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 1-6 of 6 results.

A005476 a(n) = n*(5*n - 1)/2.

Original entry on oeis.org

0, 2, 9, 21, 38, 60, 87, 119, 156, 198, 245, 297, 354, 416, 483, 555, 632, 714, 801, 893, 990, 1092, 1199, 1311, 1428, 1550, 1677, 1809, 1946, 2088, 2235, 2387, 2544, 2706, 2873, 3045, 3222, 3404, 3591
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

a(n) is half the number of ways to divide an n X n square into 3 rectangles whose side-lengths are integers. See Matthew Scroggs link. - George Witty, Feb 06 2024

Links

Crossrefs

Cf. A000217, A005475, A033994, A294833.
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488.
Cf. similar sequences listed in A022288.
Cf. A016754, A133694, A000290.

Programs

Formula

a(n) = C(5*n,2)/5 for n>=0. - Zerinvary Lajos, Jan 02 2007
a(n) = A033991(n) - A000326(n). - Zerinvary Lajos, Jun 11 2007
a(n) = a(n-1) + 5*n - 3 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = A000217(n) + A000384(n) = A000290(n) + A000326(n). - Omar E. Pol, Jan 11 2013
a(n) = A130520(5*n+1). - Philippe Deléham, Mar 26 2013
a(n) = A033994(n) - A033994(n-1). - J. M. Bergot, Jun 12 2013
From Bruno Berselli, Oct 17 2016: (Start)
G.f.: x*(2 + 3*x)/(1 - x)^3.
a(n) = A000217(3*n-1) - A000217(2*n-1). (End)
E.g.f.: x*(4 + 5*x)*exp(x)/2. - G. C. Greubel, Jul 30 2019
Sum_{n>=1} 1/a(n) = 2 * A294833. - Amiram Eldar, Nov 16 2020
From Leo Tavares, Nov 20 2021: (Start)
a(n) = A016754(n) - A133694(n+1). See Triangulated Diamonds illustration.
a(n) = A000290(n) + A000217(n) + 2*A000217(n-1)
a(n) = 2*A000217(n) + 3*A000217(n-1). (End)

A081266 Staggered diagonal of triangular spiral in A051682.

Original entry on oeis.org

0, 6, 21, 45, 78, 120, 171, 231, 300, 378, 465, 561, 666, 780, 903, 1035, 1176, 1326, 1485, 1653, 1830, 2016, 2211, 2415, 2628, 2850, 3081, 3321, 3570, 3828, 4095, 4371, 4656, 4950, 5253, 5565, 5886, 6216, 6555, 6903, 7260, 7626, 8001, 8385, 8778, 9180
Offset: 0

Views

Author

Paul Barry, Mar 15 2003

Keywords

Comments

Staggered diagonal of triangular spiral in A051682, between (0,4,17) spoke and (0,7,23) spoke.
Binomial transform of (0, 6, 9, 0, 0, 0, ...).
If Y is a fixed 3-subset of a (3n+1)-set X then a(n) is the number of (3n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007
Partial sums give A085788. - Leo Tavares, Nov 23 2023

Examples

			a(1)=9*1+0-3=6, a(2)=9*2+6-3=21, a(3)=9*3+21-3=45.
For n=3, a(3) = -0^2+1^2-2^2+3^2-4^2+5^2-6^2+7^2-8^2+9^2 = 45.

Links

Crossrefs

Cf. A000217, A000290, A005449, A014105, A022266, A033585, A062725, A144312, A144314, A218470.
Cf. A003154, A133694.

Programs

Formula

a(n) = 6*C(n,1) + 9*C(n,2).
a(n) = 3*n*(3*n+1)/2.
G.f.: (6*x+3*x^2)/(1-x)^3.
a(n) = A000217(3*n); a(2*n) = A144314(n). - Reinhard Zumkeller, Sep 17 2008
a(n) = 3*A005449(n). - R. J. Mathar, Mar 27 2009
a(n) = 9*n+a(n-1)-3 for n>0, a(0)=0. - Vincenzo Librandi, Aug 08 2010
a(n) = A218470(9n+5). - Philippe Deléham, Mar 27 2013
a(n) = Sum_{k=0..3n} (-1)^(n+k)*k^2. - Bruno Berselli, Aug 29 2013
E.g.f.: 3*exp(x)*x*(4 + 3*x)/2. - Stefano Spezia, Jun 06 2021
From Amiram Eldar, Aug 11 2022: (Start)
Sum_{n>=1} 1/a(n) = 2 - Pi/(3*sqrt(3)) - log(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(3*sqrt(3)) + 4*log(2)/3 - 2. (End)
From Leo Tavares, Nov 23 2023: (Start)
a(n) = 3*A000217(n) + 3*A000290(n).
a(n) = A003154(n+1) - A133694(n+1). (End)

A166873 a(n) = a(n-1) + 12*n for n > 1; a(1) = 1.

