OFFSET
0,3
COMMENTS
Numbers k for which 184*k + 441 is a square. - Bruno Berselli, Jul 10 2018
Partial sums of A317321. - Omar E. Pol, Jul 28 2018
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
From Colin Barker, Jul 10 2018: (Start)
G.f.: x*(1 + 21*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = n*(23*n + 42)/8 for n even.
a(n) = (23*n - 19)*(n + 1)/8 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
(End)
Sum_{n>=1} 1/a(n) = 46/441 + 2*Pi*cot(2*Pi/23)/21. - Amiram Eldar, Mar 01 2022
MAPLE
seq(coeff(series(x*(x^2+21*x+1)/((1-x)^3*(1+x)^2), x, n+1), x, n), n=0..50); # Muniru A Asiru, Jul 10 2018
MATHEMATICA
CoefficientList[Series[x (1 + 21 x + x^2)/((1 - x)^3*(1 + x)^2), {x, 0, 49}], x] (* or *)
Array[PolygonalNumber[25, (1 - 2 Boole[EvenQ@ #]) Ceiling[#/2]] &, 50, 0] (* Michael De Vlieger, Jul 10 2018 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 22, 25, 67}, 50] (* Robert G. Wilson v, Jul 15 2018 *)
PROG
(PARI) concat(0, Vec(x*(1 + 21*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jul 10 2018
(GAP) a:=[0, 1, 22, 25, 67];; for n in [6..50] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]-a[n-4]+a[n-5]; od; a; # Muniru A Asiru, Jul 10 2018
CROSSREFS
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), this sequence (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Jul 10 2018
STATUS
approved