A005920 -id:A005920 - cp's OEIS frontend

A187673 Partial sums of the tricapped prism numbers A005920.

1, 10, 43, 125, 290, 581, 1050, 1758, 2775, 4180, 6061, 8515, 11648, 15575, 20420, 26316, 33405, 41838, 51775, 63385, 76846, 92345, 110078, 130250, 153075, 178776, 207585, 239743, 275500, 315115, 358856

Offset: 0

Jonathan Vos Post, Mar 12 2011

Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A005920.

Accumulate[LinearRecurrence[{4,-6,4,-1},{1,9,33,82},40]] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,10,43,125,290},40] (* Harvey P. Dale, Feb 15 2015 *)

a(n) = Sum_{i=0..n} A005920(i).
a(n) = (n+2)*(n+1)*(9*n^2 + 19*n + 12)/24.
a(n) = A002419(n+1) + A050534(n+1).
G.f.: ( -1-5*x-3*x^2 ) / (x-1)^5. - R. J. Mathar, Mar 29 2011

Typo in formula fixed by Colin Barker, Apr 19 2013

A005945 Number of n-step mappings with 4 inputs.

Original entry on oeis.org

0, 1, 15, 60, 154, 315, 561, 910, 1380, 1989, 2755, 3696, 4830, 6175, 7749, 9570, 11656, 14025, 16695, 19684, 23010, 26691, 30745, 35190, 40044, 45325, 51051, 57240, 63910, 71079, 78765, 86986, 95760, 105105, 115039, 125580, 136746

Offset: 0

N. J. A. Sloane

a(n) is the coefficient of x^4/4! in n-th iteration of exp(x)-1.

			G.f. = x + 15*x^2 + 60*x^3 + 154*x^4 + 315*x^5 + 561*x^6 + 910*x^7 + ...

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)
B. A. Huberman, T. H. Hogg, & N. J. A. Sloane, Correspondence, 1985
Pierpaolo Natalini, Paolo E. Ricci, Integer Sequences Connected with Extensions of the Bell Polynomials, Journal of Integer Sequences, 2017, Vol. 20, #17.10.2.
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, A005920, A006003 and the terms T(2, n) or T(3, n) in the sequence A125860. [Bruno Berselli, Apr 25 2010]

Cf. A094952.

I:=[0, 1, 15, 60]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi Jun 18 2012

LinearRecurrence[{4,-6,4,-1},{0,1,15,60},50] (* Vincenzo Librandi, Jun 18 2012 *)
a[ n_] := 3 n^3 - 5/2 n^2 + 1/2 n; (* Michael Somos, Jun 10 2015 *)

{a(n) = 3*n^3 - 5/2*n^2 + 1/2*n}; /* Michael Somos, Jan 23 2014 */

G.f.: x*(1+11*x+6*x^2)/(1-x)^4. a(n)=n*(3*n-1)*(2*n-1)/2.
For n>0, a(n) = n*A000567(n) - A000217(n-1). - Bruno Berselli, Apr 25 2010; Feb 01 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 18 2012
a(n) = -A094952(-n) for all n in Z. - Michael Somos, Jan 23 2014

Edited by Michael Somos, Oct 29 2002

A125860 Rectangular table where column k equals row sums of matrix power A097712^k, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 12, 4, 1, 1, 86, 69, 22, 5, 1, 1, 698, 612, 178, 35, 6, 1, 1, 9551, 8853, 2251, 365, 51, 7, 1, 1, 226592, 217041, 46663, 5990, 651, 70, 8, 1, 1, 9471845, 9245253, 1640572, 161525, 13131, 1057, 92, 9, 1, 1, 705154187

Offset: 0

Paul D. Hanna, Dec 13 2006

Triangle A097712 satisfies: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1) for n > 0, k > 0, with A097712(n,0)=A097712(n,n)=1 for n >= 0. Column 1 equals A016121, which counts the sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.