Original entry on oeis.org

1, 25, 61, 109, 169, 241, 325, 421, 529, 649, 781, 925, 1081, 1249, 1429, 1621, 1825, 2041, 2269, 2509, 2761, 3025, 3301, 3589, 3889, 4201, 4525, 4861, 5209, 5569, 5941, 6325, 6721, 7129, 7549, 7981, 8425, 8881, 9349, 9829, 10321, 10825, 11341, 11869
Offset: 1

Views

Author

Klaus Brockhaus, Oct 22 2009

Keywords

Comments

Binomial transform of 1,24,12,0,0,0,....

Links

Crossrefs

Cf. A008594 (multiples of 12).
A000217, A028387, A133694, A059993, A166137, A166143, A166146, A166147, A166148, A166150, A166144 have recurrence a(n-1)+k*n with a(1)=1 or a(0)=1 for k = 1..11 resp.

Programs

  • Magma
    [ n eq 1 select 1 else Self(n-1)+12*n: n in [1..44] ];
  • Mathematica
    LinearRecurrence[{3,-3,1},{1,25,61},50] (* G. C. Greubel, May 27 2016 *)
  • PARI
    a(n)=6*n^2+6*n-11 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 6*n^2 + 6*n - 11.
a(n) = 2*a(n-1) - a(n-2) + 12.
G.f.: x*(1 + 22*x - 11*x^2)/(1-x)^3.
a(n) - a(n-1) = A008594(n) for n > 1.
From G. C. Greubel, May 27 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (-11 + 12*x + 6*x^2)*exp(x) + 11. (End)

A179436 a(n) = (3*n+7)*(3*n+2)/2.

Original entry on oeis.org

7, 25, 52, 88, 133, 187, 250, 322, 403, 493, 592, 700, 817, 943, 1078, 1222, 1375, 1537, 1708, 1888, 2077, 2275, 2482, 2698, 2923, 3157, 3400, 3652, 3913, 4183, 4462, 4750, 5047, 5353, 5668, 5992, 6325, 6667, 7018, 7378, 7747, 8125, 8512, 8908, 9313, 9727, 10150
Offset: 0

Views

Author

Paul Curtz, Jan 12 2011

Keywords

Comments

Trisection of A055998.

Links

Crossrefs

Cf. A000217, A016777, A016789, A055998, A133694, A153466.

Programs

Formula

G.f.: (-7-4*x+2*x^2)/(x-1)^3.
a(n) = a(n-1) + 9*(n+1) = (14 + 27*n + 9*n^2)/2.
a(n) = 2*a(n-1) - a(n-2) + 9.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) mod 9 = A153466(n) mod 9 = 7.
Sum_{n>=0} 1/a(n) = 1/2-2*Pi*sqrt(3)/45 = 0.2581600... - R. J. Mathar, Apr 07 2011
a(n) = A133694(n+1) + 6*A000217(n+1). - Leo Tavares, Mar 24 2022
Sum_{n>=0} (-1)^n/a(n) = 3/10 - 4*log(2)/15. - Amiram Eldar, Mar 27 2022
From Elmo R. Oliveira, Oct 30 2024: (Start)
E.g.f.: exp(x)*(7 + 18*x + 9*x^2/2).
a(n) = A016777(n+2)*A016789(n)/2. (End)

A133981 Triangle read by rows: A000012 * A127701 + A127701 * A000012 - A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 4, 3, 5, 6, 5, 6, 7, 8, 7, 7, 8, 9, 10, 9, 8, 9, 10, 11, 12, 11, 9, 10, 11, 12, 13, 14, 13, 10, 11, 12, 13, 14, 15, 16, 15, 11, 12, 13, 14, 15, 16, 17, 18, 17, 12, 13, 14, 15, 16, 17, 18, 19, 20, 19, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 23
Offset: 1

Views

Author

Gary W. Adamson, Sep 30 2007

Keywords

Comments

Row sums = A133694: (1, 7, 16, 28, 43, 61, ...).

Examples

			First few rows of the triangle:
   1;
   4,  3;
   5,  6,  5;
   6,  7,  8,  7;
   7,  8,  9, 10,  9;
   8,  9, 10, 11, 12, 11;
   9, 10, 11, 12, 13, 14, 13;
  10, 11, 12, 13, 14, 15, 16, 15;
  11, 12, 13, 14, 15, 16, 17, 18, 17;
  ...

Crossrefs

Cf. A127701, A133694.

Extensions

a(26) = 13 corrected and more terms from Georg Fischer, Jun 07 2023

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.
Showing 1-6 of 6 results.

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

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