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A339073 Number of strings of Hebrew letters with a gematria value equal to n.

%I A339073 #36 Nov 30 2020 08:51:28
%S A339073 1,2,4,8,16,32,64,128,256,512,1023,2045,4088,8172,16336,32656,65280,
%T A339073 130496,260864,521473,1042434,2083846,4165649,8327214,16646264,
%U A339073 33276208,66519792,132974368,265818368,531376129,1062231296,2123421181,4244760561,8485359561,16962400080,33908170232,67783096912
%N A339073 Number of strings of Hebrew letters with a gematria value equal to n.
%C A339073 A051596-restricted compositions of n.
%H A339073 Robert Israel, <a href="/A339073/b339073.txt">Table of n, a(n) for n = 1..3300</a>
%H A339073 Wikipedia, <a href="http://en.wikipedia.org/wiki/Gematria">Gematria</a>
%F A339073 From _Doron Zeilberger_, Nov 23 2020: (Start)
%F A339073 G.f.: Sum(a(n)*x^n, n=0..infinity) =
%F A339073 1/(1-add(x^i,i=1..9)-add(x^(10*i),i=1..9)-add(x^(100*i),i=1..4))
%F A339073 = 1/(1-x-...-x^9 - x^10- ... -x^90 - x^100-x^200-x^300-x^400).
%F A339073 Asymptotics:
%F A339073 a(n) ~ 0.50221591060212746248115807725009875743325273964521...*(1.9990196005347377028156443471636402056440270173905...)^n
%F A339073 If alpha is the smallest positive root of P:=1-x-...-x^9 - x^10- ... -x^90 - x^100-x^200-x^300-x^400=0
%F A339073 then the above asymptotic formula is exactly  -(alpha*P'(alpha))* (1/alpha)^n.
%F A339073 (End)
%e A339073 The four strings with a gematria of 3 are:
%e A339073 אאא (111)
%e A339073 אב (12)
%e A339073 בא (21)
%e A339073 ג (3)
%e A339073 Note: Hebrew is written right-to-left, which is why the order of the digits appears to be reversed.
%p A339073 g:= 1/(1-add(x^i,i=1..9)-add(x^(10*i),i=1..9)-add(x^(100*i),i=1..4)):
%p A339073 S:= series(g,x,101):
%p A339073 seq(coeff(S,x,n),n=1..100); # _Robert Israel_, Nov 25 2020
%t A339073 Table[SeriesCoefficient[1/(1 - Sum[x^j, {j, Join[Range[9], 10 Range[9], 100 Range[4]]}]), {x, 0, n}], {n, 100}] (* _Jan Mangaldan_, Nov 27 2020 *)
%o A339073 (C#)int[] gematrias = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400 };
%o A339073 char[] letters = { 'א', 'ב', 'ג', 'ד', 'ה', 'ו', 'ז', 'ח', 'ט', 'י', 'כ', 'ל', 'מ', 'נ', 'ס', 'ע', 'פ', 'צ', 'ק', 'ר', 'ש', 'ת' };
%o A339073 //Calculates a(n) when you call CountStrings(n), and populates ListOfStrings with the list of valid strings that have gematria equal to n.
%o A339073 int CountStrings (int leftover, string stringOfLetters = "")
%o A339073 {
%o A339073      int count = 0;
%o A339073      foreach (int value in gematrias)
%o A339073    {
%o A339073           string secondString = "";
%o A339073           if (value == leftover)
%o A339073           {
%o A339073                count++;
%o A339073                secondString += letters[Array.IndexOf(gematrias, value)];
%o A339073                ListOfStrings.Items.Add(stringOfLetters + secondString);
%o A339073           }
%o A339073           else if (value < leftover)
%o A339073           {
%o A339073                secondString += letters[Array.IndexOf(gematrias, value)];
%o A339073                count += CountStrings(leftover-value,stringOfLetters + secondString);
%o A339073           }
%o A339073      }
%o A339073      return count;
%o A339073 }
%Y A339073 Cf. A051596, A000079.
%K A339073 nonn,word
%O A339073 1,2
%A A339073 _Daniel Sterman_, Nov 22 2020
%E A339073 More terms from _Robert Israel_, Nov 25 2020