[seqfan] Re: Arrays that are second differences of a 0..k array

Ron Hardin rhhardin at att.net
Mon Jul 1 15:58:22 CEST 2013


All differences seem to have the same column recurrences (at least for 1,2,3) with different limits


/tmp/dip
T(n,k)=Number of first differences of arrays of length n+1 of numbers in 0..k

(=A047969 transposed)
T(n,k)=(k+1)^(n+1)-k^(n+1)

Table starts
....3.....5......7.......9.......11........13........15.........17.........19
....7....19.....37......61.......91.......127.......169........217........271
...15....65....175.....369......671......1105......1695.......2465.......3439
...31...211....781....2101.....4651......9031.....15961......26281......40951
...63...665...3367...11529....31031.....70993....144495.....269297.....468559
..127..2059..14197...61741...201811....543607...1273609....2685817....5217031
..255..6305..58975..325089..1288991...4085185..11012415...26269505...56953279
..511.19171.242461.1690981..8124571..30275911..93864121..253202761..612579511
.1023.58025.989527.8717049.50700551.222009073.791266575.2413042577.6513215599

Empirical for column k:
k=1: a(n)=3*a(n-1)-2*a(n-2)
k=2: a(n)=5*a(n-1)-6*a(n-2)
k=3: a(n)=7*a(n-1)-12*a(n-2)
k=4: a(n)=9*a(n-1)-20*a(n-2)
k=5: a(n)=11*a(n-1)-30*a(n-2)
k=6: a(n)=13*a(n-1)-42*a(n-2)
k=7: a(n)=15*a(n-1)-56*a(n-2)
Empirical for row n:
n=1: a(n) = 2*n + 1
n=2: a(n) = 3*n^2 + 3*n + 1
n=3: a(n) = 4*n^3 + 6*n^2 + 4*n + 1
n=4: a(n) = 5*n^4 + 10*n^3 + 10*n^2 + 5*n + 1
n=5: a(n) = 6*n^5 + 15*n^4 + 20*n^3 + 15*n^2 + 6*n + 1
n=6: a(n) = 7*n^6 + 21*n^5 + 35*n^4 + 35*n^3 + 21*n^2 + 7*n + 1
n=7: a(n) = 8*n^7 + 28*n^6 + 56*n^5 + 70*n^4 + 56*n^3 + 28*n^2 + 8*n + 1

/tmp/din
T(n,k)=Number of second differences of arrays of length n+2 of numbers in 0..k

Table starts
....5......9......13.......17........21.........25........29........33.......37
...15.....49.....103......177.......271........385.......519.......673......847
...31....199.....625.....1429......2731.......4651......7309.....10825....15319
...63....665....3151.....9705.....23351......47953.....88215....149681...238735
..127...2059...14053....58141....176851.....439927....951049...1854553..3342151
..255...6305...58975...320481...1225631....3693505...9399615..21108545.43067071
..511..19171..242461..1688101...8006491...29066311..86929081.224817481.........
.1023..58025..989527..8717049..50556551..219071473.766106895...................
.2047.175099.4017157.44633821.313882531.1609259287.............................

Empirical for column k:
k=1: a(n)=3*a(n-1)-2*a(n-2) for n>3
k=2: a(n)=5*a(n-1)-6*a(n-2) for n>5
k=3: a(n)=7*a(n-1)-12*a(n-2) for n>7
k=4: a(n)=9*a(n-1)-20*a(n-2) for n>9
k=5: a(n)=11*a(n-1)-30*a(n-2) for n>11
k=6: a(n)=13*a(n-1)-42*a(n-2) for n>13
k=7: a(n)=15*a(n-1)-56*a(n-2) for n>15
Empirical for row n:
n=1: a(n) = 4*n + 1
n=2: a(n) = 10*n^2 + 4*n + 1
n=3: a(n) = 20*n^3 + 9*n^2 + 1*n + 1
n=4: a(n) = 35*n^4 + 14*n^3 - 17*n^2 + 30*n + 1
n=5: a(n) = 56*n^5 + 14*n^4 - 108*n^3 + 289*n^2 - 125*n + 1
n=6: a(n) = 84*n^6 - 402*n^4 + 1656*n^3 - 1860*n^2 + 776*n + 1
n=7: a(n) = 120*n^7 - 42*n^6 - 1158*n^5 + 6945*n^4 - 13980*n^3 + 13512*n^2 - 4887*n + 1

/tmp/dio
T(n,k)=Number of third differences of arrays of length n+3 of numbers in 0..k

Table starts
....9.....17.......25........33.........41..........49.........57.........65
...31....143......319.......565........881........1267.......1723.......2249
...63....621.....2511......6419......12947.......22727......36471......54851
..127...2059....13933.....53315.....141989......310425.....596591....1045439
..255...6305....58911....315601....1161855.....3298681....7795501...16171769
..511..19171...242461...1688101....8003363....28791007...83538705..206640895
.1023..58025...989527...8717049...50554951...218845881..761638071.2238075697
.2047.175099..4017157..44633821..313882531..1609259287.6537612649...........
.4095.527345.16245775.227363409.1932641711.11658284065......................

Empirical for column k:
k=1: a(n)=3*a(n-1)-2*a(n-2) for n>3
k=2: a(n)=5*a(n-1)-6*a(n-2) for n>5
k=3: a(n)=7*a(n-1)-12*a(n-2) for n>7
k=4: a(n)=9*a(n-1)-20*a(n-2) for n>8
k=5: a(n)=11*a(n-1)-30*a(n-2) for n>10
k=6: a(n)=13*a(n-1)-42*a(n-2) for n>12
Empirical for row n:
n=1: a(n) = 8*n + 1
n=2: a(n) = 35*n^2 + 1*n + 1 for n>1
n=3: a(n) = 112*n^3 - 34*n^2 - 38*n - 13 for n>4
n=4: a(n) = 294*n^4 - 256*n^3 - 489*n^2 + 477*n + 415 for n>8


 
rhhardin at mindspring.com
rhhardin at att.net (either)



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