OFFSET
1,1
COMMENTS
5th polygonal numbers for polygons of 5^n sides divided by 5: p(5,5^x)/5, where p(n,k) = (n/2)*(n*k - k + 4 - 2*n).
This sequence exhibits periodic digit repetition; e.g. the last digit repeats as 7, the penultimate as 4 and the antepenultimate as 2, all with a period of 1; the fourth-to-last digit repeats the sequence 1, 6 with a period of 2; the fifth-to-last repeats the sequence 3, 5, 8, 0; the sixth-to-last repeats 1, 7, 9, 5, 6, 2, 4, 0. And so on, it seems, for the other digits as the numbers grow.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (6,-5).
FORMULA
From Vincenzo Librandi, Nov 12 2011: (Start)
a(n) = 5*a(n-1) + 12.
a(n) = 6*a(n-1) - 5*a(n-2).
G.f.: (2 - 5*x + 15*x^2)/((1-x)*(1-5*x)). (End)
E.g.f.: 1 - 3*exp(x) + 2*exp(5*x). - Elmo R. Oliveira, Mar 05 2026
MAPLE
p := proc(n, k) (n/2)*(n*k-k+4-2*n) end: for x from 1 to 19 do p(5, 5^x)/5 od;
q := proc(x) 2*5^x-3 end: for x from 1 to 19 do q(x) od;
MATHEMATICA
2*5^Range[25] - 3 (* Paolo Xausa, Mar 12 2026 *)
PROG
(PARI) p(n, k) = (n/2)*(n*k-k+4-2*n);
a(n) = p(5, 5^n)/5;
(PARI) a(n) = 2*5^n - 3; \\ Harry J. Smith, Sep 13 2009
(Magma) [2*5^n-3: n in [1..30]]; // Vincenzo Librandi, Nov 12 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Daniel Dockery (drd(AT)peritus.virtualave.net), Sep 16 2001
STATUS
approved