login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=0..71.

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.

EXAMPLE

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

...

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A118032 A089692 A066201 * A193820 A216368 A123956

Adjacent sequences:  A303270 A303271 A303272 * A303274 A303275 A303276

KEYWORD

nonn,tabl

AUTHOR

Franck Maminirina Ramaharo, Apr 20 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 05:14 EST 2019. Contains 329839 sequences. (Running on oeis4.)