A391917 Number of irredundant sets in the n-double cone graph.

2, 10, 17, 122, 257, 577, 1850, 5626, 16901, 47525, 140626, 414737, 1223237, 3583450, 10523537, 30946970, 90973445, 267322501, 785456677, 2308418117, 6784157957, 19937157602, 58589654810, 172181842705, 506006018282, 1487038791365, 4370064820901, 12842626162922

Offset

1,1

Comments

Sequence extended to n = 1 using the recurrence/formula. - Eric W. Weisstein, Jan 04 2026

Links

Table of n, a(n) for n=1..28.

Eric Weisstein's World of Mathematics, Double Cone Graph.

Eric Weisstein's World of Mathematics, Irredundant Set.

Index entries for linear recurrences with constant coefficients, signature (3,1,-5,10,-11,-29,42,-1,-13,22,-23,1,4,-7,17,-10,-3,5,-4,1,2,-1).

Formula

a(n) = A290493(n)^2 + 1. - Christian Sievers, Dec 31 2025

From Eric W. Weisstein, Jan 04 2026: (Start)

G.f.: -((x*(-2-4*x+15*x^2-71*x^3+96*x^4+309*x^5-470*x^6+43*x^7+95*x^8-363*x^9+441*x^10-37*x^11-14*x^12+146*x^13-447*x^14+268*x^15+77*x^16-151*x^17+138*x^18-34*x^19-73*x^20+37*x^21))/((-1+x)*(1-3*x-x^2+4*x^3-x^4-2*x^5+x^6)*(-1-x-x^2-2*x^3+4*x^4+8*x^5+2*x^6+5*x^7-5*x^8-4*x^9-5*x^10-3*x^11+x^12+x^13+x^14+x^15))).

a(n) = 3*a(n-1)+a(n-2)-5*a(n-3)+10*a(n-4)-11*a(n-5)-29*a(n-6)+42*a(n-7)-a(n-8)-13*a(n-9)+22*a(n-10)-23*a(n-11)+a(n-12)+4*a(n-13)-7*a(n-14)+17*a(n-15)-10*a(n-16)-3*a(n-17)+5*a(n-18)-4*a(n-19)+a(n-20)+2*a(n-21)-a(n-22). (End)

Mathematica

Table[RootSum[1 - #^2 - #^4 - #^5 + #^6 &, #^n &]^2 + 1, {n, 20}]
Table[1 + 2 RootSum[-1 - # - #^2 - #^3 + 3 #^4 + 5 #^5 + 4 #^6 + 5 #^7 - 5 #^8 - 2 #^9 - 8 #^10 - 4 #^11 + 2 #^12 + #^13 + #^14 + #^15 &, #^n &] + RootSum[1 - 2 # - #^2 + 4 #^3 - #^4 - 3 #^5 + #^6 &, #^n &], {n, 20}]
LinearRecurrence[{3, 1, -5, 10, -11, -29, 42, -1, -13, 22, -23, 1, 4, -7, 17, -10, -3, 5, -4, 1, 2, -1}, {2, 10, 17, 122, 257, 577, 1850, 5626, 16901, 47525, 140626, 414737, 1223237, 3583450, 10523537, 30946970, 90973445, 267322501, 785456677, 2308418117, 6784157957, 19937157602}, 30]
CoefficientList[Series[-((-2 - 4 x + 15 x^2 - 71 x^3 + 96 x^4 + 309 x^5 - 470 x^6 + 43 x^7 + 95 x^8 - 363 x^9 + 441 x^10 - 37 x^11 - 14 x^12 + 146 x^13 - 447 x^14 + 268 x^15 + 77 x^16 - 151 x^17 + 138 x^18 - 34 x^19 - 73 x^20 + 37 x^21)/((-1 + x) (1 - 3 x - x^2 + 4 x^3 - x^4 - 2 x^5 + x^6) (-1 - x - x^2 - 2 x^3 + 4 x^4 + 8 x^5 + 2 x^6 + 5 x^7 - 5 x^8 - 4 x^9 - 5 x^10 - 3 x^11 + x^12 + x^13 + x^14 + x^15))), {x, 0, 20}], x]

Prog

(PARI) a(n)=polsym(1-x^2-x^4-x^5+x^6, n)[n+1]^2+1 \\ Christian Sievers, Dec 31 2025

Crossrefs

Cf. A290493, A290494, A392237.

Sequence in context: A258974 A366508 A077247 * A341032 A082939 A074069

Adjacent sequences: A391914 A391915 A391916 * A391918 A391919 A391920

Keyword

nonn,easy

Author

Eric W. Weisstein, Dec 23 2025

Extensions

a(16) onward from Christian Sievers, Dec 31 2025

a(1)-a(2) prepended by Eric W. Weisstein, Jan 04 2026

Status

approved