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A339073 Number of strings of Hebrew letters with a gematria value equal to n. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 3 22:03 EDT 2021. Contains 346441 sequences. (Running on oeis4.)