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 21 results. Next

A028263 Elements in 3-Pascal triangle A028262 (by row) that are not 1.

Original entry on oeis.org

3, 4, 4, 5, 8, 5, 6, 13, 13, 6, 7, 19, 26, 19, 7, 8, 26, 45, 45, 26, 8, 9, 34, 71, 90, 71, 34, 9, 10, 43, 105, 161, 161, 105, 43, 10, 11, 53, 148, 266, 322, 266, 148, 53, 11, 12, 64, 201, 414, 588, 588, 414, 201, 64, 12, 13, 76, 265, 615, 1002, 1176, 1002, 615, 265, 76, 13
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Comments

Rows of triangle formed using Pascal's rule except begin and end n-th row with n+3.

Examples

			The triangle T(n,k) begins
n\k  0  1   2   3    4    5    6   7   8  9 10 ...
0:   3
1:   4  4
2:   5  8   5
3:   6 13  13   6
4:   7 19  26  19    7
5:   8 26  45  45   26    8
6:   9 34  71  90   71   34    9
7:  10 43 105 161  161  105   43  10
8:  11 53 148 266  322  266  148  53  11
9:  12 64 201 414  588  588  414 201  64 12
10: 13 76 265 615 1002 1176 1002 615 265 76 13
... Reformatted. - _Wolfdieter Lang_, Jun 28 2015

Links

Crossrefs

Row sums give A051633(n).

Programs

  • Haskell
    a028263 n k = a028263_tabl !! n !! k
a028263_row n = a028263_tabl !! n
a028263_tabl = zipWith (zipWith (+)) a007318_tabl a014410_tabl
-- Reinhard Zumkeller, Mar 12 2012

Formula

T(n,k) = A007318(n,k) + A014410(n+2,k+1). [Reinhard Zumkeller, Mar 12 2012]

Extensions

More terms from James Sellers

A028270 Central elements in 3-Pascal triangle A028262 (by row).

Original entry on oeis.org

1, 3, 8, 26, 90, 322, 1176, 4356, 16302, 61490, 233376, 890188, 3409588, 13104756, 50517200, 195234120, 756197910, 2934686610, 11408741520, 44420399100, 173191792620, 676104403260, 2642356838160, 10337529691320, 40481034410700
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Comments

Or, start with Pascal's triangle; a(n) is the sum of the numbers on the periphery of the n-th central triangle containing exactly 3 numbers. The first three triangles are
...1...........2.........6
.1...1.......3...3.....10..10
and the corresponding sums are 3, 8 and 26. - Amarnath Murthy, Mar 25 2003
This sequence starting at a(n+2) has Hankel transform A000032(2n+1)*2^n (empirical observation). - Tony Foster III, May 20 2016

Crossrefs

Cf. A081494, A081495, A081496, A000984.
Cf. A000108.

Programs

  • Maple
    seq(binomial(2*n,n)+binomial(2*n-2,n-1),n=0..24);
seq(2*binomial(2*n-1,n-1)+binomial(2*n-2,n-1),n=1..24);

Formula

G.f.: (x+1)/sqrt(1-4*x). - Vladeta Jovovic, Jan 08 2004
a(n) = binomial(2n, n)+binomial(2n-2, n-1)=A000984(n)+A000984(n-1). - Emeric Deutsch, Apr 20 2004
a(n) = 2binomial(2n-1, n-1)+binomial(2n-2, n-1). - Emeric Deutsch, Apr 20 2004
a(n) = (n+1)*C(n) + n*C(n-1), C = Catalan number (A000108). - Gary W. Adamson, Dec 28 2007
G.f.: G(0) where G(k)= 1 + x/(1 - (4*k+2)/((4*k+2) + (k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 24 2012
D-finite with recurrence n*a(n) -3*n*a(n-1) +2*(-2*n+5)*a(n-2)=0. - R. J. Mathar, May 01 2024

Extensions

More terms from James Sellers

A028266 Even elements in 3-Pascal triangle A028262 (by row).

