica
    Join[{1},Table[3*Binomial[2n-1,n],{n,30}]] (* Harvey P. Dale, Aug 11 2015 *)
  • PARI
    concat([1], for(n=1, 50, print1(3*binomial(2*n-1,n), ", "))) \\ G. C. Greubel, Jan 23 2017

Formula

a(n) = 3 * binomial(2n-1, n) (n>0). - Len Smiley, Nov 03 2001
a(n) = 3*A001700(n-1), (n>=1).
G.f.: (1+xC(x))/(1-2xC(x)), C(x) the g.f. of A000108. - Paul Barry, Dec 17 2004
a(n) = A003409(n), n>0. - R. J. Mathar, Oct 23 2008
a(n) = Sum_{k=0..n} A039599(n,k)*A000034(k). - Philippe Deléham, Oct 29 2008
a(n) = (3/2)*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(1+n))-0^n/2. - Peter Luschny, Dec 16 2015
a(n) ~ (3/2)*4^n*(1-(1/8)/n+(1/128)/n^2+(5/1024)/n^3-(21/32768)/n^4)/sqrt(n*Pi). - Peter Luschny, Dec 16 2015
a(n) = 2^(1-n)*Sum_{k=0..n} binomial(k+n,k)*binomial(2*n-1,n-k), n>0, a(0)=1. - Vladimir Kruchinin, Nov 23 2016
E.g.f.: (3*exp(2*x)*BesselI(0,2*x) - 1)/2. - Ilya Gutkovskiy, Nov 23 2016
a(n) = A143398(2n,n) = A145460(2n,n). - Alois P. Heinz, Sep 09 2018
a(n) = [x^n] C(-x)^(-3*n), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Oct 16 2024

Extensions

More terms from David W. Wilson

A029638 Numbers in the (1,2)-Pascal triangle A029635 that are different from 1.

Original entry on oeis.org

2, 2, 3, 2, 4, 5, 2, 5, 9, 7, 2, 6, 14, 16, 9, 2, 7, 20, 30, 25, 11, 2, 8, 27, 50, 55, 36, 13, 2, 9, 35, 77, 105, 91, 49, 15, 2, 10, 44, 112, 182, 196, 140, 64, 17, 2, 11, 54, 156, 294, 378, 336, 204, 81, 19, 2, 12, 65, 210, 450, 672, 714, 540, 285, 100, 21, 2, 13, 77, 275, 660, 1122
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Examples

			Triangle begins:
  2;
  3,  2;
  4,  5,  2;
  5,  9,  7,  2;
  6, 14, 16,  9,  2;
  7, 20, 30, 25, 11,  2;
  ...

References

  • D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; triangle on page 6, denominators.

Links

Crossrefs

Cf. A048601, A029656.

Extensions

More terms from David W. Wilson
Leading 2 inserted as consequence of change in A029635 by Sean A. Irvine, Mar 01 2020

A251634 Numerators of inverse Riordan triangle of Riordan triangle A029635. Riordan (1/(1-x), x/(1+2*x)). Triangle read by rows for 0 <= m <= n.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, 3, -3, 1, 1, -5, 9, -5, 1, 1, 11, -23, 19, -7, 1, 1, -21, 57, -61, 33, -9, 1, 1, 43, -135, 179, -127, 51, -11, 1, 1, -85, 313, -493, 433, -229, 73, -13, 1, 1, 171, -711, 1299, -1359, 891, -375, 99, -15, 1, 1, -341, 1593, -3309, 4017, -3141, 1641, -573, 129, -17, 1
Offset: 0

Views

Author

Wolfdieter Lang, Jan 09 2015

Keywords

Comments

The denominators are given by 2*A130321(n,m).
The rational lower triangular matrix with entries R(n,m) = T(n,m)/(2*A130321(n,m)) = T(n,m)/2^(n-m+1) for n >= m >= 0 and 0 otherwise is the inverse of the Riordan matrix A029635.
R is the rational Riordan triangle (1/(2-x), x/(1+x)).
The numerator triangle T is the Riordan array (1/(1-x), x/(1+2*x)). From the o.g.f. of the column sequences of R and T(n,m) = 2^(n-m+1)*R(n,m).
Row sums of the rational triangle R are [1/2, seq(3/2^(n+1), for n >= 1)].
Row sums of the present triangle T give [repeat(1,2,)].
Alternating row sums of the rational triangle R give (-1)^n*A102900(n)/2^(n+1), n >= 0: 1/2, -1/4, 7/8, -25/16, 103/32, -409/64, 1639/128, -6553/256, 26215/512, ... .
Alternating row sums of the present triangle T give A084567.
The inverse of the T Riordan matrix is ((1-3*x)/(1-2*x), x/(1-2*x)) = A251636.
Equals A248810 when the first column (m = 0) of ones is removed. - Georg Fischer, Jul 26 2023

Examples

			The triangle T(n,m) begins:
  n\m  0    1    2     3     4     5    6    7    8   9 ...
  0:   1
  1:   1    1
  2:   1   -1    1
  3:   1    3   -3     1
  4:   1   -5    9    -5     1
  5:   1   11  -23    19    -7     1
  6:   1  -21   57   -61    33    -9    1
  7:   1   43 -135   179  -127    51  -11    1
  8:   1  -85  313  -493   433  -229   73  -13    1
  9:   1  171 -711  1299 -1359   891 -375   99  -15   1
  ...
The rational Riordan triangle R(n,m) begins:
  n\m  0      1      2      3     4    5  ...
  0:  1/2
  1:  1/4    1/2
  2:  1/8   -1/4    1/2
  3:  1/16   3/8   -3/4    1/2
  4:  1/32  -5/16   9/8   -5/4   1/2
  5:  1/64  11/3  -23/1   19/8  -7/4  1/2
  ...
For more rows see the link.

