OFFSET
1,1
COMMENTS
From each ordered partition of the numbers (10+j) with 0<j<5 one removes the first part z(1) and adds 1 to the next z(1) parts to get a new partition until a period is reached.
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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<j<5.
s(n) = n*(25*n - 5 + 2*j)/2 and 0<j<5.
The general formula for numbers with periods of length k: a(k,j,n) = n*(k^2*n - k + 2*j)/2 with 0<j<k.
For j=1 and j=(k-1) the numbers have 1 period.
For 1<j<(k-1) the numbers have A092964(k-4,j-1) periods.
G.f.: (binomial(k,2)*(1+x) + j + (k-j)*x)/(1-x)^3.
EXAMPLE
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)].
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {11, 47, 108}, 50] (* Harvey P. Dale, Jan 14 2019 *)
Table[n*(25*n-3)/2, {n, 1, 50}] (* G. C. Greubel, Jan 30 2019 *)
PROG
(PARI) a(n)=n*(25*n-3)/2 \\ Charles R Greathouse IV, Jun 18 2017
(Magma) [n*(25*n-3)/2: n in [1..50]]; // G. C. Greubel, Jan 30 2019
(Sage) [n*(25*n-3)/2 for n in (1..50)] # G. C. Greubel, Jan 30 2019
(GAP) List([1..50], n -> n*(25*n-3)/2); # G. C. Greubel, Jan 30 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, Dec 24 2010
STATUS
approved