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A257171
Sum of numbers on n-th segment of Ulam's spiral.
1
1, 5, 9, 13, 27, 36, 62, 78, 120, 145, 207, 243, 329, 378, 492, 556, 702, 783, 965, 1065, 1287, 1408, 1674, 1818, 2132, 2301, 2667, 2863, 3285, 3510, 3992, 4248, 4794, 5083, 5697, 6021, 6707, 7068, 7830, 8230, 9072, 9513, 10439, 10923, 11937, 12466, 13572, 14148, 15350
OFFSET
0,2
COMMENTS
From Ulam's spiral, consider successive segments s(0) = [1]; s(1) = [2, 3]; s(2) = [4, 5]; s(3) = [6, 7]; s(4) = [8, 9, 10] and so on. a(n) is sum of numbers of the segment s(n). The first differences are A000290 interleaved with 2*A002061. See illustration in the links.
FORMULA
a(0) = 1; a(1) = 5; a(2) = 9; for n >= 3, a(n) = a(n-1)+((n-1)/2+1)^2, if n = even; otherwise a(n) = a(n-1)+2*((n-1)^2/4+(n-1)+7/4).
From Colin Barker, Apr 17 2015: (Start)
a(n) = (n^3+5*n^2+14*n+16)/8 for n even and n>1.
a(n) = (n^3+4*n^2+11*n+8)/8 for n odd and n>1.
G.f.: (2*x^8-3*x^7-5*x^6+9*x^5+5*x^4-8*x^3+x^2+4*x+1) / ((x-1)^4*(x+1)^3).
(End)
MATHEMATICA
f[n_] := Block[{t = {5, 9}}, For[i = 3, i <= n, i++, If[OddQ@ i, AppendTo[t, t[[i - 1]] + ((i - 1)/2 + 1)^2], AppendTo[t, t[[i - 1]] + 2 ((i - 1)^2/4 + (i - 1) + 7/4)]]]; {1}~Join~t]; f@ 48(* Michael De Vlieger, Apr 17 2015 *)
PROG
(PARI) a(n) = if(n<=0, 1, if(n<=1, 5, if(n<=2, 9, if(Mod(n, 2)==0, a(n-1)+2*((n-1)^2/4+(n-1)+7/4), a(n-1)+((n-1)/2+1)^2))))
for (n=0, 100, print1(a(n), ", "))
(PARI) Vec((2*x^8-3*x^7-5*x^6+9*x^5+5*x^4-8*x^3+x^2+4*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Apr 18 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Apr 17 2015
STATUS
approved