cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A022288 a(n) = n*(31*n-1)/2.

Original entry on oeis.org

0, 15, 61, 138, 246, 385, 555, 756, 988, 1251, 1545, 1870, 2226, 2613, 3031, 3480, 3960, 4471, 5013, 5586, 6190, 6825, 7491, 8188, 8916, 9675, 10465, 11286, 12138, 13021, 13935, 14880, 15856, 16863, 17901
Offset: 0

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000217, A022289.
Cf. similar sequences of the form n*((2*k+1)*n - 1)/2: A161680 (k=0), A000326 (k=1), A005476 (k=2), A022264 (k=3), A022266 (k=4), A022268 (k=5), A022270 (k=6), A022272 (k=7), A022274 (k=8), A022276 (k=9), A022278 (k=10), A022280 (k=11), A022282 (k=12), A022284 (k=13), A022286 (k=14), this sequence (k=15).

Programs

  • Mathematica
    Table[n (31 n - 1)/2, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 15, 61}, 40] (* Harvey P. Dale, Mar 31 2014 *)
  • PARI
    a(n)=n*(31*n-1)/2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 31*n + a(n-1) - 16 for n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
a(0)=0, a(1)=15, a(2)=61; for n>2, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Mar 31 2014
G.f.: x*(15 + 16*x)/(1 - x)^3. - R. J. Mathar, Sep 02 2016
a(n) = A000217(16*n-1) - A000217(15*n-1). In general, n*((2*k+1)*n - 1)/2 = A000217((k+1)*n-1) - A000217(k*n-1), and the ordinary generating function is x*(k + (k+1)*x)/(1 - x)^3. - Bruno Berselli, Oct 14 2016
E.g.f.: (x/2)*(31*x + 30)*exp(x). - G. C. Greubel, Aug 24 2017

A178572 Numbers with ordered partitions that have periods of length 5.

Original entry on oeis.org

11, 47, 108, 194, 305, 441, 602, 788, 999, 1235, 1496, 1782, 2093, 2429, 2790, 3176, 3587, 4023, 4484, 4970, 5481, 6017, 6578, 7164, 7775, 8411, 9072, 9758, 10469, 11205, 11966, 12752, 13563, 14399, 15260, 16146, 17057, 17993, 18954, 19940, 20951
Offset: 1

Views

Author

Paul Weisenhorn, Dec 24 2010

Keywords

Comments

From each ordered partition of the numbers (10+j) with 0
The a(n) sequence begins with 11 and each member has 1 period; the b(n) = A022282(n) sequence begins with 12 and each member has 2 periods; the c(n) = A022283(n) sequence begins with 13 and each member has 2 periods; the d(n) = n*(25*n + 3)/2 sequence begins with 14 and each member has 1 period of length 5.

Examples

			For n=11 the period is [(4,3,2,1,1), (4,3,2,2), (4,3,3,1), (4,4,2,1), (5,3,2,1)].
For n=47 the period is [(9,8,7,6,6,4,3,2,1,1), (9,8,7,7,5,4,3,2,2), (9,8,8,6,5,4,3,3,1), (9,9,7,6,5,4,4,2,1), (10,8,7,6,5,5,3,2,1)].
For n=12 the 2 periods are [(4,3,2,2,1), (4,3,3,2), (4,4,3,1), (5,4,2,1), (5,3,2,1,1)] and [(4,3,3,1,1), (4,4,2,2), (5,3,3,1), (4,4,2,1,1), (5,3,2,2)].
For n=49 the 2 periods are [(9,8,7,7,6,4,3,2,2,1), (9,8,8,7,5,4,3,3,2), (9,9,8,6,5,4,4,3,1), (10,9,7,6,5,5,4,2,1), (10,8,7,6,6,5,3,2,1,1)] and [(9,8,8,6,6,4,3,3,1,1), (9,9,7,7,5,4,4,2,2),(10,8,8,6,5,5,3,3,1), (9,9,7,6,6,4,4,2,1,1), (10,8,7,7,5,5,3,2,2)].

Links

Crossrefs

Cf. A092964, A037306.

Programs

Formula

G.f. for a(n): (11 + 14*x)/(1-x)^3.
for b(n): (12 + 13*x)/(1-x)^3.
for c(n): (13 + 12*x)/(1-x)^3.
for d(n): (14 + 11*x)/(1-x)^3.
All sequences have the same recurrence
s(n+3) = 3*s(n+2) - 3*s(n+1) + s(n)
with s(0)=0, s(1) = 10 + j, s(2) = 45 + 2*j and 0
s(n) = n*(25*n - 5 + 2*j)/2 and 0
The general formula for numbers with periods of length k: a(k,j,n) = n*(k^2*n - k + 2*j)/2 with 0
For j=1 and j=(k-1) the numbers have 1 period.
For 1A092964(k-4,j-1) periods.
G.f.: (binomial(k,2)*(1+x) + j + (k-j)*x)/(1-x)^3.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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