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 1-4 of 4 results.

A351153 Triangle read by rows: T(n, k) = n*(k - 1) - k*(k - 3)/2 with 0 < k <= n.

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 8, 10, 1, 6, 10, 13, 15, 1, 7, 12, 16, 19, 21, 1, 8, 14, 19, 23, 26, 28, 1, 9, 16, 22, 27, 31, 34, 36, 1, 10, 18, 25, 31, 36, 40, 43, 45, 1, 11, 20, 28, 35, 41, 46, 50, 53, 55, 1, 12, 22, 31, 39, 46, 52, 57, 61, 64, 66, 1, 13, 24, 34, 43, 51, 58, 64, 69, 73, 76, 78
Offset: 1

Views

Author

Stefano Spezia, Feb 02 2022

Keywords

Comments

Except for the number 2, it contains all the positive integers.

Examples

			Triangle begins:
  1;
  1, 3;
  1, 4,  6;
  1, 5,  8, 10;
  1, 6, 10, 13, 15;
  1, 7, 12, 16, 19, 21;
  1, 8, 14, 19, 23, 26, 28;
  ...

Crossrefs

Cf. A000012 (1st column), A000217 (leading diagonal), A005843 (3rd column), A006007 (sum of the first n rows), A006527 (row sums).
Cf. A087401, A141418, A351154.

Programs

  • Mathematica
    Flatten[Table[n(k-1)-k(k-3)/2,{n,12},{k,n}]]

Formula

T(n, k) = 1 + Sum_{i=1..k-1} (n - i + 1).
From R. J. Mathar, Feb 07 2022: (Start)
G.f.: x*y*(1 - x + y*x^2 + y^2*x^3)/((1 - x)^2*(1 - y*x)^3).
T(n, k) = 1 + A141418(n+1, k-1) = 1 + A087401(n+1, k-1). (End)

A138666 Numbers n such that every sum of consecutive positive numbers ending in n is not prime.

Original entry on oeis.org

1, 8, 14, 18, 20, 25, 26, 28, 32, 33, 35, 38, 39, 44, 46, 48, 50, 56, 58, 60, 62, 63, 65, 68, 72, 74, 77, 78, 80, 81, 85, 86, 88, 92, 93, 94, 95, 98, 102, 104, 105, 108, 110, 111, 116, 118, 119, 122, 123, 124, 125, 128, 130, 133, 134, 138, 140, 143, 144, 145, 146, 148
Offset: 1

Views

Author

T. D. Noe, Mar 26 2008

Keywords

Comments

Also numbers n such that all terms in row n of A087401 are not prime. Also the index of the triangular numbers in A076768. See A087572 for the least prime, if it exists. David Wasserman points out (in A087572) that n is in this sequence if and only if n and 2n-1 are both not prime. This sequence is infinite because 2k^2 is a term for all k>1.

Examples

			8 is in this sequence because 8, 15=7+8, 21=6+7+8, 26=5+6+7+8, 30=4+5+6+7+8, 33=3+4+5+6+7+8, 35=2+3+4+5+6+7+8 and 36=1+2+3+4+5+6+7+8 are all composite.

Links

Crossrefs

Cf. A010051.

Programs

  • Haskell
    a138666 n = a138666_list !! (n-1)
a138666_list = map (head . tail) $
   filter (all (== 0) . map a010051 . tail) $ drop 2 a087401_tabl
-- Reinhard Zumkeller, Oct 03 2012
  • Mathematica
    Select[Range[200], !PrimeQ[ # ] && !PrimeQ[2#-1] &]
Select[Range[150],AllTrue[Accumulate[Reverse[Range[#]]],!PrimeQ[#]&]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 18 2017 *)
  • Python
    from sympy import isprime
from itertools import accumulate
def ok(n): return all(not isprime(s) for s in accumulate(range(n, 0, -1)))
def aupto(nn): return [m for m in range(1, nn+1) if ok(m)]
print(aupto(148)) # Michael S. Branicky, Jan 08 2021

A141418 Triangle read by rows: T(n,k) = k * (2*n - k - 1) / 2, 1 <= k <= n.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 3, 5, 6, 6, 4, 7, 9, 10, 10, 5, 9, 12, 14, 15, 15, 6, 11, 15, 18, 20, 21, 21, 7, 13, 18, 22, 25, 27, 28, 28, 8, 15, 21, 26, 30, 33, 35, 36, 36, 9, 17, 24, 30, 35, 39, 42, 44, 45, 45
Offset: 1

Views

Author

Roger L. Bagula, Aug 05 2008

Keywords

Comments

From Reinhard Zumkeller, Aug 04 2014: (Start)
n-th row = half of Dynkin diagram weights for the Cartan Groups D_n.
n-th row = partial sums of n-th row of A025581. (End)

Examples

			Triangle begins as:
  0;
  1,  1;
  2,  3,  3;
  3,  5,  6,  6;
  4,  7,  9, 10, 10;
  5,  9, 12, 14, 15, 15;
  6, 11, 15, 18, 20, 21, 21;
  7, 13, 18, 22, 25, 27, 28, 28;
  8, 15, 21, 26, 30, 33, 35, 36, 36;
  9, 17, 24, 30, 35, 39, 42, 44, 45, 45;

References

  • R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

Links

Crossrefs

Cf. A025581, A087401, A141419.

Programs

  • Haskell
    a141418 n k = k * (2 * n - k - 1) `div` 2
a141418_row n = a141418_tabl !! (n-1)
a141418_tabl = map (scanl1 (+)) a025581_tabl
-- Reinhard Zumkeller, Aug 04 2014, Nov 18 2012
  • Magma
    [k*(2*n-k-1)/2: k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 30 2021
  • Maple
    A141418:= (n,k)-> k*(2*n-k-1)/2; seq(seq(A141418(n,k), k=1..n), n=1..12); # G. C. Greubel, Mar 30 2021
  • Mathematica
    T[n_, k_]= k*(2*n-k-1)/2; Table[T[n, k], {n,12}, {k,n}]//Flatten
  • Sage
    flatten([[k*(2*n-k-1)/2 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 30 2021

Formula

T(n, K) = k*(2*n - k - 1)/2.
Sum_{k=1..n} T(n, k) = 2*binomial(n+1, 3) = A007290(n+1). - Reinhard Zumkeller, Aug 04 2014

Extensions

Edited by Reinhard Zumkeller, Nov 18 2012

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 1-4 of 4 results.

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