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.

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A176798 Triangle read by rows: T(n,m)=1 + n*(2*m + 1 + n)/2, 0<=m<=n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 7, 10, 13, 16, 11, 15, 19, 23, 27, 16, 21, 26, 31, 36, 41, 22, 28, 34, 40, 46, 52, 58, 29, 36, 43, 50, 57, 64, 71, 78, 37, 45, 53, 61, 69, 77, 85, 93, 101, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 56, 66, 76, 86, 96, 106, 116, 126, 136, 146, 156
Offset: 0

Views

Author

Roger L. Bagula, Apr 26 2010

Keywords

Examples

			1;
2, 3;
4, 6, 8;
7, 10, 13, 16;
11, 15, 19, 23, 27;
16, 21, 26, 31, 36, 41;
22, 28, 34, 40, 46, 52, 58;
29, 36, 43, 50, 57, 64, 71, 78;
37, 45, 53, 61, 69, 77, 85, 93, 101;
46, 55, 64, 73, 82, 91, 100, 109, 118, 127;
56, 66, 76, 86, 96, 106, 116, 126, 136, 146, 156;

Links

Crossrefs

Cf. A081435 (row sums), A104249 (diagonal).

Programs

  • Maple
    A176798 := proc(n,m)
    1+n*(2*m+1+n)/2 ;
end proc: # R. J. Mathar, Feb 18 2016
  • Mathematica
    t[n_, m_] = 1 + n*(2*m + 1 + n)/2;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41
Offset: 0

Views

Author

Franck Maminirina Ramaharo, Apr 20 2018

Keywords

Comments

Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
Antidiagonals sums yield A116731.

Examples

			The array T(n,k) begins
1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012
1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027
2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843
4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777
7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767
11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861
16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957
22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993
29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770
37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173
46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341
56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401
67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605
79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991
...
The inverse binomial transforms of the columns are
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    1    2    3    4    5    6    7    8    9   10   11   12  ...
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1   1
1   2   2
1   3   4   4
1   4   6   7   7
1   5   8  10  11  11
1   6  10  13  15  16  16
1   7  12  16  19  21  22  22
1   8  14  19  23  26  28  29  29
1   9  16  22  27  31  34  36  37  37
1  10  18  25  31  36  40  43  45  46  46
...

References

  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Crossrefs

Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.

Programs

  • Maple
    T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;
  • Mathematica
    Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)
  • Maxima
    T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
    print(makelist(T(n, k), k, 0, 20));
  • PARI
    T(n,k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018

Formula

G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,0) = A152947(n+1).
T(n,1) = A000124(n).
T(n,2) = A000217(n).
T(n,3) = A034856(n+1).
T(n,4) = A052905(n).
T(n,5) = A051936(n+4).
T(n,6) = A246172(n+1).
T(n,7) = A302537(n).
T(n,8) = A056121(n+1) + 1.
T(n,9) = A056126(n+1) + 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,11) = A101859(n) + 1.
T(n,12) = A132754(n+1) + 1.
T(n,13) = A132755(n+1) + 1.
T(n,14) = A132756(n+1) + 1.
T(n,15) = A132757(n+1) + 1.
T(n,16) = A132758(n+1) + 1.
T(n,17) = A212427(n+1) + 1.
T(n,18) = A212428(n+1) + 1.
T(n,n) = A143689(n) = A300192(n,2).
T(n,n+1) = A104249(n).
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(n,n+3) = A000326(n+1).
T(n,n+4) = A095794(n+1).
T(n,n+5) = A133694(n+1).
T(n+2,n) = A005449(n+1).
T(n+3,n) = A115067(n+2).
T(n+4,n) = A133694(n+2).
T(2*n,n) = A054556(n+1).
T(2*n,n+1) = A054567(n+1).
T(2*n,n+2) = A033951(n).
T(2*n,n+3) = A001107(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
T(n,2*n) = A140066(n+1).
T(n+1,2*n) = A005891(n).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
T(2*n,2*n) = A143689(2*n).
T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
T(2*n+1,2*n) = A085473(n).
a(n+1,5*n+1)=A051865(n+1) + 1.
a(n,2*n+1) = A116668(n).
a(2*n+1,n) = A054569(n+1).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n,3*n) = A140063(n+1).
T(n+1,3*n) = A069099(n+1).
T(n,4*n) = A276819(n).
T(4*n,n) = A154106(n-1), n > 0.
T(2^n,2) = A028401(n+2).
T(1,n)*T(n,1) = A006000(n).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
T(n*(n+1)+1,n) = A294259(n+1).
T(n,n^2+1) = A081423(n).
T(n,A000217(n)) = A158842(n), n > 0.
T(n,A152947(n+1)) = A060354(n+1).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.

A319390 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=2, a(2)=3, a(3)=6, a(4)=8.

