几个vb小程序

ysr

n=1 d=8n^2-7=1^2, w=0
n=2 d=8n^2-7=5^2, w=3
n=4 d=8n^2-7=11^2, w=15
n=11 d=8n^2-7=31^2, w=120
n=23 d=8n^2-7=65^2, w=528
n=64 d=8n^2-7=181^2, w=4095
n=134 d=8n^2-7=379^2, w=17955
n=373 d=8n^2-7=1055^2, w=139128
n=781 d=8n^2-7=2209^2, w=609960

ysr

Private Sub Command1_Click()
Dim a, b, c
a = Val(Text1)
b = 0

Do While b <= a
b1 = Int((b + 1) / 2)
w = Val(((4 + Sqr(2)) * (1 + (-1) ^ b * Sqr(2)) ^ (2 * b1) + (4 - Sqr(2)) * (1 - (-1) ^ b * Sqr(2)) ^ (2 * b1)) ^ 2 / 64 - 1)
s1 = s1 & " w=" & w & vbCrLf
b = Val(b + 1)

Loop
Text2 = s1
End Sub
Private Sub Command2_Click()
Text1 = ""
Text2 = ""
End Sub

ysr

w=0
w=3
w=15
w=120
w=528
w=4095
w=17955
w=139128
w=609959.999999999
w=4726275
w=20720703
w=160554240
w=703893959.999999
w=5454117902.99999
w=23911673955
w=185279454480
w=812293020527.999
w=6294047334434.99
w=27594051024014.9
w=213812329916328
w=937385441795998

ysr

设 w 为三角数，且 w+1 =完全平方数，求 w 的通解公式（蔡家雄命题）

n=1 d=8n^2-7=1^2, w=0
n=2 d=8n^2-7=5^2, w=3
n=4 d=8n^2-7=11^2, w=15
n=11 d=8n^2-7=31^2, w=120
n=23 d=8n^2-7=65^2, w=528
n=64 d=8n^2-7=181^2, w=4095
n=134 d=8n^2-7=379^2, w=17955
n=373 d=8n^2-7=1055^2, w=139128
n=781 d=8n^2-7=2209^2, w=609960

其中m=(√d-1)/2，w=m(m+1)/2
我不会推导w的通项公式

ysr

w=0
w=3
w=15
w=120
w=528
w=4095
w=17955
w=139128
w=609959.999999999
w=4726275
w=20720703
w=160554240
w=703893959.999999
w=5454117902.99999
w=23911673955
w=185279454480
w=812293020527.999
w=6294047334434.99
w=27594051024014.9
w=213812329916328
w=937385441795998
w=7.26332516982072E+15
w=3.18435109700399E+16
w=2.46739243443988E+17
w=1.08174198753956E+18
w=8.38187095192578E+18
w=3.6747384065375E+19
w=2.84736873122032E+20
w=1.24832931623521E+21
w=9.67267181519717E+21
w=4.24064493679318E+22

ysr

w=0
w=3
w=15
w=120
w=528
w=4095
w=17955
w=139128
w=609959.999999999
w=4726275
w=20720703
w=160554240
w=703893959.999999
w=5454117902.99999
w=23911673955
w=185279454480
w=812293020527.999
w=6294047334434.99
w=27594051024014.9
w=213812329916328
w=937385441795998
w=7.26332516982072E+15
w=3.18435109700399E+16
w=2.46739243443988E+17
w=1.08174198753956E+18
w=8.38187095192578E+18
w=3.6747384065375E+19
w=2.84736873122032E+20
w=1.24832931623521E+21
w=9.67267181519717E+21
w=4.24064493679318E+22
w=3.28586104843582E+23
w=1.44057094919345E+24
w=1.11622548928666E+25
w=4.89370058232092E+25
w=3.7918808025262E+26
w=1.66241762703992E+27
w=1.28812324736962E+28
w=5.64732623135341E+28
w=4.37582716025418E+29
w=1.91842850103312E+30
w=1.48649311123905E+31
w=6.51700957728124E+31
w=5.04970075105252E+32
w=2.21386482777459E+33
w=1.71541176224662E+34
w=7.52062340485632E+34
w=5.82735029088745E+35
w=2.55479809282337E+36
w=1.97958368713949E+37
w=8.67879289219461E+37
w=6.72475718598336E+38
w=2.94823478525335E+39
w=2.2844378595472E+40
w=1.00153194769692E+41
w=7.76036396527451E+41
w=3.40226038738427E+42
w=2.63623931033378E+43
w=1.15576699976295E+44
w=8.95545329116959E+44
w=3.92620553880666E+45
w=3.04221787968733E+46
w=1.3337541161945E+47
w=1.03345853376457E+48
w=4.53083778952249E+48
w=3.51071679691986E+49
w=1.53915109432145E+50
w=1.19261025241899E+51
w=5.22858288290341E+51
w=4.05136414142764E+52
w=1.77617902909284E+53
w=1.37627119783298E+54
w=6.03378011603275E+54
w=4.6752707084907E+55
w=2.04970906042204E+56
w=1.588215769689E+57
w=6.96297702531891E+57
w=5.39525834623412E+58
w=2.36536247954801E+59
w=1.83279962194991E+60
w=8.0352694534379E+60
w=6.22612345628347E+61
w=2.72962625168934E+62
w=2.11504917551443E+63
w=9.2726939862903E+63
w=7.18494107329277E+64
w=3.14998632908701E+65
w=2.44076491574403E+66
w=1.07006808249096E+67
w=8.2914157724564E+67
w=3.63508149414016E+68
w=2.81664059771943E+69
w=1.23485763992516E+70
w=9.5682866164736E+70
w=4.19488089425141E+71
w=3.25040080900331E+72
w=1.42502464640556E+73
w=1.10417944639948E+74
w=4.84088891688464E+74
w=3.75095971694922E+75
w=1.64447720709437E+76
输入值是100以内的，包括0应该是101项。

ysr

设 w 为三角数，且 7w+1 =完全平方数，求 w 的通解公式。
{120, 107880, 96876240，96876240, 86994755760, 78121193796360}

7w+1为完全平方数的情况，如下是程序得到的前21项：（据朋友搞出来的通项公式编程的）

w=0
w=120
w=107880
w=96876240
w=86994755760
w=78121193796360
w=7.01527450343756E+16
w=6.29970869196755E+19
w=5.65713139011235E+22
w=5.0800976886122E+25
w=4.56192206724236E+28
w=4.09660093628595E+31
w=3.67874307886272E+34
w=3.30350718821779E+37
w=2.96654577627649E+40
w=2.6639548035891E+43
w=2.39222844707724E+46
w=2.14821848152055E+49
w=1.92909780417701E+52
w=1.73232767993247E+55
w=1.55562832748156E+58

ysr

代码如下：

Private Sub Command1_Click()
Dim a, b, c
a = Val(Text1)
b = 0

Do While b <= a
b1 = Int((b + 1) / 2)
w = Val(((15 + 4 * Sqr(14)) ^ (2 * b + 1) + (15 - 4 * Sqr(14)) ^ (2 * b + 1) - 30) / 224)
s1 = s1 & " w=" & w & vbCrLf
b = Val(b + 1)

Loop
Text2 = s1
End Sub
Private Sub Command2_Click()
Text1 = ""
Text2 = ""
End Sub

ysr

验证：7*0+1=1，√1=1，
7*120+1=841，√841=29，
7*107880+1=755161，√755161=869.

ysr

5w+1为完全平方数的情况：

n=1d=40n^2-15=5^2, w=0
n=4d=40n^2-15=25^2, w=3
n=34d=40n^2-15=215^2, w=231
n=151d=40n^2-15=955^2, w=4560
n=1291d=40n^2-15=8165^2, w=333336