Links

Crossrefs

Cf. A029635, A084567, A102900, A130321, A251636, A248810, A331333.

Programs

  • Maple
    A251634 := proc(n, k) local S; S := proc(n, k) option remember; `if`(k = 0, 1,
`if`(k > n, 0, S(n-1, k-1)/k - 2*S(n-1, k))) end: k!*S(n, k) end:
seq(seq(A251634(n, k), k=0..n)), n=0..9); # Peter Luschny, Jan 19 2020

Formula

O.g.f. of the row polynomials P(n,x) = Sum_{m=0..n} R(n,m)*x^m of the rational triangle R: G(z,x) = Sum_{n>=0} P(n,x)*z^n = (1+z)/((2-z)*(1+(1-x)*z)).
O.g.f. column m of the rational triangle R: (1/(2-x))*(x/(1+x))^m, m >= 0 (Riordan property of R).
O.g.f. column m of the numerator triangle T: (1/(1-x))*(x/(1+2*x))^m, m >= 0. (Riordan property of T).
T(n, k) = k!*S(n, k) where S(n, k) is recursively defined by:
if k = 0 then 1 else if k > n then 0 else S(n-1, k-1)/k - 2*S(n-1, k). - Peter Luschny, Jan 19 2020

A029637 Numbers in the (1,2)-Pascal triangle A029635 that are different from 2.

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 5, 9, 7, 1, 6, 14, 16, 9, 1, 7, 20, 30, 25, 11, 1, 8, 27, 50, 55, 36, 13, 1, 9, 35, 77, 105, 91, 49, 15, 1, 10, 44, 112, 182, 196, 140, 64, 17, 1, 11, 54, 156, 294, 378, 336, 204, 81, 19, 1, 12, 65, 210, 450, 672, 714, 540, 285, 100, 21, 1, 13, 77, 275, 660
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from David W. Wilson
Leading 1 removed as consequence of change in A029635 by Sean A. Irvine, Mar 01 2020

A029640 Even numbers in the (1,2)-Pascal triangle A029635.

Original entry on oeis.org

2, 2, 2, 4, 2, 2, 6, 14, 16, 2, 20, 30, 2, 8, 50, 36, 2, 2, 10, 44, 112, 182, 196, 140, 64, 2, 54, 156, 294, 378, 336, 204, 2, 12, 210, 450, 672, 714, 540, 100, 2, 660, 1122, 1386, 1254, 2, 14, 90, 352, 1782, 2508, 2640, 1210, 506, 144, 2, 104, 442, 4290, 5148, 1716, 650, 2, 16
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from David W. Wilson
Leading 2 inserted as consequence of change in A029635 by Sean A. Irvine, Mar 01 2020

A029641 Even numbers in the (1,2)-Pascal triangle A029635 that are different from 2.

Original entry on oeis.org

4, 6, 14, 16, 20, 30, 8, 50, 36, 10, 44, 112, 182, 196, 140, 64, 54, 156, 294, 378, 336, 204, 12, 210, 450, 672, 714, 540, 100, 660, 1122, 1386, 1254, 14, 90, 352, 1782, 2508, 2640, 1210, 506, 144, 104, 442, 4290, 5148, 1716, 650, 16, 546, 4004, 9438, 8008, 2366, 196
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from David W. Wilson

A029643 Even numbers to the right of the central numbers of the (1,2)-Pascal triangle A029635.

Original entry on oeis.org

2, 2, 2, 2, 16, 2, 2, 36, 2, 2, 196, 140, 64, 2, 336, 204, 2, 714, 540, 100, 2, 1254, 2, 2640, 1210, 506, 144, 2, 1716, 650, 2, 8008, 2366, 196, 2, 2, 37180, 30888, 20384, 10556, 4200, 1240, 256, 2, 68068, 51272, 30940, 14756, 5440, 1496, 2, 140998, 119340, 82212
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from David W. Wilson

A029647 Even numbers to the left of the central elements of the (1,2)-Pascal triangle A029635.

Original entry on oeis.org

4, 6, 14, 20, 8, 50, 10, 44, 112, 182, 54, 156, 294, 12, 210, 450, 672, 660, 1122, 14, 90, 352, 1782, 2508, 104, 442, 4290, 16, 546, 4004, 9438, 18, 152, 800, 2940, 8008, 16744, 27456, 35750, 170, 952, 3740, 10948, 24752, 44200, 63206, 20, 1122, 4692
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers

A029636 Distinct numbers in the (1,2)-Pascal triangle A029635.

Original entry on oeis.org

2, 1, 3, 4, 5, 9, 7, 6, 14, 16, 20, 30, 25, 11, 8, 27, 50, 55, 36, 13, 35, 77, 105, 91, 49, 15, 10, 44, 112, 182, 196, 140, 64, 17, 54, 156, 294, 378, 336, 204, 81, 19, 12, 65, 210, 450, 672, 714, 540, 285, 100, 21, 275, 660, 1122, 1386, 1254, 825, 385, 121, 23, 90, 352, 935
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from David W. Wilson
First two terms swapped by Sean A. Irvine, Mar 01 2020
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