A226492 a(n) = n*(11*n-5)/2.

0, 3, 17, 42, 78, 125, 183, 252, 332, 423, 525, 638, 762, 897, 1043, 1200, 1368, 1547, 1737, 1938, 2150, 2373, 2607, 2852, 3108, 3375, 3653, 3942, 4242, 4553, 4875, 5208, 5552, 5907, 6273, 6650, 7038, 7437, 7847, 8268, 8700, 9143, 9597, 10062, 10538, 11025, 11523

Offset: 0

Author

Bruno Berselli, Jun 11 2013

Keywords

• nonn
• easy

Comments

Sequences of numbers of the form n*(n*k - k + 6)/2:

. k from 0 to 10, respectively: A008585, A055998, A005563, A045943, A014105, A005475, A033428, A022264, A033991, A062741, A147874;

. k=11: a(n);

. k=12: A094159;

. k=13: 0, 3, 19, 48, 90, 145, 213, 294, 388, 495, 615, 748, 894, ...;

. k=14: 0, 3, 20, 51, 96, 155, 228, 315, 416, 531, 660, 803, 960, ...;

. k=15: A152773;

. k=16: A139272;

. k=17: 0, 3, 23, 60, 114, 185, 273, 378, 500, 639, 795, 968, ...;

. k=18: A152751;

. k=19: 0, 3, 25, 66, 126, 205, 303, 420, 556, 711, 885, 1078, ...;

. k=20: 0, 3, 26, 69, 132, 215, 318, 441, 584, 747, 930, 1133, ...;

. k=21: A152759;

. k=22: 0, 3, 28, 75, 144, 235, 348, 483, 640, 819, 1020, 1243, ...;

. k=23: 0, 3, 29, 78, 150, 245, 363, 504, 668, 855, 1065, 1298, ...;

. k=24: A152767;

. k=25: 0, 3, 31, 84, 162, 265, 393, 546, 724, 927, 1155, 1408, ...;

. k=26: 0, 3, 32, 87, 168, 275, 408, 567, 752, 963, 1200, 1463, ...;

. k=27: A153783;

. k=28: A195021;

. k=29: 0, 3, 35, 96, 186, 305, 453, 630, 836, 1071, 1335, 1628, ...;

. k=30: A153448;

. k=31: 0, 3, 37, 102, 198, 325, 483, 672, 892, 1143, 1425, 1738, ...;

. k=32: 0, 3, 38, 105, 204, 335, 498, 693, 920, 1179, 1470, 1793, ...;

. k=33: A153875.

Also:

a(n) - n = A180223(n);

a(n) + n = n*(11*n-3)/2 = 0, 4, 19, 45, 82, 130, 189, 259, ...;

a(n) - 2*n = A051865(n);

a(n) + 2*n = A022268(n);

a(n) - 3*n = A152740(n-1);

a(n) + 3*n = A022269(n);

a(n) - 4*n = n*(11*n-13)/2 = 0, -1, 9, 30, 62, 105, 159, 224, ...;

a(n) + 4*n = A254963(n);

a(n) - n*(n-1)/2 = A147874(n+1);

a(n) + n*(n-1)/2 = A094159(n) (case k=12);

a(n) - n*(n-1) = A062741(n) (see above, this is the case k=9);

a(n) + n*(n-1) = n*(13*n-7)/2 (case k=13);

a(n) - n*(n+1)/2 = A135706(n);

a(n) + n*(n+1)/2 = A033579(n);

a(n) - n*(n+1) = A051682(n);

a(n) + n*(n+1) = A186030(n);

a(n) - n^2 = A062708(n);

a(n) + n^2 = n*(13*n-5)/2 = 0, 4, 21, 51, 94, 150, 219, ..., etc.

Sum of reciprocals of a(n), for n > 0: 0.47118857003113149692081665034891...

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. sequences in Comments lines.

First differences are in A017425.

Cf. A033584, A180223.

Programs

Magma

[n*(11*n-5)/2: n in [0..50]];

Magma

I:=[0,3,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..46]]; // Vincenzo Librandi, Aug 18 2013

Mathematica

Table[n (11 n - 5)/2, {n, 0, 50}]
CoefficientList[Series[x (3 + 8 x) / (1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{3,-3,1},{0,3,17},50] (* Harvey P. Dale, Jan 14 2019 *)

PARI

a(n)=n*(11*n-5)/2 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: x*(3+8*x)/(1-x)^3.

a(n) + a(-n) = A033584(n).

From Elmo R. Oliveira, Dec 27 2024: (Start)

E.g.f.: exp(x)*x*(6 + 11*x)/2.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.

a(n) = n + A180223(n). (End)