A026037 -id:A026037 - cp's OEIS frontend

A026039 a(n) = (d(n) - r(n))/5, where d = A026037 and r is the periodic sequence with fundamental period (1,2,0,2,0).

2, 4, 8, 13, 21, 31, 44, 61, 81, 106, 135, 169, 209, 254, 306, 364, 429, 502, 582, 671, 768, 874, 990, 1115, 1251, 1397, 1554, 1723, 1903, 2096, 2301, 2519, 2751, 2996, 3256, 3530, 3819, 4124, 4444, 4781, 5134, 5504, 5892, 6297, 6721, 7163, 7624, 8105, 8605, 9126, 9667, 10229, 10813, 11418, 12046, 12696, 13369

Offset: 3

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 3..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,1,-3,3,-1).

Cf. A026037.

[Round((2*n-1)*(n^2-n+6)/30): n in [3..60]]; // Vincenzo Librandi, Jun 25 2011

f[n_] := Round[(2 n - 1)*(n^2 - n + 6)/30]; Array[f, 57, 3]
LinearRecurrence[{3,-3,1,0,1,-3,3,-1},{2,4,8,13,21,31,44,61},60] (* Harvey P. Dale, Sep 05 2023 *)

a(n) = (n + 3)*(2*n^2 + 9*n + 22)/30 - 1/5 - (-1/25*((5 - 5^(1/2))^(1/2) - (5 + 5^(1/2))^(1/2))*2^(1/2))*sin(2*n*Pi/5) - (1/25*((5 - 5^(1/2))^(1/2) + (5 + 5^(1/2))^(1/2))*2^(1/2))*sin(4*n*Pi/5). - Richard Choulet, Dec 14 2008
a(n) = round((2*n-1)*(n^2-n+6)/30) = floor((2*n^3-3*n^2+13*n)/30) = ceiling((n-1)*(2*n^2-n+12)/30) = round((n-1)*(2*n^2-n+12)/30). - Mircea Merca, Dec 03 2010
From R. J. Mathar, May 24 2010: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8).
G.f.: -x^3*(-2+2*x-2*x^2+x^3-2*x^4+3*x^5-3*x^6+x^7) / ( (x^4+x^3+x^2+x+1)*(x-1)^4 ). (End)
a(n) = a(n-5) + n^2 - 6*n + 13, n > 5, a(1)=0, a(2)=1. - Mircea Merca, Dec 03 2010

A026038 a(n) = (d(n)-r(n))/2, where d = A026037 and r is the periodic sequence with fundamental period (1,0,0,1).

Original entry on oeis.org

5, 11, 20, 33, 52, 78, 111, 152, 203, 265, 338, 423, 522, 636, 765, 910, 1073, 1255, 1456, 1677, 1920, 2186, 2475, 2788, 3127, 3493, 3886, 4307, 4758, 5240, 5753, 6298, 6877, 7491, 8140, 8825, 9548, 10310, 11111, 11952, 12835, 13761, 14730, 15743, 16802, 17908, 19061, 20262, 21513, 22815, 24168

Offset: 3

Clark Kimberling

nonn

Index entries for linear recurrences with constant coefficients, signature (4,-7,8,-7,4,-1).

G.f. x^3*( 5-9*x+11*x^2-10*x^3+7*x^4-2*x^5 ) / ( (x^2+1)*(x-1)^4 ). - R. J. Mathar, Jun 22 2013

A094414 Triangle T read by rows: dot product <1,2,...,r> * .

Original entry on oeis.org

1, 5, 4, 14, 11, 11, 30, 24, 22, 24, 55, 45, 40, 40, 45, 91, 76, 67, 64, 67, 76, 140, 119, 105, 98, 98, 105, 119, 204, 176, 156, 144, 140, 144, 156, 176, 285, 249, 222, 204, 195, 195, 204, 222, 249, 385, 340, 305, 280, 265, 260, 265, 280, 305, 340, 506, 451, 407, 374, 352, 341, 341, 352, 374, 407, 451

Offset: 0

Ralf Stephan, May 02 2004

Offset for r (the rows) is 1, for s (the columns) it is 0.

			Triangle begins as:
   1;
   5,  4;
  14, 11, 11;
  30, 24, 22, 24;
  55, 45, 40, 40, 45;
  91, 76, 67, 64, 67, 76;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Columns 0-6 are A000330, A006527, A026037, A026040, A026043, A026046, A026049.
Row sums are A000537.
See also A094415, A088003.

Flat(List([0..12], n-> List([0..n-1], k-> n*((n+1)*(2*n+1) -3*k*(n-k))/6 ))); # G. C. Greubel, Oct 30 2019

[n*((n+1)*(2*n+1) -3*k*(n-k))/6: k in [0..n-1], n in [0..12]]; // G. C. Greubel, Oct 30 2019

T:=proc(r,s) if s>=r then 0 else r*(2*r^2+3*r+1-3*r*s+3*s^2)/6 fi end: for r from 1 to 11 do seq(T(r,s),s=0..r-1) od; # yields sequence in triangular form # Emeric Deutsch, Nov 27 2006

Table[n*((n+1)*(2*n+1) -3*k*(n-k))/6, {n,0,12}, {k,0,n-1}]//Flatten (* G. C. Greubel, Oct 30 2019 *)

T(n,k) = n*((n+1)*(2*n+1) -3*k*(n-k))/6;
for(n=0,12, for(k=0,n-1, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019

[[n*((n+1)*(2*n+1) -3*k*(n-k))/6 for k in (0..n-1)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

T(r, s) = r*(2*r^2 + 3*r - 3*r*s + 1 + 3*s^2)/6, r >= 1, 0 <= s <= r-1.

More terms from G. C. Greubel, Oct 30 2019

cp's OEIS Frontend

A026039 a(n) = (d(n) - r(n))/5, where d = A026037 and r is the periodic sequence with fundamental period (1,2,0,2,0).

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A026038 a(n) = (d(n)-r(n))/2, where d = A026037 and r is the periodic sequence with fundamental period (1,0,0,1).

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A094414 Triangle T read by rows: dot product <1,2,...,r> * .

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