[seqfan] Re: Resurrect A090566?

Sun Nov 22 23:21:18 CET 2015

Dear David,

	Now that we know that our algorithm is working properly in Mathematica, how about the following sequences:

A090566: a(1) = 1, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
  1, 6, 25, 281, 961, 6201, 59409, 187600, 730641, 4429444, 28600025, 85336064, 468650384, 4590568025, 23901253604, 36922256164, 228378872384, 519390415729, 3999576229761, 22053449580964, 52752598923921, 67153745961316, 346596997521321, 2205389504844676, 32117901134901281 

a(1) = 2, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
{2, 5, 29, 241, 1809, 6516, 27729, 70281, 191236, 537636, 5052601, 24352064, 50491721, 335176900, 816286736, 1584582656, 5835352241, 31064957504, 299026078001, 368254999225, 916181280225, 6283970794161, 31966212255489, 247575988078441, 558234718638336} 

a(1) = 3, {3, 6, 25, 281, 961, merges into A090566

a(1) = 4, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
{4, 9, 61, 504, 4516, 47504, 382025, 3975209, 33057329, 80214016, 454665681, 4507966404, 44168848384, 69005350809, 163894140625, 784386132324, 5954843762641, 7954794246144, 53996843222416, 69176076458289, 379510987739761, 1641640879622564, 7593632535763529, 31733339799107600, 187824533234499236}

a(1) = 5, {5, 29, 241, 1809, -> a(1) = 2.

a(1) = 6, {6, 25, 281, 961, -> a(1) = 1.

a(1) = 7, {7, 29, 241, 1809, -> a(1) = 2.

a(1) = 8, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
{8, 41, 209, 764, 5225, 8441, 9344, 63761, 82201, 477264, 3191044, 4038489, 34656049, 61233321, 271005625, 3465072801, 36565416324, 83511106624, 222222321476, 425286636356, 2743260628100, 9534841632400, 33984728488004, 128198574830929, 741089622057984}

a(1) = 9, {9, 61, 504, -> a(1) = 4.

a(1) = 10, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
{10, 24, 336, 400, 689, 5876, 7556, 8249, 53284, 335556, 4512400, 25092921, 165947209, 496186596, 3891489129, 6897736129, 10128495225, 18547234816, 81770476100, 203672467856, 909690622025, 6063906517681, 14045408555225, 50912872680100, 145763131189824}

a(1) = 11, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
{11, 56, 169, 744, 769, 5076, 5625, 43524, 390625, 1827776, 2562500, 8273225, 37136225, 38371001, 43037561, 258421444, 792669636, 2928667041, 38512058944, 260125180889, 405701529401, 688085041025, 5890084946609, 22508111494025, 64017148660004}

a(1) = 12, {12, 25, 281, 961, -> a(1) = 1.

a(1) = 13, {13, 69, 169, 744, 769, -> a(1) = 11

a(1) = 14, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
{14, 44, 89, 401, 956, 6649, 17796, 58596, 432489, 4211044, 22847241, 34268944, 85740489, 530152900, 718608036, 3266783209, 33250749225, 96733442161, 617288020224, 5959324297569, 20015258667081, 123104551223296, 420105398760804, 552382701059344, 967075372931216}

a(1) = 15, {15, 44, 89, 401, -> a(1) = 14.

a(1) = 16, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
{16, 44, 89, 401, 956, 6649, 17796, 58596, 432489, 4211044, 22847241, 34268944, 85740489, 530152900, 718608036, 3266783209, 33250749225, 96733442161, 617288020224, 5959324297569, 20015258667081, 123104551223296, 420105398760804, 552382701059344, 967075372931216}

a(1) = 17, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
{17, 64, 516, 961, 6201, 59409, 187600, 730641, 4429444, 28600025, 85336064, 468650384, 4590568025, 23901253604, 36922256164, 228378872384, 519390415729, 3999576229761, 22053449580964, 52752598923921, 67153745961316, 346596997521321, 2205389504844676, 32117901134901281, 91982666507893316}

a(1) = 18, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
{18, 49, 284, 2596, 9216, 68881, 70025, 332129, 758249, 4698689, 8380601, 17735824, 26222084, 65910016, 145350521, 689363225, 5040271504, 46302098001, 99834040576, 323967691556, 475037410304, 4346495992025, 27724410287321, 51737510512400, 355073484557009}

a(1) = 19, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
{19, 36, 481, 636, 804, 2896, 5924, 35600, 142400, 569600, 768961, 2190225, 9600225, 70661361, 202914304, 394770609, 860830225, 5071887281, 42699729296, 230304881489, 626445840100, 1464915433361, 2000710639616, 4927757542400, 8444293553081}

a(1) = 20, {20, 25, 281, 961, -> a(1) = 1.

Integers, n, which are unique starting points for the algorithm described in A090566,i.e.; a(n) is the initial term and subsequent terms are the smallest number greater that the previous term such that the concatenation of the two is a square.
{1, 2, 4, 7, 8, 10, 11, 14, 16, 17, 18, 19, 21, 22, 23, …, .

	An even ten, not counting the initial a(1) = 1. We could stop at a(1) is 16 if you feel that the rest are not necessary. 

	If agreed I put them in an internal format for your inspection.

Sincerely yours, Bob.

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of David Wilson
Sent: Sunday, November 22, 2015 2:31 PM
To: 'Sequence Fanatics Discussion list'
Subject: [seqfan] Re: Resurrect A090566?

RWG:

Indeed, I did not correctly transcribe my algorithm. Try this one:

x = a(n);
d = number of digits in x;
p = (10^d + 1)*x = x concatenated with x.
q = (floor(sqrt(p)) + 1)^2 = smallest square > p; If (q < (10^d)(x + 1))
	a(n+1) = q mod (10^d) = last d digits of q.
else
	p = (10^d)*(10x+1) = x concatenated with 10^d;
	q =  (floor(sqrt(p)) + 1)^2 = smallest square > p;
	a(n+1) = q mod (10^(d+1)) = last d+1 digits of q.

> -----Original Message-----
> From: Robert G. Wilson v [mailto:rgwv at rgwv.com]
> Sent: Sunday, November 22, 2015 1:28 PM
> To: David Wilson
> Subject: RE: [seqfan] Re: Resurrect A090566?
> 
> Dear David,
> 
> 	Thank you for the reply. We are very close, but I cannot implement 
> your algorithm successfully.
> 
> Example: a(2) = 6; p = 66; q = 81 and the number of digits is even so a(3) = 1?
> 
> Sincerely yours, Bob.
>
> [cascade elided]

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