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

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172, 16336, 32656, 65280, 130496, 260864, 521473, 1042434, 2083846, 4165649, 8327214, 16646264, 33276208, 66519792, 132974368, 265818368, 531376129, 1062231296, 2123421181, 4244760561, 8485359561, 16962400080, 33908170232, 67783096912

Offset

1,2

Comments

A051596-restricted compositions of n.

Links

Robert Israel, Table of n, a(n) for n = 1..3300

Wikipedia, Gematria

Formula

From Doron Zeilberger, Nov 23 2020: (Start)

G.f.: Sum(a(n)*x^n, n=0..infinity) =

1/(1-add(x^i,i=1..9)-add(x^(10*i),i=1..9)-add(x^(100*i),i=1..4))

= 1/(1-x-...-x^9 - x^10- ... -x^90 - x^100-x^200-x^300-x^400).

Asymptotics:

a(n) ~ 0.50221591060212746248115807725009875743325273964521...*(1.9990196005347377028156443471636402056440270173905...)^n

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

then the above asymptotic formula is exactly -(alpha*P'(alpha))* (1/alpha)^n.

(End)

Example

The four strings with a gematria of 3 are:
אאא (111)
אב (12)
בא (21)
ג (3)
Note: Hebrew is written right-to-left, which is why the order of the digits appears to be reversed.

Maple

g:= 1/(1-add(x^i, i=1..9)-add(x^(10*i), i=1..9)-add(x^(100*i), i=1..4)):
S:= series(g, x, 101):
seq(coeff(S, x, n), n=1..100); # Robert Israel, Nov 25 2020

Mathematica

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 *)

Prog

(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 };
char[] letters = { 'א', 'ב', 'ג', 'ד', 'ה', 'ו', 'ז', 'ח', 'ט', 'י', 'כ', 'ל', 'מ', 'נ', 'ס', 'ע', 'פ', 'צ', 'ק', 'ר', 'ש', 'ת' };
//Calculates a(n) when you call CountStrings(n), and populates ListOfStrings with the list of valid strings that have gematria equal to n.
int CountStrings (int leftover, string stringOfLetters = "")
{
     int count = 0;
     foreach (int value in gematrias)
   {
          string secondString = "";
          if (value == leftover)
          {
               count++;
               secondString += letters[Array.IndexOf(gematrias, value)];
               ListOfStrings.Items.Add(stringOfLetters + secondString);
          }
          else if (value < leftover)
          {
               secondString += letters[Array.IndexOf(gematrias, value)];
               count += CountStrings(leftover-value, stringOfLetters + secondString);
          }
     }
     return count;
}

Crossrefs

Cf. A051596, A000079.

Sequence in context: A172319 A234591 A122265 * A194633 A243088 A113699

Adjacent sequences: A339070 A339071 A339072 * A339074 A339075 A339076

Keyword

nonn,word

Author

Daniel Sterman, Nov 22 2020

Extensions

More terms from Robert Israel, Nov 25 2020

Status

approved