T(2, n) = (n+1)*A005408(n) - Sum_{i=0..n} A001477(i) = (n+1)*(2*n+1) - A000217(n) = (n+1)*(3*n+2)/2; T(3, n) = (n+1)*A001106(n+1) - Sum_{i=0..n} A001477(i) = (n+1)*((n+1)*(7*n+2)/2) - A000217(n) = (n+1)*(7*n^2 + 8*n + 2)/2. - Bruno Berselli, Apr 25 2010

			Recurrence is illustrated by:
  T(4,1) = T(3,1) + T(3,2) = 17 + 69 = 86;
  T(4,2) = T(3,2) + T(3,3) + T(3,4) = 69 + 178 + 365 = 612;
  T(4,3) = T(3,3) + T(3,4) + T(3,5) + T(3,6) = 178 + 365 + 651 + 1057 = 2251.
Rows of this table begin:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,...;
  1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, ...;
  1, 17, 69, 178, 365, 651, 1057, 1604, 2313, 3205, 4301, 5622, 7189,..;
  1, 86, 612, 2251, 5990, 13131, 25291, 44402, 72711, 112780, 167486,..;
  1, 698, 8853, 46663, 161525, 435801, 996583, 2025458, 3768273, ...;
  1, 9551, 217041, 1640572, 7387640, 24530016, 66593821, 156664796, ...;
  1, 226592, 9245253, 100152049, 586285040, 2394413286, 7713533212, ...;
  1, 9471845, 695682342, 10794383587, 82090572095, 412135908606, ...;
  1, 705154187, 93580638024, 2079805452133, 20540291522675, ...;
  1, 94285792211, 22713677612832, 723492192295786, 9278896006526795,...;
  1, 22807963405043, 10025101876435413, 458149292979837523, ...;
  ...
where column k equals the row sums of matrix power A097712^k for k >= 0.
Triangle A097712 begins:
  1;
  1,      1;
  1,      3,       1;
  1,      8,       7,       1;
  1,     25,      44,      15,       1;
  1,    111,     346,     208,      31,      1;
  1,    809,    4045,    3720,     912,     63,     1;
  1,  10360,   77351,   99776,   35136,   3840,   127,   1;
  1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255; ...
where A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1);
e.g., A097712(5,2) = A097712(4,2) + [A097712^2](4,1) = 44 + 302 = 346.
Matrix square A097712^2 begins:
     1;
     2,     1;
     5,     6,     1;
    17,    37,    14,     1;
    86,   302,   193,    30,    1;
   698,  3699,  3512,   881,   62,   1;
  9551, 73306, 96056, 34224, 3777, 126, 1; ...
Matrix cube A097712^3 begins:
       1;
       3,      1;
      12,      9,      1;
      69,     87,     21,      1;
     612,   1146,    447,     45,    1;
    8853,  22944,  12753,   2019,   93,   1;
  217041, 744486, 549453, 120807, 8595, 189, 1; ...

Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 16.

Cf. A097712; columns: A016121, A125862, A125863, A125864, A125865; A125861 (diagonal), A125859 (antidiagonal sums). Variants: A125790, A125800.

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) and similar: A081436, A005920, A005945, A006003. - Bruno Berselli, Apr 25 2010

T[n_, k_] := T[n, k] = If[Or[n == 0, k == 0], 1, Sum[T[n - 1, j + k], {j, 0, k}]];
Table[T[#, k] &[n - k + 1], {n, 0, 9}, {k, 0, n + 1}] (* Michael De Vlieger, Dec 10 2024, after PARI *)

T(n,k)=if(n==0 || k==0,1,sum(j=0,k,T(n-1,j+k)))

T(n,k) = Sum_{j=0..k} T(n-1, j+k) for n > 0, with T(0,n)=T(n,0)=1 for n >= 0.

A093445 The triangular triangle.

Original entry on oeis.org

1, 3, 3, 6, 9, 6, 10, 18, 17, 10, 15, 30, 33, 27, 15, 21, 45, 54, 51, 39, 21, 28, 63, 80, 82, 72, 53, 28, 36, 84, 111, 120, 114, 96, 69, 36, 45, 108, 147, 165, 165, 150, 123, 87, 45, 55, 135, 188, 217, 225, 215, 190, 153, 107, 55, 66, 165, 234, 276, 294, 291, 270, 234

Offset: 1

Amarnath Murthy, Apr 02 2004

The n-th row of the triangular table begins by considering n triangular numbers (A000217) in order. Now segregate them into n groups beginning with n members in the first group, n-1 members in the second group, etc. Now sum each group. Thus the first term is the sum of first n numbers = n(n+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), etc. and the last term is simply n(n+1)/2. This triangle can be called a triangular triangle. The sequence contains the triangle by rows.

			Triangle begins:
   1
   3,  3
   6,  9,   6
  10, 18,  17,  10
  15, 30,  33,  27,  15
  21, 45,  54,  51,  39, 21
  28, 63,  80,  82,  72, 53, 28
  36, 84, 111, 120, 114, 96, 69, 36
The row for n = 4 is (1+2+3+4), (5+6+7), (8+9), 10 => 10 18 17 10.

