A139097 formula and extension correct?

[Jonathan Post]

The recent re-editing of this and related seqs by njas makes me enquire:
Is this extension and formula correct?

 A139097 0,4,8,13,21,30,36,45,54,63,73,85,95,105,119,137
add these:
148, 168, 188, 209, 223, 244, 263,283, 304, 320, 338, 361, 381, 402,
416, 434, 455, 475, 497, 519, 538, 560, 576, 597

a(0) = 0, a(n+1) = a(n) + A005589(a(n)).

If so, the French and other language analogues should have analogous formulae.

Is it worth my submitting something (antidiagonals, or main diagonal)
of the array
A(k,0) = k, A(k,n+1) = A(k,n) + A005589(A(k,n))? That eliminates the
arbitrariness of the "seed" a(0).

Thanks,

Jonathan Vos Post




jvp> From seqfan-owner at ext.jussieu.fr  Sun Jun 15 03:45:41 2008
jvp> Date: Sat, 14 Jun 2008 18:44:35 -0700
jvp> From: "Jonathan Post" <jvospost3 at gmail.com>
jvp> To: "Sequence Fans" <seqfan at ext.jussieu.fr>
jvp> Subject: A139097 formula and extension correct?
jvp> 
jvp> The recent re-editing of this and related seqs by njas makes me enquire:
jvp> Is this extension and formula correct?
jvp> 
jvp>  A139097 0,4,8,13,21,30,36,45,54,63,73,85,95,105,119,137
jvp> add these:
jvp> 148, 168, 188, 209, 223, 244, 263,283, 304, 320, 338, 361, 381, 402,
jvp> 416, 434, 455, 475, 497, 519, 538, 560, 576, 597
jvp> 
jvp> a(0) = 0, a(n+1) = a(n) + A005589(a(n)).
jvp> ...

Using the formula and b005589.txt of the fifth Wilson,
I get a different extension of A139097, that is

0,4,8,13,21,30,36,45,54,63,73,85,95,105,119,137,158,178,200,211,227,248,268,288,
309,325,347,369,390,408,424,445,465,485,506,520,537,559,579,601,614,632,651,669,
688,709,725,747,769,790,808,825,847,869,890,908,924,945,965,985,1006,1020,1037,1059,
1079,1101,1125,1156,1185,1216,1244,1274,1306,1332,1364,1396,1428,1461,1491,1522,
1553,1585,1617,1647,1678,1711,1740,1768,1801,1827,1861,1892,1924,1956,1986,2017,
2037,2059,2079,2101,2125,2156,2185,2216,2244,2274,2306,2332,2364,2396,2428,2461,
2491,2522,2553,2585,2617,2647,2678,2711,2740,2768,2801,2827,2861,2892,2924,2956,
2986,3017,3039,3062,3083,3107,3135,3168,3201,3227,3261,3292,3324,3359,3393,3429,
3463,3497,3532,3565,3598,3633,3667,3700,3726,3760,3790,3821,3855,3889,3924,3958,

The next term after 137 is 137+A005589(137) = 137+21=158 etc.

Richard