cp's OEIS Frontend

Empirical: a(n) = a(n-1) + a(n-2) + a(n-3) for n > 4.

G.f.: 1 - 1/x - 1/x^2 + 1/x^2/G(0), where G(k)= 1 - (2*k+1)*x/(1 - x/(x - (2*k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 09 2013

Empirical: a(n) = 2*(A001590(n) + A001590(n-1) + A001590(n-2)) for n > 1. - Peter M. Chema, Feb 03 2017

From Gregory L. Simay, Jun 23 2017: (Start)

a(n) = A000073(n+2) - A000073(n-2), the difference of two tribonacci numbers. The corresponding g.f. is (1 - x^4)/(1 - x - x^2 - x^3). E.g.: a(10) = A000073(12) - A000073(8) = 274 - 24 = 250.

The tribonacci formula arises from considering the number of compositions of n where only the order of parts 1,2,3 matters (part of an upcoming paper), which may be denoted by C(n [4). We are convolving the number of partitions of n with parts >3 with the tribonacci numbers. The number of partitions of n with parts greater than 3 is P(n) - P(n-1) - P(n-2) + P(n-4) + P(n-5) - P(n-6). (Derived from the corresponding gf which is (1-x)(1-x^2)(1-x^3)gfP(x).) The rest is algebra. It looks like C(n, [4) = P(n) + Sum_{j=0..n-3} P(n-3-j)*A196700(j+1). (End)

Empirical: a(n) = a(n-1) +4*a(n-2) +16*a(n-3) +9*a(n-4) -a(n-5) +2*a(n-6) +2*a(n-7) for n>8.

Empirical g.f.: x*(2 + 9*x + 19*x^2 + 22*x^3 + 6*x^4 + 7*x^5 + x^6 - 2*x^7) / (1 - x - 4*x^2 - 16*x^3 - 9*x^4 + x^5 - 2*x^6 - 2*x^7). - Colin Barker, May 09 2018

Empirical: a(n) = 2*a(n-1) +16*a(n-2) +54*a(n-3) +114*a(n-4) +239*a(n-5) +95*a(n-6) -1499*a(n-7) -3689*a(n-8) -5467*a(n-9) -343*a(n-10) +12342*a(n-11) +11754*a(n-12) -2955*a(n-13) +3006*a(n-14) +15969*a(n-15) +11712*a(n-16) +381*a(n-17) +5853*a(n-18) +8837*a(n-19) +2089*a(n-20) -4023*a(n-21) +2697*a(n-22) -1309*a(n-23) +137*a(n-24) -187*a(n-25) +46*a(n-26) -41*a(n-27) -4*a(n-28) -a(n-29) for n>30

Empirical: a(n) = 3*a(n-1) +44*a(n-2) +418*a(n-3) +2123*a(n-4) +9666*a(n-5) +17867*a(n-6) -31103*a(n-7) -197975*a(n-8) -230889*a(n-9) -1128188*a(n-10) -10294716*a(n-11) -25363963*a(n-12) +10109804*a(n-13) +103009787*a(n-14) -41521218*a(n-15) -540648165*a(n-16) -490445197*a(n-17) +298496144*a(n-18) +581581325*a(n-19) -791636376*a(n-20) -6650364072*a(n-21) +7765218732*a(n-22) +63237542944*a(n-23) +47129253858*a(n-24) -50164097213*a(n-25) +221166151571*a(n-26) -68300476076*a(n-27) -1507350365119*a(n-28) +2286027931208*a(n-29) -1615011002135*a(n-30) -2064723761796*a(n-31) +10965745389067*a(n-32) -12557336858803*a(n-33) +12100326294111*a(n-34) -2941555694638*a(n-35) -5301314787748*a(n-36) -521040003542*a(n-37) -2211986829512*a(n-38) -684539722837*a(n-39) -11363446274967*a(n-40) +15750252315140*a(n-41) -2531664165706*a(n-42) -4802852996620*a(n-43) +18524903969559*a(n-44) -9899802947654*a(n-45) +9044730601964*a(n-46) +1636031611457*a(n-47) -4283599340178*a(n-48) +9289438091383*a(n-49) -13032765921092*a(n-50) -6827048634321*a(n-51) +5970500374922*a(n-52) -16177927247769*a(n-53) +5186623239157*a(n-54) -2348762640686*a(n-55) +2481169040671*a(n-56) -1056794241201*a(n-57) -989891859322*a(n-58) +1027346613037*a(n-59) -670566409353*a(n-60) +543900035351*a(n-61) +84211547551*a(n-62) +18236259246*a(n-63) +38945802085*a(n-64) -4280014106*a(n-65) +4341538315*a(n-66) +11264153476*a(n-67) +8779125376*a(n-68) -3365069472*a(n-69) +3792604437*a(n-70) -1532663159*a(n-71) -385210547*a(n-72) -83439919*a(n-73) -102572613*a(n-74) -5981192*a(n-75) +9692966*a(n-76) +2503246*a(n-77) -3875313*a(n-78) +1001324*a(n-79) -542833*a(n-80) -6802*a(n-81) +11947*a(n-82) +587*a(n-83) -260*a(n-84) -10*a(n-85) for n>87