A093445 -id:A093445 - cp's OEIS frontend

A093446 Largest member of the n-th row of the triangular triangle (A093445).

1, 3, 9, 18, 33, 54, 82, 120, 165, 225, 294, 378, 476, 588, 720, 865, 1035, 1221, 1430, 1662, 1914, 2197, 2499, 2835, 3195, 3585, 4008, 4456, 4947, 5463, 6021, 6612, 7239, 7910, 8610, 9366, 10153, 10989, 11868, 12788, 13764, 14775, 15850, 16965, 18135

Offset: 1

Amarnath Murthy, Apr 02 2004

nonn

The largest terms is near the middle term at about 42.265%. Lim a(n) -> 0.19245*n^3

			The row for n = 4 is (1+2+3+4), (5+6+7), (8+9), 10 => 10 18 17 10. The largest member is 18 hence a(4) = 18.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A093445, A001571.

a093446 = maximum . a093445_row  -- Reinhard Zumkeller, Oct 03 2012

T[n_] := n(n + 1)/2; TT[n_, k_] := T[k*n - T[k - 1]] - T[(k - 1)*n - T[k - 2]]; Max[ # ] & /@ Table[ TT[n, k], {n, 45}, {k, n}]

Edited, corrected and extended by Robert G. Wilson v, Apr 24 2004

A005920 Tricapped prism numbers.

Original entry on oeis.org

1, 9, 33, 82, 165, 291, 469, 708, 1017, 1405, 1881, 2454, 3133, 3927, 4845, 5896, 7089, 8433, 9937, 11610, 13461, 15499, 17733, 20172, 22825, 25701, 28809, 32158, 35757, 39615, 43741, 48144, 52833, 57817, 63105, 68706, 74629, 80883, 87477, 94420

Offset: 0

N. J. A. Sloane

a(n) = (n+1)*A000326(n+1) - Sum_{i=0...n} A001477(i) = (n+1)*((n+1)*(3*n+2)/2) - A000217(n) = (n+1)*(3*n^2+4n+2)/2. - Bruno Berselli, Apr 25 2010

Also central terms of triangle A093445: a(n) = A093445(2*n+1,n+1). - Reinhard Zumkeller, Oct 03 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. for recursive method [Ar(m) is the m-th term of a sequence in the OEIS] a(n) = n*Ar(n) - A000217(n-1) or a(n) = (n+1)*Ar(n+1) - A000217(n) or similar: A081436, A005945, A006003 and the terms T(2, n) or T(3, n) in the sequence A125860. - Bruno Berselli, Apr 25 2010

Cf. A000326, A001477, A093445.

a005920 n = (n * (n * (3 * n + 7) + 6) + 2) `div` 2
-- Reinhard Zumkeller, Oct 03 2012

[(3*n^3+7*n^2+6*n+2)/2 : n in [0..50]]; // Wesley Ivan Hurt, May 05 2021

a:=n->(3*n^3+7*n^2+6*n+2)/2: seq(a(n),n=0..60);
A005920:=(1+5*z+3*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

CoefficientList[ Series[(1+5x+3x^2)/(1-x)^4, {x, 0, 39}], x] (* Jean-François Alcover, Dec 02 2011, after Simon Plouffe *)
LinearRecurrence[{4,-6,4,-1},{1,9,33,82},40] (* Harvey P. Dale, Sep 25 2012 *)

a(n)=n*(3*n^2+7*n+6)/2+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (1/2) * (3*n^3 + 7*n^2 + 6*n + 2). - Ralf Stephan, Apr 20 2004
a(0)=1, a(1)=9, a(2)=33, a(3)=82, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Sep 25 2012
E.g.f.: exp(x)*(2 + 16*x + 16*x^2 + 3*x^3)/2. - Stefano Spezia, Jun 10 2022

More terms from Emeric Deutsch, May 09 2004

A177708 Pentagonal triangle.

Original entry on oeis.org

1, 6, 12, 18, 57, 51, 40, 156, 209, 145, 75, 330, 531, 534, 330, 126, 600, 1074, 1278, 1122, 651, 196, 987, 1895, 2488, 2559, 2081, 1162, 288, 1512, 3051, 4275, 4824, 4563, 3537, 1926, 405, 2196, 4599, 6750, 8100, 8370, 7506, 5634, 3015

Offset: 1

Jonathan Vos Post, Dec 11 2010

This is to A093445 as pentagonal numbers A000326 are to triangular numbers A000217. The n-th row of the triangular table begins by considering A000217(n) pentagonal numbers (starting with 1) in order. Now segregate them into n chunks beginning with n members in the first chunk, n-1 members in the second chunk, and so forth. Now sum each chunk. Thus the first term is the sum of first n numbers = n*(3n-1)/2, the second term is the sum of the next n-1 terms (from n+1 to 2n-1), the third term is the sum of the next n-2 terms (2n to 3n-3)... This triangle can be called the pentagonal triangle. The sequence contains the triangle by rows. The first column is A002411 (Pentagonal pyramidal numbers: n^2*(n+1)/2).