Original entry on oeis.org

4, 4, 8, 6, 6, 26, 8, 26, 26, 8, 34, 90, 34, 10, 10, 148, 266, 322, 266, 148, 12, 64, 414, 588, 588, 414, 64, 12, 76, 1002, 1176, 1002, 76, 14, 880, 2178, 2178, 880, 14, 430, 4356, 430, 16, 118, 3718, 6292, 6292, 3718, 118, 16, 134, 2184, 10010, 16302, 10010
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers

A028273 Even elements to right of central elements in 3-Pascal triangle A028262.

Original entry on oeis.org

4, 6, 26, 8, 34, 10, 266, 148, 588, 414, 64, 12, 1002, 76, 2178, 880, 14, 430, 6292, 3718, 118, 16, 10010, 2184, 134, 18, 55198, 39832, 22932, 10388, 3620, 936, 116688, 95030, 62764, 33320, 14008, 4556, 188, 20, 211718, 157794, 96084, 47328, 18564
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers

A028264 Odd elements in 3-Pascal triangle A028262 (by row).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 5, 1, 1, 13, 13, 1, 1, 7, 19, 19, 7, 1, 1, 45, 45, 1, 1, 9, 71, 71, 9, 1, 1, 43, 105, 161, 161, 105, 43, 1, 1, 11, 53, 53, 11, 1, 1, 201, 201, 1, 1, 13, 265, 615, 615, 265, 13, 1, 1, 89, 341, 1617, 1617, 341, 89, 1, 1, 15, 103, 1221, 2497, 3795, 3795
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers

A028265 Odd elements in 3-Pascal triangle A028262 (by row) that are not 1.

Original entry on oeis.org

3, 5, 5, 13, 13, 7, 19, 19, 7, 45, 45, 9, 71, 71, 9, 43, 105, 161, 161, 105, 43, 11, 53, 53, 11, 201, 201, 13, 265, 615, 615, 265, 13, 89, 341, 1617, 1617, 341, 89, 15, 103, 1221, 2497, 3795, 3795, 2497, 1221, 103, 15, 533, 1651, 8151, 8151, 1651, 533, 17, 651
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers

A028267 Number of distinct elements in 3-Pascal triangle A028262 (by row).

Original entry on oeis.org

1, 3, 4, 5, 8, 6, 13, 7, 19, 26, 45, 9, 34, 71, 90, 10, 43, 105, 161, 11, 53, 148, 266, 322, 12, 64, 201, 414, 588, 76, 265, 615, 1002, 1176, 14, 89, 341, 880, 1617, 2178, 15, 103, 430, 1221, 2497, 3795, 4356, 16, 118, 533, 1651, 3718, 6292, 8151, 17, 134, 651
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers

A028268 Distinct odd elements in 3-Pascal triangle A028262 (by row).

Original entry on oeis.org

1, 3, 5, 13, 7, 19, 45, 9, 71, 43, 105, 161, 11, 53, 201, 265, 615, 89, 341, 1617, 15, 103, 1221, 2497, 3795, 533, 1651, 8151, 17, 651, 5369, 14443, 151, 785, 2835, 7553, 15379, 24453, 30745, 169, 1105, 21, 1293, 5661, 229, 1501, 24225, 23, 251, 8455
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers

A028269 Distinct even elements in 3-Pascal triangle A028262 (by row).

Original entry on oeis.org

4, 8, 6, 26, 34, 90, 10, 148, 266, 322, 12, 64, 414, 588, 76, 1002, 1176, 14, 880, 2178, 430, 4356, 16, 118, 3718, 6292, 134, 2184, 10010, 16302, 18, 936, 3620, 10388, 22932, 39832, 55198, 61490, 20, 188, 4556, 14008, 33320, 62764, 95030, 116688
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers

A028271 Elements to right of central elements in 3-Pascal triangle A028262.

Original entry on oeis.org

1, 1, 4, 1, 5, 1, 13, 6, 1, 19, 7, 1, 45, 26, 8, 1, 71, 34, 9, 1, 161, 105, 43, 10, 1, 266, 148, 53, 11, 1, 588, 414, 201, 64, 12, 1, 1002, 615, 265, 76, 13, 1, 2178, 1617, 880, 341, 89, 14, 1, 3795, 2497, 1221, 430, 103, 15, 1, 8151, 6292, 3718, 1651, 533, 118, 16, 1
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers
Showing 1-10 of 21 results. Next

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

Content is available under The OEIS End-User License Agreement.