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Cf. A000217, A093446. TT(n, 2) = A045943. TT(n, n-1) = A014209. TT(0, k) = A027480.

Cf. A005920 (central terms), A002817 (row sums).

a093445 n k = a093445_row n !! (k-1)
a093445_row n = f [n, n - 1 .. 1] [1 ..] where
   f [] _      = []
   f (x:xs) ys = sum us : f xs vs where (us,vs) = splitAt x ys
a093445_tabl = map a093445_row [1 ..]
-- Reinhard Zumkeller, Oct 03 2012

A093445 := proc(n,k)
    A000217(k*n-A000217(k-1))-A000217((k-1)*n-A000217(k-2)) ;
end proc:
seq(seq(A093445(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Dec 09 2015

T[n_] := n(n + 1)/2; TT[n_, k_] := T[k*n - T[k - 1]] - T[(k - 1)*n - T[k - 2]]; Flatten[ Table[ TT[n, k], {n, 1, 11}, {k, 1, n}]] (* Robert G. Wilson v, Apr 24 2004 *)
Table[Total/@TakeList[Range[(n(n+1))/2],Range[n,1,-1]],{n,20}]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 15 2019 *)

T(n) = A000217(n) is the n-th Triangular number. TT(n, k) is the k-th term of the n-th row, 0 < k <= n.
TT(n, k) = T(k*n - T(k - 1)) - T((k - 1)*n - T(k - 2)).
TT(n, 1) = TT(n, n) = T(n) = A000217(n).

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

A100119 a(n) = n-th centered n-gonal number.

Original entry on oeis.org

1, 2, 7, 19, 41, 76, 127, 197, 289, 406, 551, 727, 937, 1184, 1471, 1801, 2177, 2602, 3079, 3611, 4201, 4852, 5567, 6349, 7201, 8126, 9127, 10207, 11369, 12616, 13951, 15377, 16897, 18514, 20231, 22051, 23977, 26012, 28159, 30421, 32801, 35302

Offset: 0

Jonathan Vos Post, Dec 26 2004

a(n) is n times the n-th triangular number plus 1. - Thomas M. Green, Nov 16 2009
From Gary W. Adamson, Jul 31 2010: (Start)
Equals (1, 2, 3, 4, ...) convolved with (1, 0, 4, 7, 10, 13, ...).
Example: a(5) = 76 = (6, 5, 4, 3, 2, 1) dot (1, 0, 4, 7, 10, 13) = (6 + 0 + 16 + 21 + 20 + 13). (End)

			a(2) = 2*3 + 1 = 7, a(3) = 3*6 + 1 = 19, a(4) = 4*10 + 1 = 41. - _Thomas M. Green_, Nov 16 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

See also A101357 (Cumulative sums of the n-th n-gonal numbers).
A diagonal of A101321.
Cf. A060354, A098547, A101357.
Cf. A006003, A005920.

I:=[1, 2, 7, 19]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 25 2012

Table[(n^3+n^2)/2+1,{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)
LinearRecurrence[{4,-6,4,-1},{1,2,7,19},40] (* Vincenzo Librandi, Jun 25 2012 *)

a(n) = n^2*(n+1)/2+1; \\ Altug Alkan, Sep 21 2018

a(n) = 1 + n*(n + n^2)/2 = 1 + (1/2)*n^2 + (1/2) * n^3 = 1 + mean(n^2, n^3). - Joshua Zucker, May 03 2006
Equals A002411(n) + 1. - Olivier Gérard, Jun 20 2007
G.f.: (1 - 2*x + 5*x^2 - x^3) / (x-1)^4. - R. J. Mathar, Apr 04 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 25 2012
a(n) = (A098547(n)+1)/2. - Richard Turk, Jul 18 2017
a(n) = A060354(n+2) - A000290(n+1) = A006003(n+1) - A005563(n) and for n>0 A005920(n) - A068601(n+1). - Bruce J. Nicholson, Jun 23 2018

Corrected and extended by Joshua Zucker, May 03 2006

cp's OEIS Frontend

A187673 Partial sums of the tricapped prism numbers A005920.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A005945 Number of n-step mappings with 4 inputs.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A125860 Rectangular table where column k equals row sums of matrix power A097712^k, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A093445 The triangular triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A100119 a(n) = n-th centered n-gonal number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions