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

Author

Daniel Sterman, Nov 22 2020

Keywords

• nonn
• word

Comments

A051596-restricted compositions of n.

Examples

			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.
		

Links

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

Crossrefs

Cf. A051596, A000079.

Programs

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 *)

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)

Extensions

More terms from Robert Israel, Nov 25 2020