A026066 -id:A026066 - cp's OEIS frontend

A026068 (d(n)-r(n))/5, where d = A026066 and r is the periodic sequence with fundamental period (0,3,1,0,1).

21, 33, 49, 68, 90, 116, 145, 179, 217, 259, 306, 357, 414, 476, 543, 616, 694, 779, 870, 967, 1071, 1181, 1299, 1424, 1556, 1696, 1843, 1999, 2163, 2335, 2516, 2705, 2904, 3112, 3329, 3556, 3792, 4039, 4296, 4563, 4841, 5129, 5429, 5740, 6062, 6396, 6741

Offset: 7

Clark Kimberling

Index entries for linear recurrences with constant coefficients, signature (3, -3, 1, 0, 1, -3, 3, -1).

Cf. A152892.

LinearRecurrence[{3,-3,1,0,1,-3,3,-1},{21,33,49,68,90,116,145,179},60] (* Harvey P. Dale, Sep 10 2014 *)

a(n)=(n + 7)*(n^2 + 35*n + 90)/30 - 1/5*(1 + ( - 1/2 + 3/10*5^(1/2))*cos(2*n*Pi/5) + (1/5*2^(1/2)*(5 + 5^(1/2))^(1/2) + 1/10*2^(1/2)*(5 - 5^(1/2))^(1/2))*sin(2*n*Pi/5) + ( - 1/2 - 3/10*5^(1/2))*cos(4*n*Pi/5) + ( - 1/10*2^(1/2)*(5 + 5^(1/2))^(1/2) + 1/5*2^(1/2)*(5 - 5^(1/2))^(1/2))*sin(4*n*Pi/5)) - Richard Choulet, Dec 14 2008

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^7*(-21+30*x-13*x^2+x^3+20*x^5-29*x^6+11*x^7)/( (x^4+x^3+x^2+x+1) * (x-1)^4). - R. J. Mathar, Oct 05 2009

Corrected by T. D. Noe, Dec 11 2006

A026067 a(n) = (d(n)-r(n))/2, where d = A026066 and r is the periodic sequence with fundamental period (1,0,0,0).

Original entry on oeis.org

52, 84, 123, 170, 225, 290, 364, 448, 542, 648, 765, 894, 1035, 1190, 1358, 1540, 1736, 1948, 2175, 2418, 2677, 2954, 3248, 3560, 3890, 4240, 4609, 4998, 5407, 5838, 6290, 6764, 7260, 7780, 8323, 8890, 9481, 10098, 10740, 11408, 12102, 12824, 13573, 14350, 15155, 15990, 16854

Offset: 7

Clark Kimberling

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-3,3,-1).

Cf. A026066.

G.f.: x^7*( 52-72*x+27*x^2+x^3+74*x^5-28*x^6-52*x^4 ) / ( (1+x)*(x^2+1)*(x-1)^4 ). - R. J. Mathar, Jun 23 2013

A094415 Triangle T read by rows: dot product * .

Original entry on oeis.org

1, 4, 5, 10, 13, 13, 20, 26, 28, 26, 35, 45, 50, 50, 45, 56, 71, 80, 83, 80, 71, 84, 105, 119, 126, 126, 119, 105, 120, 148, 168, 180, 184, 180, 168, 148, 165, 201, 228, 246, 255, 255, 246, 228, 201, 220, 265, 300, 325, 340, 345, 340, 325, 300, 265, 286, 341

Offset: 0

Ralf Stephan, May 02 2004

			Triangle begins as:
   1;
   4,  5;
  10, 13, 13;
  20, 26, 28, 26;
  35, 45, 50, 50, 45;
  56, 71, 80, 83, 80, 71;

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

Columns 0-6 are A000292, A008778, A026054, A026057, A026060, A026063, A026066.
Half-diagonal is A050410.
Row sums are A000537.
See also A094414, A088003.

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

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

seq(seq( (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 , k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019

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

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

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

T(n, k) = n*(n^2 + 3*n*(1+k) + 2 - 3*k^2)/6 for n >= 0, 0 <= k <= n.

cp's OEIS Frontend

A026068 (d(n)-r(n))/5, where d = A026066 and r is the periodic sequence with fundamental period (0,3,1,0,1).

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A026067 a(n) = (d(n)-r(n))/2, where d = A026066 and r is the periodic sequence with fundamental period (1,0,0,0).

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A094415 Triangle T read by rows: dot product * .

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