A254963 a(n) = n*(11*n + 3)/2.

0, 7, 25, 54, 94, 145, 207, 280, 364, 459, 565, 682, 810, 949, 1099, 1260, 1432, 1615, 1809, 2014, 2230, 2457, 2695, 2944, 3204, 3475, 3757, 4050, 4354, 4669, 4995, 5332, 5680, 6039, 6409, 6790, 7182, 7585, 7999, 8424, 8860, 9307, 9765, 10234, 10714, 11205, 11707

Offset: 0

Author

Bruno Berselli, Feb 11 2015

Keywords

• nonn
• easy

Comments

This sequence provides the first differences of A254407 and the partial sums of A017473.

Also:

a(n) - n = A022269(n);

a(n) + n = n*(11*n+5)/2: 0, 8, 27, 57, 98, 150, 213, 287, ...;

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

a(n) + 2*n = n*(11*n+7)/2: 0, 9, 29, 60, 102, 155, 219, 294, ...;

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

a(n) + 3*n = A211013(n);

a(n) - 4*n = A226492(n);

a(n) + 4*n = A152740(n);

a(n) - 5*n = A180223(n);

a(n) + 5*n = n*(11*n+13)/2: 0, 12, 35, 69, 114, 170, 237, 315, ...;

a(n) - 6*n = A051865(n);

a(n) + 6*n = n*(11*n+15)/2: 0, 13, 37, 72, 118, 175, 243, 322, ...;

a(n) - 7*n = A152740(n-1) with A152740(-1) = 0;

a(n) + 7*n = n*(11*n+17)/2: 0, 14, 39, 75, 122, 180, 249, 329, ...;

a(n) - n*(n-1)/2 = A168668(n);

a(n) + n*(n-1)/2 = A049453(n);

a(n) - n*(n+1)/2 = A202803(n);

a(n) + n*(n+1)/2 = A033580(n).

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. A008729 and A218530 (seventh column); A017473, A254407.

Cf. similar sequences of the type 4*n^2 + k*n*(n+1)/2: A055999 (k=-7, n>6), A028552 (k=-6, n>2), A095794 (k=-5, n>1), A046092 (k=-4, n>0), A000566 (k=-3), A049450 (k=-2), A022264 (k=-1), A016742 (k=0), A022267 (k=1), A202803 (k=2), this sequence (k=3), A033580 (k=4).

Cf. A069125: (2*n+1)^2 + 3*n*(n+1)/2; A147875: n^2 + 3*n*(n+1)/2.

Cf. A022268, A022269, A049453, A051865, A152740, A168668, A180223, A211013, A226492.

Programs

Magma

[n*(11*n+3)/2: n in [0..50]];

Mathematica

Table[n (11 n + 3)/2, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{0,7,25},50] (* Harvey P. Dale, Mar 25 2018 *)

Maxima

makelist(n*(11*n+3)/2, n, 0, 50);

PARI

vector(50, n, n--; n*(11*n+3)/2)

Sage

[n*(11*n+3)/2 for n in (0..50)]

Formula

G.f.: x*(7 + 4*x)/(1 - x)^3.

From Elmo R. Oliveira, Dec 15 2024: (Start)

E.g.f.: exp(x)*x*(14 + 11*x)/2.

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