Original entry on oeis.org

1, 2, 3, 6, 8, 13, 16, 23, 27, 36, 41, 52, 58, 71, 78, 93, 101, 118, 127, 146, 156, 177, 188, 211, 223, 248, 261, 288, 302, 331, 346, 377, 393, 426, 443, 478, 496, 533, 552, 591, 611, 652, 673, 716, 738, 783, 806, 853, 877, 926, 951, 1002
Offset: 0

Views

Author

Paul Curtz, Sep 18 2018

Keywords

Comments

The bisections A104249(n) = 1, 3, 8, ... and A143689(n+1) = 2, 6, 13, 23, ... are in the following hexagonal spiral:
29--28--28--27--27
/ \
29 17--17--16--16 26
/ / \ \
30 18 9---8---8 15 26
/ / / \ \ \
30 18 9 3---3 7 15 25
/ / / / \ \ \ \
31 19 10 4 1 2 7 14 25
/ / / / / / / /
19 10 4 1---2 6 14 24
\ \ \ / / /
20 11 5---5---6 13 24
\ \ / /
20 11--12--12--13 23
\ /
21--21--22--22--23
.
a(n) mod 9 = A140265(n) mod 9.

Links

Crossrefs

Cf. A001318, A104249, A143689, A004526, A140265, A026741.

Programs

  • Mathematica
    LinearRecurrence[{1,2,-2,-1,1},{1,2,3,6,8},100] (* Paolo Xausa, Nov 13 2023 *)
  • PARI
    Vec((1 + x - x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2) + O(x^50)) \\ Colin Barker, Jun 05 2019

Formula

a(2n) = (3*n^2 + n + 2)/2. a(2n+1) = (3*n^2 + 5*n + 4)/2.
a(-n) = a(n).
a(n) = a(n-1) + A026741(n).
G.f.: (1 + x - x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2). - Colin Barker, Jun 05 2019
a(n) = 1 + A001318(n). - Peter Bala, Feb 04 2021
E.g.f.: ((8 + 7*x + 3*x^2)*cosh(x) + (9 + 5*x + 3*x^2)*sinh(x))/8. - Stefano Spezia, Feb 05 2021

A114211 a(n) = (5*n^3+12*n^2+n+6)/6.

Original entry on oeis.org

1, 4, 16, 42, 87, 156, 254, 386, 557, 772, 1036, 1354, 1731, 2172, 2682, 3266, 3929, 4676, 5512, 6442, 7471, 8604, 9846, 11202, 12677, 14276, 16004, 17866, 19867, 22012, 24306, 26754, 29361, 32132, 35072, 38186, 41479
Offset: 0

Views

Author

Paul Barry, Nov 17 2005

Keywords

Comments

Column 3 of A114202. Third differences are 1,1,7,5,5,5,5,5,... with g.f. (1+6x^2-2x^3)/(1-x).

Examples

			[1,3,9,5]=[1*1,3*1,3*3,1*5]=[C(3,0)*J(1),C(3,1)*J(2),C(3,2)*J(3),C(3,3)*J(4)].

Links

Programs

  • Mathematica
    CoefficientList[Series[(1+6x^2-2x^3)/(1-x)^4,{x,0,75}],x]  (* Harvey P. Dale, Mar 06 2011 *)

Formula

G.f.: (1+6*x^2-2*x^3)/(1-x)^4 = (1+3*(x/(1-x))+9*(x/(1-x))^2+5*(x/(1-x))^3)/(1-x).
a(n) = sum{k=0..n, C(n, k)*C(3, k)*J(k+1)} where J(n)=A001045(n).
a(0)=1, a(n)=a(n-1)+(n-1)*(n+2)+A104249(n).

A342138 Array T(n,k) = (n+k)*(3*n+3*k-5)/2 + (3*k+1), read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 3, 2, 5, 8, 7, 10, 13, 16, 15, 18, 21, 24, 27, 26, 29, 32, 35, 38, 41, 40, 43, 46, 49, 52, 55, 58, 57, 60, 63, 66, 69, 72, 75, 78, 77, 80, 83, 86, 89, 92, 95, 98, 101, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156
Offset: 0

Views

Author

Michel Marcus, Mar 01 2021

Keywords

Comments

This is an instance of a storing function on N^2 (injective) with density 1/3.

Examples

			Array begins:
   1  3   8  16  27 ...
   0  5  13  24  38 ...
   2 10  21  35  52 ...
   7 18  32  49  69 ...
  15 29  46  66  89 ...
  ...

Links

Crossrefs

Cf. A005449 (first column), A104249 (first row), A140090 (second row), A201279 (diagonal).

Programs

  • PARI
    T(n,k) = (n+k)*(3*n+3*k-5)/2 + (3*k+1);
matrix(8, 8, n, k, T(n-1, k-1))
Previous Showing 11-15 of 15 results.

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