			The row for n = 4 is (1+5+12+22), (35+51+70), (92+117), 145 => 40, 156, 209, 145.
    1;
    6,   12;
   18,   57,   51;
   40,  156,  209,   145;
   75,  330,  531,   534,   330;
  126,  600, 1074,  1278,  1122,   651;
  196,  987, 1895,  2488,  2559,  2081,  1162;
  288, 1512, 3051,  4275,  4824,  4563,  3537,  1926;
  405, 2196, 4599,  6750,  8100,  8370,  7506,  5634, 3015;
  550, 3060, 6596, 10024, 12570, 13775, 13450, 11631, 8534, 4510;

Cf. A000217, A000326, A002411, A093445, A236770 (right border).

A000326 :=proc(n) n*(3*n-1)/2 ; end proc:
A177708 := proc(n,k) kc := 1 ; nsk := n ; ns := 1 ; while kc < k do ns := ns+nsk ; kc := kc+1 ; nsk := nsk-1 ; end do: add(A000326(i),i=ns..ns+nsk-1) ; end proc: # R. J. Mathar, Dec 14 2010

Table[Total/@TakeList[PolygonalNumber[5,Range[60]],Range[n,1,-1]],{n,10}]//Flatten (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Feb 17 2018 *)

T(n,1) = A002411(n).
T(n,2) = n*(n-1)*(7*n-2)/2.
T(n,3) = (n-2)*(19*n^2-26*n+9)/2 = Sum_{i=2n..3(n-1)} A000326(i).

A093447 Triangle a(n,k) read by rows n which contain columns k=1,2,..,n, where each entry is the product of numbers (k-1)n-T(k-2)+1 through kn-T(k-1).

Original entry on oeis.org

1, 2, 3, 6, 20, 6, 24, 210, 72, 10, 120, 3024, 1320, 182, 15, 720, 55440, 32760, 4896, 380, 21, 5040, 1235520, 1028160, 175560, 13800, 702, 28, 40320, 32432400, 39070080, 7893600, 657720, 32736, 1190, 36, 362880, 980179200, 1744364160

Offset: 1

Amarnath Murthy, Apr 02 2004

This is built by starting from the sequence 1,2,....,T(n) in row n, where T(n) is the triangular number A000217(n) and packaging its first n, the next n-1, the next n-2,... up to the last number in groups and writing down the product of each group in one cell of the triangle. The first column is A000142. The second column is essentially A006963. The 3rd column is essentially A001763. The diagonal is A000217. - R. J. Mathar, Jul 26 2007

			In factorized notation the triangle starts
1;
1*2, 3;
1*2*3, 4*5, 6;
1*2*3*4, 5*6*7, 8*9, 10;
1*2*3*4*5, 6*7*8*9, 10*11*12, 13*14, 15;
which gives
1;
2, 3;
6, 20, 6;
24, 210, 72, 10;
120, 3024, 1320, 182, 15;
720,55440,32760, 4896, 380, 21;

Cf. A093445, A093446, A093448.

A000217 := proc(n) n*(n+1)/2 ; end: A093447 := proc(n,k) factorial(k*n-A000217(k-1))/factorial((k-1)*n-A000217(k-2)) ; end: for n from 1 to 16 do for k from 1 to n do printf("%d, ",A093447(n,k)) ; od ; od: # R. J. Mathar, Jul 26 2007

a(n,k)= [k*n-T(k-1)]!/[(k-1)*n-T(k-2)]! where T(n)=A000217(n). - R. J. Mathar, Jul 26 2007

More terms from R. J. Mathar, Jul 26 2007

A093448 Rows sums of the triangle A093447.

Original entry on oeis.org

1, 5, 32, 316, 4661, 94217, 2458810, 80128082, 3193424921, 153067911301, 8685693546692, 574691476630760, 43735137898763917, 3784250198022172001, 368841694500041857646, 40194470526005627873182

Offset: 1

Amarnath Murthy, Apr 02 2004

nonn

			The row for n = 4 is
(1*2*3*4), (5*6*7), (8*9), 10 or
24 210 72 10.
hence a(4) = 24 +210 +72 +10 = 316.

Cf. A093445, A093446, A093447.

A000217 := proc(n) n*(n+1)/2 ; end: A093447 := proc(n,k) factorial(k*n-A000217(k-1))/factorial((k-1)*n-A000217(k-2)) ; end: A093448 := proc(n) add( A093447(n,k),k=1..n) ; end: for n from 1 to 26 do printf("%d, ",A093448(n)) ; od: # R. J. Mathar, Jul 27 2007

More terms from R. J. Mathar, Jul 27 2007

cp's OEIS Frontend

A093446 Largest member of the n-th row of the triangular triangle (A093445).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A005920 Tricapped prism numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

A177708 Pentagonal triangle.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A093447 Triangle a(n,k) read by rows n which contain columns k=1,2,..,n, where each entry is the product of numbers (k-1)*n-T(k-2)+1 through k*n-T(k-1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A093448 Rows sums of the triangle A093447.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Extensions

A093447 Triangle a(n,k) read by rows n which contain columns k=1,2,..,n, where each entry is the product of numbers (k-1)n-T(k-2)+1 through kn-T(k-1).