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A055432 Number of points in Z^n of norm <= 8. 2
1, 17, 197, 2109, 20185, 176377, 1395261, 10248133, 70554353, 458690081, 2839094517, 16837397901, 95964034121, 526432799625, 2784251496685, 14233010034069, 70491253578465, 338968561343793, 1585620669607461 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Index entries for linear recurrences with constant coefficients, signature (65, -2080, 43680, -677040, 8259888, -82598880, 696190560, -5047381560, 31966749880, -179013799328, 895068996640, -4027810484880, 16421073515280, -60992558771040, 207374699821536, -648045936942300, 1867897112363100, -4981058966301600, 12321566916640800, -28339603908273840, 60727722660586800, -121455445321173600, 227068876035237600, -397370533061665800, 651687674221131912, -1002596421878664480, 1448194831602515360, -1965407271460556560, 2507588587725537680, -3009106305270645216, 3397378086595889760, -3609714217008132870, 3609714217008132870, -3397378086595889760, 3009106305270645216, -2507588587725537680, 1965407271460556560, -1448194831602515360, 1002596421878664480, -651687674221131912, 397370533061665800, -227068876035237600, 121455445321173600, -60727722660586800, 28339603908273840, -12321566916640800, 4981058966301600, -1867897112363100, 648045936942300, -207374699821536, 60992558771040, -16421073515280, 4027810484880, -895068996640, 179013799328, -31966749880, 5047381560, -696190560, 82598880, -8259888, 677040, -43680, 2080, -65, 1).
FORMULA
From Chai Wah Wu, Jun 24 2024: (Start)
a(n) = 65*a(n-1) - 2080*a(n-2) + 43680*a(n-3) - 677040*a(n-4) + 8259888*a(n-5) - 82598880*a(n-6) + 696190560*a(n-7) - 5047381560*a(n-8) + 31966749880*a(n-9) - 179013799328*a(n-10) + 895068996640*a(n-11) - 4027810484880*a(n-12) + 16421073515280*a(n-13) - 60992558771040*a(n-14) + 207374699821536*a(n-15) - 648045936942300*a(n-16) + 1867897112363100*a(n-17) - 4981058966301600*a(n-18) + 12321566916640800*a(n-19) - 28339603908273840*a(n-20) + 60727722660586800*a(n-21) - 121455445321173600*a(n-22) + 227068876035237600*a(n-23) - 397370533061665800*a(n-24) + 651687674221131912*a(n-25) - 1002596421878664480*a(n-26) + 1448194831602515360*a(n-27) - 1965407271460556560*a(n-28) + 2507588587725537680*a(n-29) - 3009106305270645216*a(n-30) + 3397378086595889760*a(n-31) - 3609714217008132870*a(n-32) + 3609714217008132870*a(n-33) - 3397378086595889760*a(n-34) + 3009106305270645216*a(n-35) - 2507588587725537680*a(n-36) + 1965407271460556560*a(n-37) - 1448194831602515360*a(n-38) + 1002596421878664480*a(n-39) - 651687674221131912*a(n-40) + 397370533061665800*a(n-41) - 227068876035237600*a(n-42) + 121455445321173600*a(n-43) - 60727722660586800*a(n-44) + 28339603908273840*a(n-45) - 12321566916640800*a(n-46) + 4981058966301600*a(n-47) - 1867897112363100*a(n-48) + 648045936942300*a(n-49) - 207374699821536*a(n-50) + 60992558771040*a(n-51) - 16421073515280*a(n-52) + 4027810484880*a(n-53) - 895068996640*a(n-54) + 179013799328*a(n-55) - 31966749880*a(n-56) + 5047381560*a(n-57) - 696190560*a(n-58) + 82598880*a(n-59) - 8259888*a(n-60) + 677040*a(n-61) - 43680*a(n-62) + 2080*a(n-63) - 65*a(n-64) + a(n-65) for n > 64.
G.f.: (-84067061*x^64 - 594125181016*x^63 - 358570175617596*x^62 - 54327645405793088*x^61 - 3007970153301973684*x^60 - 70751321058233331176*x^59 - 708431535121140613196*x^58 - 2104975430248540461904*x^57 + 7171233759979214996564*x^56 + 26759036706964389777864*x^55 - 84577727700353397526076*x^54 - 73337885044523755892064*x^53 + 585766983871140782043948*x^52 - 749552695685318041092424*x^51 - 663783692725740379098444*x^50 + 3481577378759511387361808*x^49 - 5029057516240412205421408*x^48 + 2295741278263309504970632*x^47 + 4465915792456106045467604*x^46 - 11106570988074196912272256*x^45 + 13029987601836099080685500*x^44 - 9070951669662352149834568*x^43 + 2002580090771753282869636*x^42 + 4165095558738342937158448*x^41 - 7062324054366106555437196*x^40 + 6773432434485734087377832*x^39 - 4796019421500025574195564*x^38 + 2635532999045053804031968*x^37 - 1085317387474474986646692*x^36 + 259933601983184071258712*x^35 + 56307130989018479346756*x^34 - 116766236029728805998320*x^33 + 89058999733157906895070*x^32 - 50464950061171488516488*x^31 + 23885136813307640501836*x^30 - 9854425855777159898944*x^29 + 3617070497318070669412*x^28 - 1195315679370871226424*x^27 + 359133844367687460892*x^26 - 99368370792899666288*x^25 + 25879804048463556508*x^24 - 6579045200791562280*x^23 + 1710956693102496716*x^22 - 471046038930846112*x^21 + 136593427213299332*x^20 - 40151798043444568*x^19 + 11465968341859484*x^18 - 3084886129169232*x^17 + 767059307842552*x^16 - 173956391417768*x^15 + 35556089847068*x^14 - 6452545668608*x^13 + 1013173798292*x^12 - 129833639064*x^11 + 11162193836*x^10 + 177547664*x^9 - 332491236*x^8 + 86590648*x^7 - 15352100*x^6 + 2104096*x^5 - 227340*x^4 + 19016*x^3 - 1172*x^2 + 48*x - 1)/(x - 1)^65. (End)
MATHEMATICA
a[n_] := SeriesCoefficient[1/(1-x) Sum[x^(i^2), {i, -8, 8}]^n, {x, 0, 64}];
a /@ Range[0, 19] (* Jean-François Alcover, Sep 29 2019, from A302997 *)
CROSSREFS
Row n=8 of A302997.
Sequence in context: A069361 A177135 A130817 * A154276 A021379 A177075
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified July 21 05:28 EDT 2024. Contains 374463 sequences. (Running on oeis4.)